The
W-method as presented by Steihaug & Wolfbrandt involves the matrix multiplication of the Jacobian with the ki’s at each step. Shampine (Implementation of Rosenbrock Algorithms, ACM Transactions on Mathematical Software, Vol. 8, No. 2, 1982, pp 93-113) also suggests “conditioning” the A matrix to avoid difficulties during the LU-decomposition. Shampine recommends working with the following format for A:
To avoid the matrix multiplication on the right hand side, the following re-arranged format is typically used for coding the 2-stage Rosenbrock method:
An intermediate step is required where the matrix g2 is calculated, from which k2 is later determined.
[…] for the A matrix. Conditioning the above equations in a similar format as the one recommended by Shampine would once again avoid the matrix multiplication with the Jacobian for […]
ReplyDelete[…] settling times for each of the variables, thus making it numerically challenging to solve. The W-method with the error estimate and the adaptive step-size control was put to test to solve the Oregonator […]
ReplyDelete[…] single-step formulations that require an inversion of the A matrix at each time step. As indicated earlier, the LU-decomposition & LU-back substitution routines come in handy for these […]
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