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Mesh-robust stability and convergence of variable-step deferred correction methods based on the BDF2 formula

Jiahe Yue, Hong-lin Liao, Nan Liu

TL;DR

This paper addresses the stability and convergence of variable-step BDF2-based deferred correction methods (BDF2-DC) for first-order ODE systems by employing a discrete orthogonal convolution (DOC) kernel framework. It derives mesh-robust stability and convergence results for the BDF2-DC3, BDF2-DC3-DC4, and extended BDF2-DC4 schemes, explicitly accounting for the numerical effects of starting values and showing that high-order accuracy can be achieved on arbitrary time grids with relaxed step-ratio conditions. The analysis demonstrates that, under appropriate starting data (e.g., second- or third-order Runge–Kutta methods), BDF2-DC3 attains third-order and BDF2-DC3-DC4 attains fourth-order convergence in a mesh-robust sense, with an important no-aftereffect phenomenon for one-step starting schemes. Numerical experiments on graded, random, and stiff problems corroborate the theory and reveal practical benefits of adaptive time stepping, highlighting the methods’ suitability for multi-scale and dissipative problems in nonlinear parabolic contexts.

Abstract

We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory.

Mesh-robust stability and convergence of variable-step deferred correction methods based on the BDF2 formula

TL;DR

This paper addresses the stability and convergence of variable-step BDF2-based deferred correction methods (BDF2-DC) for first-order ODE systems by employing a discrete orthogonal convolution (DOC) kernel framework. It derives mesh-robust stability and convergence results for the BDF2-DC3, BDF2-DC3-DC4, and extended BDF2-DC4 schemes, explicitly accounting for the numerical effects of starting values and showing that high-order accuracy can be achieved on arbitrary time grids with relaxed step-ratio conditions. The analysis demonstrates that, under appropriate starting data (e.g., second- or third-order Runge–Kutta methods), BDF2-DC3 attains third-order and BDF2-DC3-DC4 attains fourth-order convergence in a mesh-robust sense, with an important no-aftereffect phenomenon for one-step starting schemes. Numerical experiments on graded, random, and stiff problems corroborate the theory and reveal practical benefits of adaptive time stepping, highlighting the methods’ suitability for multi-scale and dissipative problems in nonlinear parabolic contexts.

Abstract

We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory.
Paper Structure (14 sections, 15 theorems, 104 equations, 4 figures, 12 tables, 1 algorithm)

This paper contains 14 sections, 15 theorems, 104 equations, 4 figures, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Stability and convergence of BDF2-DC3 scheme
  3. Stability of the BDF2-DC3 scheme
  4. Convergence of the BDF2-DC3 scheme
  5. Stability and convergence of BDF2-DC3-DC4 scheme
  6. Stability of the BDF2-DC3-DC4 scheme
  7. Convergence of the BDF2-DC3-DC4 scheme
  8. Extension to the BDF2-DC4 method
  9. Numerical experiments
  10. Tests on different meshes
  11. Effects of different starting solutions
  12. Numerical behaviors for stiff problem
  13. Adaptive time-stepping
  14. Concluding remarks

Key Result

Lemma 2.1

LiLiao:2022 The DOC kernels $\vartheta_{n-j}^{(n)}$ in eq: BDF2-DOC procedure have an explicit formula and satisfy

Figures (4)

  • Figure 1: The starting procedure (BDF1 or RK) of BDF2-DC3 scheme \ref{['eq: BDF2-DC3 scheme']}.
  • Figure 2: The starting procedure (BDF1 or RK) of BDF2-DC3-DC4 scheme \ref{['eq: BDF2-DC3-DC4 scheme']}.
  • Figure 3: The starting procedure (BDF1 or RK) of BDF2-DC4 scheme \ref{['eq: BDF2-DC4 scheme']}.
  • Figure 4: Solution curves (up) and adaptive time-steps (down) for four initial values.

Theorems & Definitions (33)

  • Definition 1: Zero stability
  • Definition 2: Stability
  • Lemma 2.1
  • Lemma 2.2
  • Remark 1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Remark 2
  • Lemma 2.3
  • ...and 23 more