tag:blogger.com,1999:blog-5874426371709142962024-03-12T20:00:27.238-04:00A Handful Of Numerical Integration Techniques(If only all DE's had a closed form solution...)Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.comBlogger102125tag:blogger.com,1999:blog-587442637170914296.post-54655999546178761752015-05-12T08:09:00.000-04:002015-05-12T08:09:00.046-04:00Numerical solution of the Rossler Equations<!--[if gte mso 9]><xml>
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Much like the Lorenz equations and its associated
non-linearities, the <a href="http://en.wikipedia.org/wiki/R%C3%B6ssler_attractor" target="_blank">Rossler </a>equations also present an “attractor” phenomenon
that is used to model equilibrium in chemical reactions. The Rossler ODE’s as
presented in <a href="http://www.springer.com/us/book/9783642280696" target="_blank">Solving ODE’s in R by Soetaert</a> et al are</div>
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<![endif]-->The initial conditions are {1,1,1}. Plotted below is the 3D
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Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-13766120555893984152015-05-05T08:02:00.000-04:002015-05-05T08:02:00.296-04:00A note on Linear Superposition of functions<!--[if gte mso 9]><xml>
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Linear algebraic functions can be expressed as a sum of two
other functions, given certain constraints. These constraints are typically the
boundaries of the original function, from which the two secondary functions can
be derived. </div>
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<br /></div>
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Consider a function f(x), defined in the interval [a,b]. The
linearity theory helps define two additional functions such that </div>
<div class="separator" style="clear: both; text-align: center;">
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Several functions can be derived for f1(x) and f2(x) that
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the constant (delta)<span style="mso-text-raise: -3.0pt; position: relative; top: 3.0pt;"></span><span style="mso-fareast-font-family: "Times New Roman";"><span style="mso-spacerun: yes;"> </span>can be derived with a few generalized
assumptions. Each of the functions f1(x) and f2(x) can be constructed such that
it meets one of the boundary requirements individually. Namely, </span></div>
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Substituting either one of these conditions for f1(x) or
f2(x) into the original equation for f(x), an expression for the constant can
be obtained:
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<br />
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The original function f(x) can then be constructed as a
linear sum of f1(x) and f2(x). An example is shown below for one such
construction.This concept of linear superpositions come in handy while
solving boundary value ODE’s. </div>
<br />
<div class="MsoNormal">
</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_xDv5LY1Vqzj8zxoAPEtFzSDOFQUrRL9lFMovr6FCL5duHbQaf6Js0_nfmFivRkIdwgrZd20ajQCH7ZuVImHkn7KylipoOGyXPwkACs5g-aAFudl2xhjkNHp4d7BQbe9BDJh9bdiLkuw/s1600/New+Picture+(3).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_xDv5LY1Vqzj8zxoAPEtFzSDOFQUrRL9lFMovr6FCL5duHbQaf6Js0_nfmFivRkIdwgrZd20ajQCH7ZuVImHkn7KylipoOGyXPwkACs5g-aAFudl2xhjkNHp4d7BQbe9BDJh9bdiLkuw/s1600/New+Picture+(3).png" height="290" width="400" /></a></div>
<br />
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-61160475109128346122015-04-28T18:01:00.000-04:002015-04-28T18:01:00.474-04:00Monte Carlo analysis of the Lorenz equations – Part II<!--[if gte mso 9]><xml>
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Cont'd from here. <br />
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<br /></div>
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The non-linearity in the model is clear from this scatter
plot. For small perturbations in the inputs from the mean values (depicted as
the square dots below), the solutions can end up being anywhere in the space
depicted by the scatter plots. </div>
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<br /></div>
<div class="MsoNormal">
Stated in Lorenz’s words, a flap of the butterfly’s wings
for the input conditions can create a drastic change in outputs, possibly
resulting in a tornado. In Meteorological terms, this is one of the main
reasons why longer term weather forecasts remain a difficulty. Even a
statistical analysis such as the one performed here is worthless in predicting
the longer term effects of weather, since conditions such as temperatures can
lie anywhere in the contour of the scatter plots and thus mean nothing in terms
of accurate predictions. This is the crux of Lorenz’s work; that accurate
longer term predictions cannot be made when it comes to weather. Further, he
leaves the question of the butterfly creating a tornado in a different
hemisphere of the world unanswered. </div>
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<br /></div>
<div class="MsoNormal">
Predictability of phenomena is more reliable for linear
systems. While short term predictions can be made for non-linear systems, exact
longer term predictions are both unreasonable and impossible. All that one can
predict precisely about the location of the <a href="https://www.youtube.com/watch?v=AH4k_YG_sGI" target="_blank">electronic butterfly</a> in response to a tap on the jar is that it can be found some place inside the
bottle after a few seconds. In a similar (and a rather cynical) vein, all that
the Meteorologist can predict with certainty for longer term forecasts is that
it will be summer in July in the Northern hemisphere. It only takes a mental
extrapolation to extend this analogy to even longer term predictions with
climate change, and the reasons for the discussions that ensue! <span style="mso-spacerun: yes;"> </span></div>
<div class="MsoNormal">
<br /></div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-53965678103491890492015-04-20T17:59:00.000-04:002015-04-20T18:20:24.099-04:00Monte Carlo analysis of the Lorenz equations – Part I<!--[if gte mso 9]><xml>
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Cont'd from <a href="http://numinteg.blogspot.com/2015/04/edward-lorenz-and-butterfly-theory.html" target="_blank">here</a>.
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<br /></div>
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Solving a differential equation on a computer in the early
70’s was by no means an easy task, let alone solving a set of 12 aperiodic coupled
non-linear ODE’s. Lorenz states that he got a break when he met with Dr. Barry
Saltzman, whose work on thermal convection resulted in a set of 7-equations, 4
of which approached zero. Lorenz was able to modify the other three equations
and solve for the aperiodicity of the outputs. These three equations are
commonly referred to as the Lorenz model. </div>
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<br /></div>
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How does one study the aperiodicity of the Lorenz model? By
following Lorenz’s lead and introducing minor perturbations to the initial
values. And that is exactly where the Monte Carlo technique comes in useful. As
stated <a href="http://numinteg.blogspot.com/2015/02/on-monte-carlo-simulations.html" target="_blank">earlier</a>, the MC technique works on the premise of allocating randomized
probabilities for perturbed inputs and studying the behavior of the system. The
inputs used for the process were as follows:</div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh53du7iZrJV8ae6bgBqrRUF2zYQ693Bw3N2CXTOFgnO6IrrA5zKY-89bmp06GXRB5lGvi12y-o6Baz6J8k70BfRNvLibdkAUtSzL5tv5YaIshD-CZ7HofYNpFP2yWMlB5Rwwmb31M7ZG4/s1600/New+Picture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh53du7iZrJV8ae6bgBqrRUF2zYQ693Bw3N2CXTOFgnO6IrrA5zKY-89bmp06GXRB5lGvi12y-o6Baz6J8k70BfRNvLibdkAUtSzL5tv5YaIshD-CZ7HofYNpFP2yWMlB5Rwwmb31M7ZG4/s1600/New+Picture.png" /></a></div>
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Randomized initial values were thrown as inputs to the model
within the bounds shown above. After 2000 such throws, the outputs were
obtained for a specific instant of time, 15 time units in this case. The
results of the scatter phase-plots are shown below.To be cont'd. </div>
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Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-86412268271248147022015-04-13T17:55:00.000-04:002015-04-20T18:18:39.726-04:00Edward Lorenz and the Butterfly Theory<!--[if gte mso 9]><xml>
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Predictability; Does the flap of a butterfly’s wings in
Brazil set off a tornado in Texas? This was the question proposed by Dr. Edward
Lorenz at the 135<sup>th</sup> Meeting of the American Association for the
Advancement of Science in 1972. Dr. Lorenz can be stated as one of the pioneers
of modern day Meteorology, and laid the foundations for equations in weather
prediction. He went on to research on whether the behavior of atmosphere is
“unstable” with respect to small perturbations in amplitudes when it comes to
long term weather predictions.
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In his other seminal work, “<a href="http://link.springer.com/chapter/10.1007%2FBFb0069804" target="_blank">On the Prevalence ofAperiodicity in Simple Systems</a>”,
Lorenz states that weather predictions can be likened to a set of non-linear
differential equations, whose solutions are heavily governed by the initial
values. He states that periodic solutions would be naïve in weather
forecasting, since if there were indeed a periodicity, the statistical
prediction would be a trivial one. </div>
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It turns out that Lorenz started out with solving a set of
12 differential equations and stumbled upon the aperiodicity in the solutions
with minor changes to input conditions. He noticed that re-solving the
equations with small differences in the initial values resulted in vastly
different outputs over a period of time. To quote Lorenz, “These perturbations
were amplified quasi-exponentially, doubling in about four simulated days, so
that after two months the solutions were going their separate ways”. Meaning,
the instability in the solution was “due to the lack of its periodicity”. Lorenz’s
work on non-periodic equations for weather predictions has been the founding
blocks for Chaos Theory and the “<a href="http://en.wikipedia.org/wiki/Butterfly_effect" target="_blank">Butterfly Effect</a>”
. </div>
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<br /></div>
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To be <a href="http://numinteg.blogspot.com/2015/04/monte-carlo-analysis-of-lorenz.html" target="_blank">cont'd</a>. </div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-15923230498057100732015-04-05T18:39:00.000-04:002015-04-05T18:39:00.158-04:00Implicit ODE’s, Rosenbrock Formulation & the Josephson’s Junction:<!--[if gte mso 9]><xml>
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ODE’s of the form </div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1e8jVf08qt8BgEWrrv8MIWg7MguiyDBXTjlKbdEtXc8EDLG1cljKw_hNSpaiDGgZ278i8KWazXTl8KX4O1kKKO1MMRErg02Iive-jKmG6l46vhbdk_mey__EkskIXFwFy0BskYS5BaW4/s1600/New+Picture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1e8jVf08qt8BgEWrrv8MIWg7MguiyDBXTjlKbdEtXc8EDLG1cljKw_hNSpaiDGgZ278i8KWazXTl8KX4O1kKKO1MMRErg02Iive-jKmG6l46vhbdk_mey__EkskIXFwFy0BskYS5BaW4/s1600/New+Picture.png" /></a></div>
<br />
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are referred to as implicit ODE’s. While a matrix inversion
can be used to convert these to an explicit ODE, the form of the Rosenbrock
integration scheme can be modified to directly solve for an implicit 1<sup>st</sup>
order set of ODE’s. </div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi31OQOB4zL30NbC3qsQhVnrnpCvI5sXG1roTjHXZDsvVLshU1TjBx1bRJbHUFP8iECafsw1hJE7Tz-jzjJo9k8qZiYZtalE0Rkwp7ErSdX8SuXo4y5Ec9XXBjLT1AE9rMDXxytPSsDaiM/s1600/New+Picture+(1).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi31OQOB4zL30NbC3qsQhVnrnpCvI5sXG1roTjHXZDsvVLshU1TjBx1bRJbHUFP8iECafsw1hJE7Tz-jzjJo9k8qZiYZtalE0Rkwp7ErSdX8SuXo4y5Ec9XXBjLT1AE9rMDXxytPSsDaiM/s1600/New+Picture+(1).png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5BBOKUYKHePuLIHnDuY-lzEBWNAK_HMWN1bQn8IavC-m5r2aI1AyuXOdIR1-6ScR3RlhAB4MaUQsTMuvHb2TvWeYO2BfAObH8UwHbZglyVQeyI3NjpmZT73lFeTXTfBxBISK6Q2lec6M/s1600/New+Picture+(2).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5BBOKUYKHePuLIHnDuY-lzEBWNAK_HMWN1bQn8IavC-m5r2aI1AyuXOdIR1-6ScR3RlhAB4MaUQsTMuvHb2TvWeYO2BfAObH8UwHbZglyVQeyI3NjpmZT73lFeTXTfBxBISK6Q2lec6M/s1600/New+Picture+(2).png" /></a></div>
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The complete integration can be performed using the same
procedures described <a href="http://numinteg.blogspot.com/2014/07/the-generalized-rosenbrock-formulation.html" target="_blank">earlier</a>. </div>
<br />
<div class="MsoNormal">
Getting back to numerically solving the <a href="http://numinteg.blogspot.com/2015/03/2nd-order-ode-of-josephsons-junction.html" target="_blank">Josephson’s Junction</a>
equations, which is an implicit ODE, the above formulation can also be used as
an alternative and avoid the matrix inversion mentioned earlier. </div>
<div class="MsoNormal">
<br /></div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-75090696503415284992015-03-27T18:31:00.000-04:002015-03-27T18:31:00.314-04:002nd order ODE of Josephson’s Junction <!--[if gte mso 9]><xml>
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The system of two coupled non-linear 2<sup>nd</sup> order
differential equations that represent the Josephson’s Junction phenomenon on
superconductors is presented here, as adapted from <a href="http://link.springer.com/book/10.1007%2F978-3-540-78862-1" target="_blank">Hairer et al</a>
. </div>
<br />
<div class="separator" style="clear: both; text-align: center;">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji7wUv446HdirJl0x36S8TDm_ekCLL0ww0aXmKNMUqKOdw-x40qkx_E-c7PxhVzbqr_4Wt_p_xS-zXP0eFNjq0pYJQEdWxok04c3ciw7LV_zw-aQL8jLte25NXyGgOfcq5LdXFbSwRvKQ/s1600/New+Picture+(1).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji7wUv446HdirJl0x36S8TDm_ekCLL0ww0aXmKNMUqKOdw-x40qkx_E-c7PxhVzbqr_4Wt_p_xS-zXP0eFNjq0pYJQEdWxok04c3ciw7LV_zw-aQL8jLte25NXyGgOfcq5LdXFbSwRvKQ/s1600/New+Picture+(1).png" /></a></div>
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The equations describe the electron-pair interactions
between two superconducting elements that are separated by a thin insulator,
under the influence of an applied current source. The obvious first step in attempting
to numerically integrate these equations is to decouple it.</div>
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</div>
<br />
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</div>
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Coupled equations can be decoupled in a variety of ways.
Perhaps the easiest one is the brute-force method, where in a few algebraic
sleights are performed to eliminate one the variables in either equation. In
the subject case, multiplying the second equation by <span style="mso-text-raise: -3.0pt; position: relative; top: 3.0pt;"></span><span style="font-size: 14.0pt; mso-fareast-font-family: "Times New Roman";"><span style="mso-spacerun: yes;">"alpha" </span></span><span style="mso-fareast-font-family: "Times New Roman";">and adding it to the first to eliminate one of the 2<sup>nd</sup>
derivative variables. Subsequently, this equation can be substituted into the
first equation to obtain the decoupled versions. </span></div>
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<br /></div>
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<span style="mso-fareast-font-family: "Times New Roman";">The
algebraic sleight works best with a limited set of equations (usually up to two
at the most). As the number of equations increases, the algebraic decoupling
process becomes rather cumbersome. An alternative is to use the matrix formulation
instead. The Josephson equations then become:</span></div>
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<br /></div>
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<span style="mso-fareast-font-family: "Times New Roman";"></span></div>
<span style="mso-fareast-font-family: "Times New Roman";">A
matrix inversion can then be performed to obtain the decoupled equations: </span><br />
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</div>
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Since the matrix that requires the inversion is a constant,
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And the two 2<sup>nd</sup> order ODE’s can be considered as
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the 2<sup>nd</sup> order ODE’s are converted to 1<sup>st</sup> order ODE’s. </div>
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Plotted below is the numerical solution of the equations, performed
using a <a href="http://numinteg.blogspot.com/2014/07/the-generalized-rosenbrock-formulation.html" target="_blank">Rosenbrock </a>technique. The initial conditions and values for the
constants from Hairer et al were used for the analysis. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTyEd0DDxINb_uJEOUqmXADwOYgFznBUvu5665t1fuIDgMEhqAjD1xF4g0pkpYScG9JmSjfHXAce9z6W3egE_NYTUXiR9Ej2vjb_Jkchq9p1GzFzSq2P4OHn16AeGYfhfgK_YKXZp_NzA/s1600/New+Picture+(4).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTyEd0DDxINb_uJEOUqmXADwOYgFznBUvu5665t1fuIDgMEhqAjD1xF4g0pkpYScG9JmSjfHXAce9z6W3egE_NYTUXiR9Ej2vjb_Jkchq9p1GzFzSq2P4OHn16AeGYfhfgK_YKXZp_NzA/s1600/New+Picture+(4).png" /></a></div>
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<br />Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-68154487782148566082015-03-20T18:55:00.000-04:002015-03-20T18:55:00.620-04:00Some observations on the Monte Carlo simulations<!--[if gte mso 9]><xml>
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Setting aside the triviality of
the premise for “calculating” pi (i.e., one needs to be cognizant of the fact
that the area of the circle is represented using a pi term in it, implying that
pi is already known to the user), this simple example illustrates the basics of
the Monte Carlo technique. The following general observations can be made from
the pi-experiment: </div>
<br />
<ul>
<li><span style="mso-bidi-font-family: Calibri;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"> </span></span></span>A mathematical model needs to be in place in
order to perform a sensitivity analysis of the system.</li>
<li><span style="mso-bidi-font-family: Calibri;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";">
</span></span></span>Given a model, a prediction can be made for the
steady state output of the system, for a set of input variables.</li>
<li><span style="mso-bidi-font-family: Calibri;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";">
</span></span></span>Should a question arise on the validity of the
outputs, the MC technique can be consulted. Specifically, the question can
arise in the form of what variation can be statistically forecast in the
outputs in the event of a variation in the inputs.</li>
<li><span style="mso-bidi-font-family: Calibri;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";">
</span></span></span>Having posed the question thusly, the parametric
statistical variation can be performed to the inputs over a bounded interval,
and the behavior of the mathematical model can be obtained.</li>
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If the system being analyzed is
linear, a bell-curve distribution can be expected for large sample sizes
randomized multiple times. One needs to be cognizant of the fact that MC
simulations are highly statistical, and depend heavily on the validity of both
the input variables and the mathematical model. In the case of the
pi-estimation experiment, a sample size of 1000 can only have a discrete number
of marbles fall within the confines of the circle. Thus, the accuracy of the
estimate for pi is limited to the 1000<sup>ths</sup> place; i.e., only three
decimal places can be predicted for pi with a 1000-sample bin. For a 10,000
size bin, only a 4-digit accuracy can be predicted. However, repeated throws and
subsequent averaging using the MC simulation can statistically result in values
of pi that include far more number of digits. This is clear from the histograms
shown earlier. </div>
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<br /></div>
<div class="MsoNormal" style="tab-stops: 372.0pt;">
That greater accuracies can be
seemingly predicted with a MC technique that is initially not available in the
model is a rather disturbing thought. Adequate understanding of the Physics of
the system being modeled, combined with proper bin sizes for the data analysis
in a real-world system is thus an absolute necessity before one can launch off
on an MC simulation to predict meaningful results. </div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-91828822437821519112015-03-14T18:52:00.000-04:002015-03-15T18:04:29.329-04:00Estimating pi using the MC technique - part 3<!--[if gte mso 9]><xml>
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Continuing the example from the <a href="http://numinteg.blogspot.com/2015/03/estimating-pi-using-mc-technique-part-2.html" target="_blank">previous </a>post, histograms
for three such throws are presented here. In each of these experiments, a VBA
macro was incorporated to determine the distribution and the predicted value of
pi. Sample sizes of 1000, 10000 and 100000 were used as a starting point, and
each sample was randomized 100 times to determine the results. </div>
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<br /></div>
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As the sample size increases, the statistical analysis
approaches closer to a bell-curve distribution, while the mean value of pi gets
closer to the “expected” value. </div>
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<br /></div>
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Incidentally, today is 3/14 - pi day. A special one, at 9:26:54 (or 53?!) am or pm. </div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-1685873660829270722015-03-07T18:50:00.000-05:002015-03-15T18:05:07.070-04:00Estimating pi using the MC technique - part 2<!--[if gte mso 9]><xml>
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As is evident from the <a href="http://numinteg.blogspot.com/2015/03/estimating-pi-using-monte-carlo.html" target="_blank">earlier </a>post, the larger the sample
size, the more is the number of digits in the estimate for pi. However, a
question now arises as to whether or not this random sampling of marbles would
hold good for every throw. The clichéd analogy is the well known problem that
the probability of obtaining a head or a tail from a coin toss is 50%. While it
is very well possible that the percentage may spike vastly from 50% for small
sample sizes (aka beginner’s luck in gambling), the average gets closer to 50%
as the number of tosses increases and stays pegged at 50% for very large sizes.
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Getting back to estimating pi, the Monte Carlo analysis looks
at analyzing the problem for several throws of the marbles, while each throw
gets perturbed for its inputs (i.e., randomized probabilities) in order to
obtain a contour map of the possibilities or regions for the occurrences of the
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However, what’s to say that a random throw of marbles (no
matter what the sample size is) that works for one throw will hold good for a 2<sup>nd</sup>
throw, or any subsequent one? A second level analysis utilizing the MC
technique can be incorporated, with multiple throws of randomized sample sizes
in order to determine a statistical parametric output for large number of
throws. </div>
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To be <a href="http://numinteg.blogspot.com/2015/03/estimating-pi-using-mc-technique-part-3.html" target="_blank">cont'd </a></div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-15586418801532411642015-03-01T18:46:00.000-05:002015-03-01T18:46:00.458-05:00Estimating pi using the Monte Carlo technique<!--[if gte mso 9]><xml>
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The <a href="http://numinteg.blogspot.com/2015/02/on-monte-carlo-simulations.html" target="_blank">MC</a>-approach to estimating pi claims the following:
Inscribe a circle of diameter D in a square of side D. The ratio of areas of the
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ratio of the areas can somehow be determined, then a statistical value for pi
can be calculated by multiplying this ratio by 4. </div>
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How does one assign probabilities to the MC technique? The
approach becomes rather intuitive when one imagines a square canvas with a
handful of marbles that can be randomly thrown in to represent the area covered
by the canvas. With a random sample thrown in, a simple count of the marbles
that fall in the area covered by the circle can be compared to the ones that
fall outside, thus providing an estimate for the ratio of the areas. </div>
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<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSFUEFP8O0MhZFZ57NNusadBBYhyphenhyphenMxmAiGpb-Yyf9CLbC-GOcD9Jtxgv_IQXYMX-X92uq1XEU09gcdHxF4BP9vqfinlJxctxYfFpg6i9AwN0_pE6g2P5KbA1rQ294x6mmunKDGVD124R0/s1600/New+Picture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSFUEFP8O0MhZFZ57NNusadBBYhyphenhyphenMxmAiGpb-Yyf9CLbC-GOcD9Jtxgv_IQXYMX-X92uq1XEU09gcdHxF4BP9vqfinlJxctxYfFpg6i9AwN0_pE6g2P5KbA1rQ294x6mmunKDGVD124R0/s1600/New+Picture.png" /></a></div>
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Having chosen the boundaries as a circle of radius 1 unit,
the coordinates of the marbles falling in the space of one quadrant can be
normalized as probabilities (since the boundaries of the space on both the
abscissa and ordinate are between 0 & 1). Random number generators can be
used to digitize the process, as shown in the figure above. </div>
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<br /></div>
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Voids in the canvas can be minimized by increasing the
sample size of the marbles in order to obtain a true representation of the
estimate for areas. A simulated table for various sample sizes, along with the
predicted values of pi, is shown below. </div>
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Size</span></b></div>
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for pi</span></b></div>
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<span style="font-size: 11.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">100</span></div>
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<span style="font-size: 11.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">1000</span></div>
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<span style="font-size: 11.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">3.112</span></div>
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<span style="font-size: 11.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">10000</span></div>
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To be cont'd. </div>
<br />
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-61332352541645039742015-02-23T18:42:00.001-05:002015-03-03T18:29:26.930-05:00On Monte Carlo Simulations<!--[if gte mso 9]><xml>
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I was recently quizzed on the application of Monte Carlo
Uncertainty/Probabilistic techniques for Engineering applications of
turbomachinery, and that deserves some thought here.
While there are several practical applications of the MC technique for various
facets of the Sciences, perhaps the one that has fascinated me the most is its
use for the estimation of pi. </div>
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Pi needs no definition whatsoever. Several centuries ago,
Archimedes had spent a lifetime working on the geometric properties of circles
and had stumbled upon the elegant fact that all circles hold a constant ratio
between its perimeter and diameter, irrespective of the size of the circle. Archimedes
had used the concept of inscribing and superscribing circles of various sizes
in polygons of increasing number of sides to arrive at an estimate for pi. His
estimation was based on a limit at approximating the curved line as a series of
straight lines. Legend has it that Archimedes was able to scribe a 180-sided
polygon to arrive at the estimation that we use to date. (Legend also has it
that he was so obsessed with is circles that when the Romans came to take him
to the King, Archimedes enraged the soldier by telling him to not disturb his
circles. The enraged soldier ended up killing Archimedes with his sword). </div>
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Monte Carlo techniques work on the premise of allocating
randomized probabilities for occurrences of events, and then studying the behavior
of the system with these perturbed inputs. An underlying assumption in
implementing this technique is that the system under scrutiny can be
represented as a mathematical model. Given such a model, randomized
perturbations can be supplied as inputs, and the outputs can be obtained to
understand the sensitivity and uncertainties of the system. </div>
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To be <a href="http://numinteg.blogspot.com/2015/03/estimating-pi-using-monte-carlo.html" target="_blank">cont'd</a>. </div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-68615277275040794042015-02-16T19:06:00.002-05:002015-02-16T19:07:09.158-05:00Taylor’s series expansion and the Finite Difference Method<!--[if !mso]>
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<![endif]-->Perhaps the easiest interpretation for a Finite Difference
formulation of numerical integration comes from the Taylor’s series expansion. Given
a continuous function f(x), the discretized locations <span style="mso-text-raise: -3.0pt; position: relative; top: 3.0pt;"></span><span style="mso-spacerun: yes;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJpKfQqeJ0uHJ0ZwSilxwiMEoMH_sBjDFYkxFwR2oYCwlm2KyCsdnZNV1LAG13bixyK_zQyYgrinCjR5Qpczsqqv3WPvpjazSjyZDpe4mFYjoLPyDN5-EGoN7ActPU9JfHDIEMuOaBg8A/s1600/New+Picture+(1).png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJpKfQqeJ0uHJ0ZwSilxwiMEoMH_sBjDFYkxFwR2oYCwlm2KyCsdnZNV1LAG13bixyK_zQyYgrinCjR5Qpczsqqv3WPvpjazSjyZDpe4mFYjoLPyDN5-EGoN7ActPU9JfHDIEMuOaBg8A/s1600/New+Picture+(1).png" /></a> </span>on the curve of f(x) that are separated by a
distance ‘h’ can be expanded as a Taylor’s series. If the <span style="mso-spacerun: yes;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJpKfQqeJ0uHJ0ZwSilxwiMEoMH_sBjDFYkxFwR2oYCwlm2KyCsdnZNV1LAG13bixyK_zQyYgrinCjR5Qpczsqqv3WPvpjazSjyZDpe4mFYjoLPyDN5-EGoN7ActPU9JfHDIEMuOaBg8A/s1600/New+Picture+(1).png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJpKfQqeJ0uHJ0ZwSilxwiMEoMH_sBjDFYkxFwR2oYCwlm2KyCsdnZNV1LAG13bixyK_zQyYgrinCjR5Qpczsqqv3WPvpjazSjyZDpe4mFYjoLPyDN5-EGoN7ActPU9JfHDIEMuOaBg8A/s1600/New+Picture+(1).png" /></a></span><span style="mso-text-raise: -3.0pt; position: relative; top: 3.0pt;"></span><span style="mso-spacerun: yes;"> </span>are <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_s6nrDPnQeKosZy_KNOwGB-honbX2UuMN1XkTEgwEMUClDoDcaMCFjahlihvv9DxjuELBg2yhT3bqQ_xkH56b2XK-dm0bWDHvnd1MkI9P6mbSH2-u7Xj90-9lytL8y0dPtW5WBt2bzPA/s1600/New+Picture+(2).png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_s6nrDPnQeKosZy_KNOwGB-honbX2UuMN1XkTEgwEMUClDoDcaMCFjahlihvv9DxjuELBg2yhT3bqQ_xkH56b2XK-dm0bWDHvnd1MkI9P6mbSH2-u7Xj90-9lytL8y0dPtW5WBt2bzPA/s1600/New+Picture+(2).png" /></a><span style="mso-text-raise: -3.0pt; position: relative; top: 3.0pt;"></span><span style="mso-spacerun: yes;"> </span>and <span style="mso-text-raise: -3.0pt; position: relative; top: 3.0pt;"></span><span style="font-size: 14.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="mso-spacerun: yes;"> </span></span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkz6yiBd06WgzYWjOcBaX9FuSB6sxcNMwuvf8oIVUkUVu0Oixf8jHMdLy8bttnHbI4XVxFooNSuLPrBkxkuSETzuc8SkkL9WNbfPfc3GJHMMYEONHDUe62l9aPgyxPVAjnCbUQ_6OAz3k/s1600/New+Picture+(9).png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkz6yiBd06WgzYWjOcBaX9FuSB6sxcNMwuvf8oIVUkUVu0Oixf8jHMdLy8bttnHbI4XVxFooNSuLPrBkxkuSETzuc8SkkL9WNbfPfc3GJHMMYEONHDUe62l9aPgyxPVAjnCbUQ_6OAz3k/s1600/New+Picture+(9).png" /></a>,
the function at i+1 can be represented in terms of the value at i.
Mathematically </span>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3tUvuF41C3YrHSqN4Hc9GQgeqTP8nTeQrJ81rBpnTCgVWAmdDlaDVt7sFlN1i_7KoFk6b9KC8K90kSX7KrDQn34TGUclvBadawT4F5MwA0UrqnVak6g0N4FXkbdqLz5h_Nvtber5pJ6A/s1600/New+Picture+(3).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3tUvuF41C3YrHSqN4Hc9GQgeqTP8nTeQrJ81rBpnTCgVWAmdDlaDVt7sFlN1i_7KoFk6b9KC8K90kSX7KrDQn34TGUclvBadawT4F5MwA0UrqnVak6g0N4FXkbdqLz5h_Nvtber5pJ6A/s1600/New+Picture+(3).png" /></a></div>
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<span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Stated in simpler terms, the first order
Taylor’s series expansion is a literal translation of the definition of the
first derivative of a continuous function f(x). Having stated thusly, the
expansion can be pursued on both the forward and backward directions. That is,
the expansion at a location </span><span style="mso-text-raise: -3.0pt; position: relative; top: 3.0pt;"></span><span style="font-size: 14.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="mso-spacerun: yes;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDxFu2GoestEZMt16SA6pEDiObOgD-smLSihyphenhyphenQPZrCgNbifPSoMVdjqPgrUYHfXJiA8FWga0-sM5X426l0-55EkFuU__SKglbbDEVjUf0ZNvKOZkqSl5T9imTEXUFwi7UW4Y9mdmbyIXg/s1600/New+Picture+(8).png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDxFu2GoestEZMt16SA6pEDiObOgD-smLSihyphenhyphenQPZrCgNbifPSoMVdjqPgrUYHfXJiA8FWga0-sM5X426l0-55EkFuU__SKglbbDEVjUf0ZNvKOZkqSl5T9imTEXUFwi7UW4Y9mdmbyIXg/s1600/New+Picture+(8).png" /></a> </span></span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">can
also be defined. Such an expansion thus becomes</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhePfn6LKTQpypZ5EeO1FarsiJKM80ZtzAbHDrmYSq1o3s8BJS0FMbNnQm4qlhJ5Ax55TItiVlXd692jea1TFNUJpQKT8dxutfwhwH7n3w0QsAL-faUZmaqI0KsIwAz4QSpCaRAZXk9plw/s1600/New+Picture+(4).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhePfn6LKTQpypZ5EeO1FarsiJKM80ZtzAbHDrmYSq1o3s8BJS0FMbNnQm4qlhJ5Ax55TItiVlXd692jea1TFNUJpQKT8dxutfwhwH7n3w0QsAL-faUZmaqI0KsIwAz4QSpCaRAZXk9plw/s1600/New+Picture+(4).png" /></a></div>
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<span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"></span>Rearranging the above two equations in an ODE format, one
arrives at the Forward and Backward Finite Difference formulations. By
eliminating <span style="mso-text-raise: -4.0pt; position: relative; top: 4.0pt;"></span><span style="font-size: 14.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="mso-spacerun: yes;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFdiVllfJxCYj3AWYt2bS56QShHbjh8NxKNuZ1L5DE_V5Vd0iSd9-QxRd4JH5xSjLgIr6QHVQxmUdcl59-rFS7qY9RmDW65jyt2o4oehYzLtPQtswvmiXaZHH6zh71W36sPTSK-Qchtsw/s1600/New+Picture+(10).png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFdiVllfJxCYj3AWYt2bS56QShHbjh8NxKNuZ1L5DE_V5Vd0iSd9-QxRd4JH5xSjLgIr6QHVQxmUdcl59-rFS7qY9RmDW65jyt2o4oehYzLtPQtswvmiXaZHH6zh71W36sPTSK-Qchtsw/s1600/New+Picture+(10).png" /></a> </span></span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">from
the above equations (by simply subtracting one from the other), the Central
Difference formulation can be derived. </span></div>
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But what if the higher order terms are put to practical use
with the Taylor’s expansion series? Inclusion of higher order terms results in </div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPJjmI0489vURtO4DaAjmWSc1cz5qXMmzygVbXHBLFITxTUSfZcm8iM-vfqG1aWZqpGA2qnm2lbjAJ_lrUMYgbP3qS7Up1iTAaxaJ3HyWX77R7pbFn3ewiV3M4uW8KJTQdz7pMUNLTk50/s1600/New+Picture+(5).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPJjmI0489vURtO4DaAjmWSc1cz5qXMmzygVbXHBLFITxTUSfZcm8iM-vfqG1aWZqpGA2qnm2lbjAJ_lrUMYgbP3qS7Up1iTAaxaJ3HyWX77R7pbFn3ewiV3M4uW8KJTQdz7pMUNLTk50/s1600/New+Picture+(5).png" /></a></div>
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and </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgv16jN16jSZpIOnjcUNDThhquhhaZLEjSYxG0JO53crmXyV2SRe5alHY_3jd5E88NJYke5dwF39mHD8g2HRlBixNQj1qsmQdP_WEdv1r74f4iw8xUB-sXpgXPEpX2ySXMlC2EFRpVvIKk/s1600/New+Picture+(6).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgv16jN16jSZpIOnjcUNDThhquhhaZLEjSYxG0JO53crmXyV2SRe5alHY_3jd5E88NJYke5dwF39mHD8g2HRlBixNQj1qsmQdP_WEdv1r74f4iw8xUB-sXpgXPEpX2ySXMlC2EFRpVvIKk/s1600/New+Picture+(6).png" /></a></div>
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<span style="mso-tab-count: 1;"> </span></div>
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By ignoring the triple-differential term and adding the
above equations, a formulation can be derived for the 2<sup>nd</sup>
derivative, which turns out to be</div>
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To state the obvious, a Finite Difference formulation for a
2<sup>nd</sup> order differential has been derived starting from the Taylor’s
expansion series. </div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-78555077234800121662014-12-06T17:53:00.002-05:002014-12-06T17:54:09.496-05:00Discretization at the boundaries for the Burgers’ eqn.<!--[if gte mso 9]><xml>
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More on the <a href="http://numinteg.blogspot.com/2014/10/the-burgers-equation.html" target="_blank">Burgers</a>’ equation! The process of discretization
mentioned earlier works at the interior points in the grid. However, at the
boundaries, the as-presented format fails to perform a numerical approximation.
Revisiting the discretized form, </div>
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The highlighted term in the above representation clearly
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And the backward difference formulation becomes</div>
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The above reformulations facilitate a complete ODE
representation of the Burgers’ PDE. From a computational perspective, however,
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equation. A good example is adiabatic boundary conditions for the heat
equation. </div>
<br />
<br />Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-85017313282089126602014-11-25T14:46:00.000-05:002014-11-25T14:46:00.304-05:00Upwind Discretization <!--[if gte mso 9]><xml>
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The <a href="http://numinteg.blogspot.com/2014/10/the-burgers-equation.html" target="_blank">Burgers’ equation</a> was previously discretized using the
<a href="http://numinteg.blogspot.com/2014/07/the-method-of-lines.html" target="_blank">MOL</a> approach, to convert the PDE’s to a set of ODE’s. This process of
discretization utilized the central difference method for the first order
partial differential term. More specifically, the Burger’s equation was written
as: </div>
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The highlighted term in the above representation is the
central difference approximation. For flow-related PDE’s (especially for ones
with high Re’s) the central difference approximation results in numerical
errors during the integration process. An <a href="http://en.wikipedia.org/wiki/Upwind_differencing_scheme_for_convection" target="_blank">alternative</a> approach is to apply the
forward difference or Upwind approximation. </div>
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<br /></div>
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The Upwind descretized format for the Burgers’ equation then
becomes:
</div>
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What looks like a seemingly minor change has profound
implications. Plotted below is a comparison of the Burgers’ equation solved using
the central difference and the Upwind methods, for 60 elements. The equations
were “stiffened” up by using a Reynolds number of 10000. </div>
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One can observe from the plots that the central difference
formulation results in a “less-smooth” output for the numerical solutions. For
the same number of elements (60 in this case), not only is the central
difference less accurate but also does it take more computational time (i.e,
more number of iterations) to arrive at the (less accurate) solution! A
compromise is to increase the number of elements in the central difference
method to reach a more accurate solution. Or, one can resort to the Upwind
method with less number of elements. </div>
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<br />
<br />
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Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-70499023018377909052014-11-10T13:33:00.000-05:002014-11-10T13:33:00.132-05:00Solving ODE’s using the SDIRK Method<!--[if gte mso 9]><xml>
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<![endif]--><a href="http://books.google.com/books/about/Solving_Ordinary_Differential_Equations.html?id=F93u7VcSRyYC" target="_blank">Hairer, N<span style="font-family: "Calibri","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: EN-US; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US;">ø</span>rsett & Wanner</a> present a series of ODE’s with
a discontinuous forcing function. The equations presented are for the Euler’s
motion of a rigid body in a rotational frame of reference. The ODE’s are:<br />
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where<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGbWDjAJqrJFtFeCWC6AEvZcQLnaMab71T3GpffRGatbGBnsYnI9Ay-woZZw80XfYjdL1H4n74eRopLn5iJGfjr0lB7jG8l0vhNopJgZu3ZLmAJQUlfN5IisdL197JIGXlBjXZ3aBnhm4/s1600/New+Picture+(3).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGbWDjAJqrJFtFeCWC6AEvZcQLnaMab71T3GpffRGatbGBnsYnI9Ay-woZZw80XfYjdL1H4n74eRopLn5iJGfjr0lB7jG8l0vhNopJgZu3ZLmAJQUlfN5IisdL197JIGXlBjXZ3aBnhm4/s1600/New+Picture+(3).png" /></a></div>
and the initial conditions are<br />
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Plotted below is the solution to the Euler’s equations,
solved using the <a href="http://numinteg.blogspot.com/2014/11/the-butcher-tableau-for-sdirk-method.html" target="_blank">SDIRK </a>method. </div>
<br />
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<br />Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-37904996708717654302014-11-03T12:59:00.000-05:002014-11-05T17:45:48.215-05:00The Butcher Tableau for the SDIRK Method<!--[if gte mso 9]><xml>
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An L-stable Butcher tableau for the <a href="http://numinteg.blogspot.com/2014/10/single-diagonally-implicit-rk-method.html" target="_blank">SDIRK </a>method of order-4
as derived by <a href="http://books.google.com/books/about/Solving_Ordinary_Differential_Equations.html?id=m7c8nNLPwaIC" target="_blank">Hairer & Wanner</a> is shown below.<br />
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<br /></div>
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The implementation of the SDIRK coefficients now becomes
straight-forward, with the <a href="http://numinteg.blogspot.com/2014/07/the-generalized-rosenbrock-formulation.html" target="_blank">Generalized</a> RK formulation. </div>
<br />
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<br /></div>
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As a side note, is the lower-diagonalized method a
semi-implicit method? Or is it semi-explicit? Dr. Butcher’s <a href="https://www.math.auckland.ac.nz/~butcher/CONFERENCES/TRONDHEIM/trondheim.pdf" target="_blank">slides</a> have a
rather humorous take on this subject. </div>
<br />
<br />
<br />
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<br /></div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-47343234033256869692014-10-27T11:07:00.000-04:002014-10-27T11:07:00.593-04:00(Single) Diagonally Implicit RK Method<!--[if gte mso 9]><xml>
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Let us now focus attention on the s x s Butcher tableau for
the Implicit method. <a href="http://link.springer.com/book/10.1007%2F978-3-642-05221-7" target="_blank">Hairer & Wanner</a>
quote Roger Alexander’s <a href="http://epubs.siam.org/doi/abs/10.1137/0714068" target="_blank">paper</a>,
where Alexander suggests the use of a lower-triangularized form of the table.
By using the lower triangular form, the need for iterations vanishes and it
becomes a single-step process.
<br />
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
In doing so, Alexander suggests that the term ai,j becomes 0
for all i < j. He refers to such a process as a Diagonally Implicit RK
method. </div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
The Implicit RK formulation necessitates the solution of
ki’s via Newton-type iterations at each time step. A reformulation of the ki’s
can be written as follows (Hairer & Wanner):</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgulcAfjPP1eGo0ShWfFdH1XVD7xWOPPQmeUIdKM4LwVbrYQFj500keCjRyMvqUqtVGiQOxHgBmSFQbxZ0G4eHkXGi2-qvqv6FUX_SmgkbOC3yEPauQEp29HI3PNWMTqqvCPyvkuMMepmw/s1600/New+Picture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgulcAfjPP1eGo0ShWfFdH1XVD7xWOPPQmeUIdKM4LwVbrYQFj500keCjRyMvqUqtVGiQOxHgBmSFQbxZ0G4eHkXGi2-qvqv6FUX_SmgkbOC3yEPauQEp29HI3PNWMTqqvCPyvkuMMepmw/s1600/New+Picture.png" /></a></div>
<div class="MsoNormal">
Rearranging the terms to a form that has the independent
variable ki on the left hand side, the common term that arises in the iteration
is:</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK1MLD4YXBjSSxpQwmiYpq15v2EgXdEb44mrAbDOSWbnhhqPIES6iB7K4VLfUBJ2eIX4oN6uz148wtBnbfTQSpB34vka6SgvFN53Jmqk5FO7F_SmSfzQBBHGdzsyJ0X0RE56RF-zwzpEE/s1600/New+Picture+(1).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK1MLD4YXBjSSxpQwmiYpq15v2EgXdEb44mrAbDOSWbnhhqPIES6iB7K4VLfUBJ2eIX4oN6uz148wtBnbfTQSpB34vka6SgvFN53Jmqk5FO7F_SmSfzQBBHGdzsyJ0X0RE56RF-zwzpEE/s1600/New+Picture+(1).png" /></a></div>
<div class="MsoNormal">
The partial derivative is equivalent to the Jacobian. </div>
<br />
<br />
<div class="MsoNormal">
Alexander proceeds further to state that if all the ai,i
terms are equal, (i.e., the outer diagonal terms of the lower triangular
matrix), then the above matrix with the Jacobian needs to be evaluated only
once at each time step. Such a formulation is referred to as the Single
Diagonally Implicit RK method, or the SDIRK. </div>
<div class="MsoNormal">
<br /></div>
The Butcher tableau for the SDIRK method thus takes the
form: <br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
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<div class="MsoNormal">
<br /></div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-27870429619709679372014-10-20T10:47:00.000-04:002014-10-20T10:47:00.427-04:00Implicit RK Methods:<!--[if gte mso 9]><xml>
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My <a href="http://numinteg.blogspot.com/2014/04/explicit-vs-implicit-techniques.html" target="_blank">previous</a> post touched on the differences between Implicit & Explicit methods. Implicit methods (such as the <a href="http://numinteg.blogspot.com/2014/04/the-backward-euler-method.html" target="_blank">backward Euler’s</a> method)
require an iterative procedure at each step to arrive at a solution before
proceeding to the next time step. The advantage, however, is that implicit
methods are truly stable methods. Irrespective of the time step, convergence
with Implicit methods are much better compared to Explicit ones.
<br />
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<br /></div>
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A higher order Implicit-RK (IRK) method can be represented in the
following form (<a href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-047004294X.html" target="_blank">Atkinson et al</a>):</div>
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The result of the summation of all cj terms for each ki is a
square-matrix of ALL terms (as opposed to just the lower triangular matrix with
the case of an <a href="http://numinteg.blogspot.com/2014/04/generalized-runge-kutta-formulation.html" target="_blank">Explicit RK</a> method). The Butcher tableau for the Implicit method
thus takes the form:</div>
<br />
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For example, the ki terms for a 2-stage IRK method would be:</div>
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A Newton-type iterative solver would have to be employed to
solve the above set of equations to obtain k1 & k2. In general, an s-order
system will require the solution of ns coupled equations (Hairer & Wanner).
On equations of several orders of magnitude, this can significantly slow down
the process. Unless the set of equations to be solved are considerably stiff,
where it becomes a choice between stability vs execution time. </div>
<br />
<br />Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-80265771472562238222014-10-16T10:00:00.000-04:002014-10-16T10:00:05.196-04:00Solving the Burgers’ Equation with a “double-step” initial condition. <!--[if gte mso 9]><xml>
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<![endif]-->As the last illustration for the <a href="http://numinteg.blogspot.com/2014/10/the-burgers-equation.html" target="_blank">Burgers’ equation</a>, case
study-1 presented in <a href="http://books.google.com/books/about/Adaptive_Method_of_Lines.html?id=wrNKMVpGYfgC" target="_blank">Wouwer et al</a> has been considered. The initial & boundary values are defined by the
equation (which is also an exact solution of the Burgers’ equation):<br />
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where<br />
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The initial values at time t=0, and the boundary conditions
for t>0 can be calculated from the above equations. The <a href="http://numinteg.blogspot.com/2014/07/the-method-of-lines.html" target="_blank">Method of Lines</a> can
then be used to solve the discretized ODE form of the Burgers’ equations. A
120-element model was used to solve the “double-step” input. I have referred to
it as the double-step, as it will be evident from the solution. At t=0, the
drop from boundary condition of 1 (at x=0) to 0.10 (at x=1) comprises of two
step changes<span style="font-family: "Calibri","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: EN-US; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;"> (as
opposed to the single step change in the previous post)</span>. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ-H0YpukaNoyAe3webbpV5hNwTWTvTx9VLJfqnIm_Bl-g7ecaqT57mjRtDTx8x7TAtZAIHSK-TqyiDLhaGGEm3fVR8Bf7fWL5RI01fhklI_Is49QmA0cO-bVrja-qsw9_LIEqq7cqsRo/s1600/New+Picture+(4).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ-H0YpukaNoyAe3webbpV5hNwTWTvTx9VLJfqnIm_Bl-g7ecaqT57mjRtDTx8x7TAtZAIHSK-TqyiDLhaGGEm3fVR8Bf7fWL5RI01fhklI_Is49QmA0cO-bVrja-qsw9_LIEqq7cqsRo/s1600/New+Picture+(4).png" /></a></div>
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Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-5101758721467166102014-10-13T09:45:00.000-04:002014-10-13T09:45:00.254-04:00Numerical Solution of the Burgers’ Equation for a Sine Input Function<!--[if gte mso 9]><xml>
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<![endif]-->The numerical integration of the <a href="http://numinteg.blogspot.com/2014/10/the-burgers-equation.html" target="_blank">Burgers’ equation</a> with a
half-sine wave at time t=0 is shown below. 120 elements were used to integrate
the solution, with Re=1000. The <a href="http://numinteg.blogspot.com/2014/07/the-method-of-lines.html" target="_blank">MOL</a> approach was used to convert the PDE’s to
ODE’s. The ODE’s were integrated using the Rosenbrock technique with the <a href="http://numinteg.blogspot.com/2014/09/veldhuizens-coefficients-for-rosenbrock.html" target="_blank">VELDS</a>
coefficients.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-spmqCdhQJACzalO-WBXYZ2iV0K26lN6WotopgLpdzqG00r-Sx_6OwKZXKSUL8Ey_GYCQmK3PWRxTEGfKmaAiM2jd3PcqzoFsHKmaekGVtHuZ3HkvPrVf_swKrNEzRkmJEbiwt65Snj4/s1600/New+Picture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-spmqCdhQJACzalO-WBXYZ2iV0K26lN6WotopgLpdzqG00r-Sx_6OwKZXKSUL8Ey_GYCQmK3PWRxTEGfKmaAiM2jd3PcqzoFsHKmaekGVtHuZ3HkvPrVf_swKrNEzRkmJEbiwt65Snj4/s1600/New+Picture.png" height="440" width="640" /></a></div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-1007230442791359492014-10-09T12:33:00.000-04:002014-10-09T12:33:00.103-04:00The Solution of Burgers’ Equation for high Re’s<!--[if gte mso 9]><xml>
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<![endif]-->As the Reynolds number is ratcheted up to values of Re=1000,
the equation becomes highly stiff. A 60-element <a href="http://numinteg.blogspot.com/2014/07/the-method-of-lines.html" target="_blank">MOL </a>approach yields a solution
that looks like the one shown below. The solution was plotted using the
<a href="http://numinteg.blogspot.com/2014/07/the-generalized-rosenbrock-formulation.html" target="_blank">Rosenbrock</a> method with the <a href="http://numinteg.blogspot.com/2014/07/rosenbrock-shampines-parameters.html" target="_blank">Shampine </a>coefficients.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlF64u8K8ZVZ2bGK6HmSvYij8dAGzp4Di7Q158s7xcfPb2dnkCTUdgRsw2OnZJymNQ7ChsdZFF3D3QbFVPIzXVklCI7Iaet5Ps4l1W4ntAnUZs_-scdHmQ_Ua4gJkT-VvBRwuxj-FDN3I/s1600/New+Picture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlF64u8K8ZVZ2bGK6HmSvYij8dAGzp4Di7Q158s7xcfPb2dnkCTUdgRsw2OnZJymNQ7ChsdZFF3D3QbFVPIzXVklCI7Iaet5Ps4l1W4ntAnUZs_-scdHmQ_Ua4gJkT-VvBRwuxj-FDN3I/s1600/New+Picture.png" height="400" width="392" /></a></div>
<br />
<br />
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Severe discontinuities appear in the solution as one marches
along the time domain. The number of elements along the spatial domain are
clearly not adequate to capture the essence of the boundary value PDE. (How
does one know what the “desired” solution is? In a case like this, one can
cheat and refer to the literature to obtain it, as in the case of <a href="http://dl.acm.org/citation.cfm?id=48209" target="_blank">Blom et al</a>’s
paper). </div>
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<br /></div>
<div class="MsoNormal">
Plotted below is the solution to the Burgers’ equation with
Re=1000, but with 160 elements. The discontinuities appear to have quieted down
with the increased number of elements. </div>
<div class="MsoNormal">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhG8PthBLU_Z8_6aie2zGuGM-4xj-BTDJoXqyENlRHj64-YPIjMH92axHhOb4NLPwhMaDXHQ3XdJsEHLKVdsNqumxHXOvSAkHmfp5MSShNKPXKxmtPmR7IFuUL7SZ5QYbe4Odrjdu-OEVU/s1600/New+Picture+(1).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhG8PthBLU_Z8_6aie2zGuGM-4xj-BTDJoXqyENlRHj64-YPIjMH92axHhOb4NLPwhMaDXHQ3XdJsEHLKVdsNqumxHXOvSAkHmfp5MSShNKPXKxmtPmR7IFuUL7SZ5QYbe4Odrjdu-OEVU/s1600/New+Picture+(1).png" height="400" width="360" /></a></div>
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Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-36275360557629912542014-10-06T11:01:00.000-04:002014-10-06T11:01:00.147-04:00A Solution of the Burgers’ Equation<!--[if !mso]>
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As intimidating as it looks, the Burgers’ equation does have
closed form solutions for a handful of cases. <a href="http://dl.acm.org/citation.cfm?id=48209" target="_blank">Blom et al</a> present a few such examples of initial value problems that do offer closed form
solutions.
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Presented here is a solution of the Burgers’ equation, for a
Reynolds number of 100. The initial values at t=0 follow the solution:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg72JBkbybg6lPVSG1KDlNRWjkjO0723Z1UI8CWZ-S27JhENPo692goR5S7BaRl1GeXvmXRmojKRfXoQUHwFxFkawSjxjpOQmhBTLXS169Sb5ts6MOMV_k8Gwwmn_bIpCDInVQHZM7-9HA/s1600/New+Picture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg72JBkbybg6lPVSG1KDlNRWjkjO0723Z1UI8CWZ-S27JhENPo692goR5S7BaRl1GeXvmXRmojKRfXoQUHwFxFkawSjxjpOQmhBTLXS169Sb5ts6MOMV_k8Gwwmn_bIpCDInVQHZM7-9HA/s1600/New+Picture.png" /></a></div>
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A 60-element MOL approach was implemented with the
<a href="http://numinteg.blogspot.com/2014/07/rosenbrock-shampines-parameters.html" target="_blank">Rosenbrock-Shampine</a> method. Plotted below is the 3-D solution. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOM2n8AlueapbVrbirDpd8EUMWTtBvVahyphenhyphenXKs57sfsPriI2D26Dznn3r3-gTqds-xr8IqfM3WbD24zhIesEyn_2JXP3AmBwD_aTofPLAQYsicvVNqo1X6nKCL2TARdJVBvcRBske0LkgA/s1600/New+Picture+(1).png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOM2n8AlueapbVrbirDpd8EUMWTtBvVahyphenhyphenXKs57sfsPriI2D26Dznn3r3-gTqds-xr8IqfM3WbD24zhIesEyn_2JXP3AmBwD_aTofPLAQYsicvVNqo1X6nKCL2TARdJVBvcRBske0LkgA/s1600/New+Picture+(1).png" /></a><br />
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The real challenge, however, is to obtain solutions for Re's >= 1000! </div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-53466330937311229682014-10-03T10:51:00.000-04:002014-10-03T10:51:00.070-04:00The Burgers’ Equation <!--[if !mso]>
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<br />
My colleague Mikhail had recently introduced me to the
Burger’s equation, and had posed the question of how the Rosenbrock techniques
fare at solving the Burger’s equation for large values of Reynolds number.<br />
The Burgers’ equation is a partial differential equation
that is used to describe shock phenomenon in gas flows, first presented in <a href="http://www.sciencedirect.com/science/article/pii/S0065215608701005" target="_blank">J.M.Burgers</a>’ 1948 paper. The general form of the Burgers’ equation is <div class="MsoNormal">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6D-W9h1d2AqRMfRM8VJZ73itxmps9FZgqh4WnkByjoj28GG1uSQ1MHMJWZkcBqbpRCMVgUuwBjEKJdbNZQzg8hwOLACa34tKtsXGdOt0s_65jY6_rUKOFfamiKylndfTHjELMsIy8z_0/s1600/New+Picture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6D-W9h1d2AqRMfRM8VJZ73itxmps9FZgqh4WnkByjoj28GG1uSQ1MHMJWZkcBqbpRCMVgUuwBjEKJdbNZQzg8hwOLACa34tKtsXGdOt0s_65jY6_rUKOFfamiKylndfTHjELMsIy8z_0/s1600/New+Picture.png" /></a></div>
<div class="MsoNormal">
</div>
<div class="MsoNormal">
This is a 1-dimensional PDE with the Reynolds number Re
controlling the stiffness of the equation. As indicated earlier, PDE’s can be
converted into ODE’s using the Method of Lines. The <a href="http://numinteg.blogspot.com/2014/07/the-method-of-lines.html" target="_blank">MOL</a> representation for the
Burgers’ equation thus becomes: </div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip6ySDuOGGOLsU5n8_1n6p3biRrLDNTK4WgUoBbQ-0c_p5juNCEnrC6LrHOlrOj1IG1aAVKU09M8CDJc6YDIBXw0fCtE6h6oDykHcc2yCGRblI2Udx_Bnb_OEKkRKcwsVGxo_tIUoIu9s/s1600/New+Picture+(1).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip6ySDuOGGOLsU5n8_1n6p3biRrLDNTK4WgUoBbQ-0c_p5juNCEnrC6LrHOlrOj1IG1aAVKU09M8CDJc6YDIBXw0fCtE6h6oDykHcc2yCGRblI2Udx_Bnb_OEKkRKcwsVGxo_tIUoIu9s/s1600/New+Picture+(1).png" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
With a known set of initial and boundary conditions, the MOL
representation can be put to use to solve the PDEs.</div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0tag:blogger.com,1999:blog-587442637170914296.post-75172488185091273862014-09-30T18:30:00.000-04:002014-09-30T18:30:00.756-04:00The Prothero-Robinson stiff ODE model<!--[if !mso]>
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</xml><![endif]--><a href="http://www.sciencedirect.com/science/article/pii/S0377042702004764" target="_blank">Bartel & <span style="font-family: "Calibri","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: EN-US; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">Günther</span></a>
present a model for stiff systems as adapted from Kvaerno. The Prothero
Robinson model that they use is defined by the following set of non-homogeneous
equations: </div>
<div class="MsoNormal">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOq1T3NDEydHLLOIVabOxfD7GAONbBoavYCK1OG_5THo2QQX5a8InJvA6yjU1DagsSxajedCa9DE8jGOFsHU7Dq-9qhufMl2Z0ImUTSjxcVC7eY8emYDvPqnRAj9i_tNA8mWLFb_Pv9Y0/s1600/New+Picture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOq1T3NDEydHLLOIVabOxfD7GAONbBoavYCK1OG_5THo2QQX5a8InJvA6yjU1DagsSxajedCa9DE8jGOFsHU7Dq-9qhufMl2Z0ImUTSjxcVC7eY8emYDvPqnRAj9i_tNA8mWLFb_Pv9Y0/s1600/New+Picture.png" /></a></div>
<div class="MsoNormal">
</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
The above equations represent sinusoidal solutions governed
by the frequency w, with the stiffness factor governed by gamma. Bartel & <span style="font-family: "Calibri","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: EN-US; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">Günther </span>suggest -10E4 for the stiffness factor, with zero initial conditions at
t=0, w=100 and eps =0.1. The solutions of the ODE’s are thus two variables with
vastly differing settling rates, clearly justifying the classification as
stiff. <span style="mso-spacerun: yes;"><br /></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Plotted below is the solution obtained using the <a href="http://numinteg.blogspot.com/2014/07/the-generalized-rosenbrock-formulation.html" target="_blank">Generalized Rosenbrock</a> Algorithm with the <a href="http://numinteg.blogspot.com/2014/09/veldhuizens-coefficients-for-rosenbrock.html" target="_blank">VELDS</a> parameters. </div>
<div class="MsoNormal">
<span style="font-size: 16.0pt; mso-fareast-font-family: "Times New Roman";"> </span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXrc3appsS0Uwujn2CNO54O-bFFrXr4FrWJqB9x0vFdbGgEN4KZmMNBaTszC7QJAy90ulBQP4NngAvl7O939Q2vleNUyvB-W1yZ2Y3ciJHLWP0d8jPmPM7X7ibUWHlSAC0dqqfk98OO_w/s1600/New+Picture+(1).png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXrc3appsS0Uwujn2CNO54O-bFFrXr4FrWJqB9x0vFdbGgEN4KZmMNBaTszC7QJAy90ulBQP4NngAvl7O939Q2vleNUyvB-W1yZ2Y3ciJHLWP0d8jPmPM7X7ibUWHlSAC0dqqfk98OO_w/s1600/New+Picture+(1).png" /></a></div>
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</div>
Anand Srinivasanhttp://www.blogger.com/profile/03602788985242496456noreply@blogger.com0