Solving Linear Partial Differential Equations by Exponential Splitting
Qin Sheng · IMA Journal of Numerical Analysis · 1989
Let A1, A2,…,AN be square matrices which do not commute. We consider approximations to the matrix exponential M = exp [t(A1 + A2 + … + AN)] of the form ∑κ=1κγkEk where each Yk is a positive multiplying factor, and each Ek is a product of terms having the form exp (αtAn) for some α > 0 and 1 ≤n≤N. This form is relevant to semi-discretization methods for the solution of linear partial differential equations and it produces systems which are easy to solve. The accuracy and stability of the splitting approximation are studied. It is shown that, even if the number of terms and the value of K are chosen to be large, the highest order of a stable approximation is two. Numerical examples are given.