Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a new class of BDF and IMEX schemes for parabolic type equations

Fukeng Huang, Jie Shen

TL;DR

This work introduces a tunable family of BDF and IMEX schemes for parabolic-type problems by basing Taylor expansions at time $t^{n+β}$ with $β>1$, enabling higher-order methods to operate with larger time steps on stiff systems. By constructing explicit energy multipliers and proving stability via an energy framework, the authors show that linear stability regions enlarge with increasing $β$ and obtain nonlinear stability and error estimates for the IMEX variants. They derive telescoping energy identities and provide explicit results for second- to fourth-order schemes, with extensions toward fifth order supported by numerical evidence. The schemes preserve the same computational footprint as classical methods and are straightforward to implement by modifying existing BDF/IMEX codes, offering a practically impactful approach to stable, high-order time stepping for stiff parabolic equations.

Abstract

When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit (IMEX) schemes for parabolic type equations based on the Taylor expansions at time $t^{n+β}$ with $β> 1$ being a tunable parameter. These new schemes, with a suitable $β$, allow larger time steps at higher-order for stiff problems than that is allowed with a usual higher-order scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes, and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.

On a new class of BDF and IMEX schemes for parabolic type equations

TL;DR

This work introduces a tunable family of BDF and IMEX schemes for parabolic-type problems by basing Taylor expansions at time with , enabling higher-order methods to operate with larger time steps on stiff systems. By constructing explicit energy multipliers and proving stability via an energy framework, the authors show that linear stability regions enlarge with increasing and obtain nonlinear stability and error estimates for the IMEX variants. They derive telescoping energy identities and provide explicit results for second- to fourth-order schemes, with extensions toward fifth order supported by numerical evidence. The schemes preserve the same computational footprint as classical methods and are straightforward to implement by modifying existing BDF/IMEX codes, offering a practically impactful approach to stable, high-order time stepping for stiff parabolic equations.

Abstract

When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit (IMEX) schemes for parabolic type equations based on the Taylor expansions at time with being a tunable parameter. These new schemes, with a suitable , allow larger time steps at higher-order for stiff problems than that is allowed with a usual higher-order scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes, and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.
Paper Structure (18 sections, 2 theorems, 123 equations, 6 figures, 1 table)

This paper contains 18 sections, 2 theorems, 123 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. A new class of BDF and IMEX schemes
  3. The abstract setting
  4. Construction of the new schemes
  5. Linear stability regions
  6. Multipliers for the new BDF and IMEX schemes
  7. Notations and a key lemma
  8. A uniform multiplier
  9. Explicit telescoping formulae for the second and third order schemes
  10. Stability of \ref{['genBDF']} for linear parabolic type equations
  11. Stability and error analysis of \ref{['genIMEX']} for nonlinear parabolic type equations
  12. Stability
  13. Truncation errors
  14. Error estimate
  15. Comparison to the classical BDF and IMEX schemes
  16. ...and 3 more sections

Key Result

Proposition 3.1

For the second-order version of genIMEX, we have and where the coefficients are given by Moreover, we have $a_2>0$ for all $\beta \ge 1$.

Figures (6)

  • Figure 1: The pink parts show the region of absolute stability of the general third order BDF scheme with Taylor expansion at $n+\beta, \, \beta=1,3,5$.
  • Figure 2: The pink parts show the region of absolute stability of the general fourth-order BDF scheme with Taylor expansion at $n+\beta, \, \beta=1,3,5$.
  • Figure 3: Values of $\hat{a}_3$ and $a_3$ with different $\beta$.
  • Figure 6: Convergence test for the general IMEX type methods. From left to right: second order, third order and fourth order schemes with different $\beta$.
  • Figure 7: The evolution of radius $R$ with $\Delta t=0.75$ under different schemes.
  • ...and 1 more figures

Theorems & Definitions (5)

  • proof
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • proof