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Strang splitting structure-preserving high-order compact difference schemes for nonlinear convection diffusion equations

Baolin Kuang, Shusen Xie, Hongfei Fu

TL;DR

The paper develops high-order, structure-preserving schemes for nonlinear convection-diffusion equations, addressing the dual requirements of mass conservation and bound preservation in convection-dominated regimes. The authors integrate Strang operator splitting with Crank–Nicolson diffusion and SSP-RK2 transport, all coupled with fourth-order compact spatial discretization and a Lagrange multiplier framework to enforce physical bounds; these ideas extend to 2D via efficient ADI-based schemes. They introduce BP-HOC-Splitting and BP-MC-HOC-Splitting in 1D and their ADI counterparts in 2D, with rigorous error analysis for the bound-preserving ADI scheme and extensive numerical validation demonstrating accuracy, efficiency, and robust structure preservation. The work provides scalable, linear-solver-friendly schemes that maintain discrete mass and bounds, enabling reliable large-scale simulations of nonlinear convection-diffusion processes.

Abstract

In this paper, we present a class of high-order and efficient compact difference schemes for nonlinear convection diffusion equations, which can preserve both bounds and mass. For the one-dimensional problem, we first introduce a high-order compact Strang splitting scheme (denoted as HOC-Splitting), which is fourth-order accurate in space and second-order accurate in time. Then, by incorporating the Lagrange multiplier approach with the HOC-Splitting scheme, we construct two additional bound-preserving or/and mass-conservative HOC-Splitting schemes that do not require excessive computational cost and can automatically ensure the uniform bounds of the numerical solution. These schemes combined with an alternating direction implicit (ADI) method are generalized to the two-dimensional problem, which further enhance the computational efficiency for large-scale modeling and simulation. Besides, we present an optimal-order error estimate for the bound-preserving ADI scheme in the discrete $L_2$ norm. Finally, ample numerical examples are presented to verify the theoretical results and demonstrate the accuracy, efficiency, and effectiveness in preserving bounds or/and mass of the proposed schemes.

Strang splitting structure-preserving high-order compact difference schemes for nonlinear convection diffusion equations

TL;DR

The paper develops high-order, structure-preserving schemes for nonlinear convection-diffusion equations, addressing the dual requirements of mass conservation and bound preservation in convection-dominated regimes. The authors integrate Strang operator splitting with Crank–Nicolson diffusion and SSP-RK2 transport, all coupled with fourth-order compact spatial discretization and a Lagrange multiplier framework to enforce physical bounds; these ideas extend to 2D via efficient ADI-based schemes. They introduce BP-HOC-Splitting and BP-MC-HOC-Splitting in 1D and their ADI counterparts in 2D, with rigorous error analysis for the bound-preserving ADI scheme and extensive numerical validation demonstrating accuracy, efficiency, and robust structure preservation. The work provides scalable, linear-solver-friendly schemes that maintain discrete mass and bounds, enabling reliable large-scale simulations of nonlinear convection-diffusion processes.

Abstract

In this paper, we present a class of high-order and efficient compact difference schemes for nonlinear convection diffusion equations, which can preserve both bounds and mass. For the one-dimensional problem, we first introduce a high-order compact Strang splitting scheme (denoted as HOC-Splitting), which is fourth-order accurate in space and second-order accurate in time. Then, by incorporating the Lagrange multiplier approach with the HOC-Splitting scheme, we construct two additional bound-preserving or/and mass-conservative HOC-Splitting schemes that do not require excessive computational cost and can automatically ensure the uniform bounds of the numerical solution. These schemes combined with an alternating direction implicit (ADI) method are generalized to the two-dimensional problem, which further enhance the computational efficiency for large-scale modeling and simulation. Besides, we present an optimal-order error estimate for the bound-preserving ADI scheme in the discrete norm. Finally, ample numerical examples are presented to verify the theoretical results and demonstrate the accuracy, efficiency, and effectiveness in preserving bounds or/and mass of the proposed schemes.
Paper Structure (24 sections, 9 theorems, 76 equations, 2 algorithms)

This paper contains 24 sections, 9 theorems, 76 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. High-order splitting schemes for 1D model
  3. Operator splitting and temporal discretizations
  4. HOC spatial discretization
  5. Bound-preserving or/and mass-conservative HOC-Splitting schemes
  6. BP-HOC-Splitting scheme
  7. Step 1 (Predictor).
  8. Step 2 (Corrector).
  9. Step 1 (Predictor).
  10. Step 2 (Corrector).
  11. BP-MC-HOC-Splitting scheme
  12. Step 1 (Predictor).
  13. Step 2 (Corrector).
  14. High-order ADI splitting schemes for 2D model
  15. HOC-ADI-Splitting scheme
  16. ...and 9 more sections

Key Result

theorem 1

(Discrete Mass Conservation) Let $u_h^n= \{u_{i}^{n}\} \in \mathcal{V}_h$ be the solution of the HOC-Splitting scheme 1d:scheme:diffu1--1d:scheme:diffu2. Then there holds

Theorems & Definitions (22)

  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • theorem 1
  • proof
  • remark thmcounterremark
  • remark thmcounterremark
  • theorem 2
  • ...and 12 more