Correction in the dominant space: a numerical technique for a certain class of stiff initial value problems

Abstract:

Consider a stiff linear initial value problem $y\prime = A(x)y + g(x)$, where the eigenvalues of $A(x)$ may be separated into two sets, one of which dominates the other. The dominant eigenvalues and corresponding right and left eigenvectors may be computed by the power method. A technique is proposed which consists of taking one forward step by a conventional multistep method and then making a correction entirely in the subspace spanned by the eigenvectors corresponding to the dominant eigenvalues. A number of alternative corrections are proposed and discussed. It is shown that the technique is stable provided that the product of the steplength and each of the subdominant eigenvalues lies within the region of absolute stability of the multistep method. The application of the technique to nonlinear problems is discussed, and numerical results are reported.