Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations

TL;DR

This paper provides a stability and convergence analysis of the third-order variable-step BDF scheme for diffusion equations, establishing a discrete energy dissipation law and demonstrating mesh-robust error control.

Contribution

It introduces a discrete gradient structure for BDF3 with variable steps and proves mesh-robust stability and convergence under step ratio constraints.

Findings

Discrete energy dissipation law established.

Mesh-robust stability and convergence proven.

Numerical tests confirm theoretical results.

Abstract

This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see e.g. [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the $L^2$ norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical results.