Local discontinuous Galerkin method for a hidden-memory variable order reaction-diffusion equation

A high-order numerical method for a hidden-memory variable order reaction-diffusion equation is investigated in this paper. We propose the local discontinuous Galerkin method and a finite difference scheme to discrete the spatial and temporal variables, respectively. This paper provides an effective decomposition strategy for dealing with the monotonic loss of temporal discretization coefficients caused by changing order. The scheme is proved to be unconditionally stable and convergent with O(t + h(k+1)), where t is the temporal step, h is the spatial step and k is the degree of the piecewise P-k polynomial. Some numerical examples are carried out to show the effectiveness of the scheme and confirm the theoretical convergence rates.