Error estimation for the W-method

Automatic step control and error estimation for the W-method can be implemented using a 3^rd order estimate for the local truncation error, as presented by Steihaug and Wolfbrandt. Having determined k1, k2 and y(n+1) as noted earlier, the calculation proceeds with k3 and k4 as follows:

 

The error is given by

The error estimates as presented by Steihaug & Wolfbrandt do not have the “conditioned” formulation 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 k4.

Once the error estimation is determined at each time step, an adaptive step-size control can be used to advance to the next time step.