General linear and spectral Galerkin methods for the nonlinear two-sided space distributed-order diffusion equation

Abstract

A high order numerical method is constructed for the two-dimensional nonlinear two-sided space distributed-order diffusion equation, in which the distributed-order integral is approximated by the Gauss-Legendre quadrature formula, the time and space are discretized by the general linear method and spectral Galerkin method, respectively. The proposed method is proved to be stable and convergent of order p in time, when the nonlinear term satisfies the local Lipschitz condition and the general linear method with generalized stage order p is coercive and algebraically stable. Moreover, the optimal error estimates in distributed-order integral and in space are also obtained. Finally, numerical experiments are presented to illustrate the theoretical estimates.