Stability and convergence of the variable-step time filtered backward Euler scheme for parabolic equations

Abstract

This work is concerned with numerical analysis of the variable-step time filtered backward Euler scheme (see e.g. DeCaria in SIAM J Sci Comput 43(3):A2130–A2160, 2021) for linear parabolic equations. To this end, we build up a discrete gradient structure of the associated one-leg multi-step scheme of the time filtered backward Euler (FiBE) scheme, and establish the discrete energy dissipation law for the dissipative case. Furthermore, upon developing the discrete energy technique with two new classes of discrete orthogonal convolution kernels, we present the rigorous stability and convergence results for the variable-step FiBE scheme in the $L^2$ norm under a practical step-ratio constraint $1/2\le \tau _k/\tau _{k-1}\le 2$ for $k\ge 2,$ where $\tau _k$ is the associated discrete time step. This seems to be the first energy stability and $L^2$ norm error estimate for the variable-step time filtered stiff solver. We also present numerical tests to support the theoretical findings.