Splitting schemes for non-stationary problems with a rational approximation for fractional powers of the operator

Abstract

In this paper we study the numerical approximation of the solution of a Cauchy problem for a first-order-in-time differential equation involving a fractional power of a self-adjoint positive operator. One popular approach for the approximation of fractional powers of such operators is based on rational approximations. The purpose of this work is to construct special approximations in time so that the solution at a new time level is produced by solving a set of standard problems involving the self-adjoint positive operator rather than its fractional power. Stable splitting schemes with weight parameters are proposed for the additive representation of the rational approximation of the fractional power of the operator. Finally, numerical results for a two-dimensional non-stationary problem with a fractional power of the Laplace operator are also presented.