Wolfbrandt’s Rosenbrock method

A 2-stage Rosenbrock formulation was presented by Steihaug & Wolfbrandt (An attempt to avoid exact Jacobian and non-linear equations in the numerical solution of stiff DE’s, Mathematics of Computation, Vol. 33, No. 146, 1979, pp 521-534). In their famous paper, Shampine & Reichelt (The Matlab ODE Suite, SIAM Journal of Scientific Computing, Vol. 18, Issue 1, 1997) also introduce this “delightfully simple” 2-stage method, and refer to them as the W-formulas.

The two-step formulation is represented as follows:

 

 The constant

.

The matrix A is the same as the one defined earlier. The step-by-step process (which is similar to the generalized RK methods) can be summarized thusly:

1)      Frame the matrix A
2)      Create the LU-decomposed form of matrix A
3)      Use LU-back substitution with f(yn) to calculate k1
4)      Perform the matrix multiplication J*k1
5)      Calculate k2
6)      Calculate y(n+1)

As the order of the matrices increases, the computational effort for the matrix multiplication may have an impact on the computational time. Reformulation of the above method to avoid the matrix multiplication is a common technique.