Solving Linear Partial Differential Equations by Exponential Splitting

Let A₁, A₂,...,A_N be square matrices which do not commute. Weconsider approximations to the matrix exponential M = exp t(A₁+ A₂ + ... + A_N)] of the form where each Y_k is a positive multiplying factor, and each E_kis a product of terms having the form exp (tA_n) for some >0 and 1 nN. This form is relevant to semi-discretization methodsfor the solution of linear partial differential equations andit produces systems which are easy to solve. The accuracy andstability of the splitting approximation are studied. It isshown that, even if the number of terms and the value of K arechosen to be large, the highest order of a stable approximationis two. Numerical examples are given.