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

Hong-lin Liao, Tao Tang, Tao Zhou

TL;DR

The paper develops a discrete energy framework for the third-order backward differentiation formula (BDF3) with variable time steps applied to linear diffusion equations. By constructing a discrete gradient structure via a step-rescaled transform and employing discrete orthogonal convolution (DOC) kernels, the authors derive an energy dissipation law and prove mesh-robust $L^2$ stability and convergence under the adjacent step-ratio bound $R_e\approx1.4877$. The analysis decouples stability from the time-step sequence, removing the unbounded growth terms that appear in previous variable-step analyses, and is complemented by numerical experiments that confirm stability and third-order convergence on both periodic and random time meshes. The work provides a rigorous foundation for reliable and efficient multi-scale time stepping using variable-step BDF3 in diffusion-type problems.

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.

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

TL;DR

The paper develops a discrete energy framework for the third-order backward differentiation formula (BDF3) with variable time steps applied to linear diffusion equations. By constructing a discrete gradient structure via a step-rescaled transform and employing discrete orthogonal convolution (DOC) kernels, the authors derive an energy dissipation law and prove mesh-robust stability and convergence under the adjacent step-ratio bound . The analysis decouples stability from the time-step sequence, removing the unbounded growth terms that appear in previous variable-step analyses, and is complemented by numerical experiments that confirm stability and third-order convergence on both periodic and random time meshes. The work provides a rigorous foundation for reliable and efficient multi-scale time stepping using variable-step BDF3 in diffusion-type problems.

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 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.
Paper Structure (11 sections, 10 theorems, 99 equations, 1 figure, 5 tables)

This paper contains 11 sections, 10 theorems, 99 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Discrete orthogonal convolution kernels
  3. Positive definiteness and energy stability
  4. Discrete gradient structure
  5. Energy dissipation law
  6. Stability and convergence analysis
  7. Properties of DOC kernels
  8. $L^2$ norm stability
  9. $L^2$ norm convergence
  10. Numerical experiments
  11. Technical results for Lemma \ref{['lem: BDF3-DGS']}

Key Result

Lemma 3.1

Define the following functions If the step-ratios $0<r_k<R_e$, there exist two nonnegative functionals $G$ and $F$ such that where the Lyapunov-type functional and the remainder term

Figures (1)

  • Figure 1: Surfaces of $\eta(\sqrt{R_e},y,z)$ and $\zeta(\sqrt{R_e},y,z)$ over the domain $(0,\sqrt{R_{e}})^2$.

Theorems & Definitions (18)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • Theorem 4.1
  • ...and 8 more