Solution of the Wave Eqn.

The ODE’s for the wave equation were solved for the following boundary conditions:

The spatial boundary conditions represented by the partials can be converted into ODE’s using the forward and backward difference formulations:

  

The ODE’s were solved using the Rosenbrock method with the GRK4T coefficients, and the result is plotted below.

Kaps-Rentrop GRK4T - an example

The solution of problem B4 (class B) from Enright & Pryce was obtained using the Rosenbrock method with the Kaps-Rentrop GRK4T parameters. The ODE’s are:

 

 

 

 with initial conditions of y1(0)=3, y2(0)=y3(0)=0. A TOL of 1E-4 was used, with the adaptive step size algorithm. The integrated solutions are plotted below:

Rosenbrock – GRK4T parameters

Kaps & Rentrop (Generalized Runge Kutta methods of order four with stepsize control for stiff ODE’s) also provide a second set of coefficients, for gamma = 0.231. These coefficients for the GRK4T are:

A21= 0.2000000000000000E01
A31= 0.4524708207373116E01
A32= 0.4163528788597648E01
C21=-0.5071675338776316E01
C31= 0.6020152728650786E01
C32= 0.1597506846727117
C41=-0.1856343618686113E01
C42=-0.8505380858179826E01
C43=-0.2084075136023187E01
B1= 0.3957503746640777E01
B2= 0.4624892388363313E01
B3= 0.6174772638750108
B4= 0.1282612945269037E01
e1= 0.2302155402932996E01
e2= 0.3073634485392623E01
e3=-0.8732808018045032
e4=-0.1282612945269037E01
GAMMA= 0.2310000000000000
c2= 0.4620000000000000
c3= 0.8802083333333334
d1= 0.2310000000000000
d2=-0.3962966775244303E-01
d3= 0.5507789395789127
d4=-0.5535098457052764E-01

The GRK4T coefficients can also be used with the generalized Rosenbrock method.