The Burgers’ equation was previously discretized using the
MOL 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:
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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 alternative approach is to apply the
forward difference or Upwind approximation.
The Upwind descretized format for the Burgers’ equation then
becomes:
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.
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.
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