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Non-splitting Eulerian-Lagrangian WENO schemes for two-dimensional nonlinear convection-diffusion equations

Nanyi Zheng, Xiaofeng Cai, Jing-Mei Qiu, Jianxian Qiu

TL;DR

This work develops a high-order, conservative, non-splitting Eulerian-Lagrangian finite-volume WENO framework for two-dimensional convection-diffusion equations. By introducing a modified velocity field and a flux-form semi-discretization, the authors relax the time-step restriction for convection while preserving mass conservation via a remapping strategy that uses a fixed Eulerian WENO reconstruction to update moving upstream cells. The approach integrates high-order spatial WENO-ZQ reconstructions with IMEX Runge-Kutta time integration, enabling accurate treatment of both convection and diffusion on dynamic meshes, and extends naturally to nonlinear fluxes with minimal overhead. Numerical tests on linear and nonlinear problems validate third-order spatial and temporal accuracy, demonstrate robustness under large CFL numbers, and confirm mass conservation across challenging flow scenarios, highlighting the method’s potential for efficient, high-fidelity simulations of complex convection-diffusion phenomena.

Abstract

In this paper, we develop high-order, conservative, non-splitting Eulerian-Lagrangian (EL) Runge-Kutta (RK) finite volume (FV) weighted essentially non-oscillatory (WENO) schemes for convection-diffusion equations. The proposed EL-RK-FV-WENO scheme defines modified characteristic lines and evolves the solution along them, significantly relaxing the time-step constraint for the convection term. The main algorithm design challenge arises from the complexity of constructing accurate and robust reconstructions on dynamically varying Lagrangian meshes. This reconstruction process is needed for flux evaluations on time-dependent upstream quadrilaterals and time integrations along moving characteristics. To address this, we propose a strategy that utilizes a WENO reconstruction on a fixed Eulerian mesh for spatial reconstruction, and updates intermediate solutions on the Eulerian background mesh for implicit-explicit RK temporal integration. This strategy leverages efficient reconstruction and remapping algorithms to manage the complexities of polynomial reconstructions on time-dependent quadrilaterals, while ensuring local mass conservation. The proposed scheme ensures mass conservation due to the flux-form semi-discretization and the mass-conservative reconstruction on both background and upstream cells. Extensive numerical tests have been performed to verify the effectiveness of the proposed scheme.

Non-splitting Eulerian-Lagrangian WENO schemes for two-dimensional nonlinear convection-diffusion equations

TL;DR

This work develops a high-order, conservative, non-splitting Eulerian-Lagrangian finite-volume WENO framework for two-dimensional convection-diffusion equations. By introducing a modified velocity field and a flux-form semi-discretization, the authors relax the time-step restriction for convection while preserving mass conservation via a remapping strategy that uses a fixed Eulerian WENO reconstruction to update moving upstream cells. The approach integrates high-order spatial WENO-ZQ reconstructions with IMEX Runge-Kutta time integration, enabling accurate treatment of both convection and diffusion on dynamic meshes, and extends naturally to nonlinear fluxes with minimal overhead. Numerical tests on linear and nonlinear problems validate third-order spatial and temporal accuracy, demonstrate robustness under large CFL numbers, and confirm mass conservation across challenging flow scenarios, highlighting the method’s potential for efficient, high-fidelity simulations of complex convection-diffusion phenomena.

Abstract

In this paper, we develop high-order, conservative, non-splitting Eulerian-Lagrangian (EL) Runge-Kutta (RK) finite volume (FV) weighted essentially non-oscillatory (WENO) schemes for convection-diffusion equations. The proposed EL-RK-FV-WENO scheme defines modified characteristic lines and evolves the solution along them, significantly relaxing the time-step constraint for the convection term. The main algorithm design challenge arises from the complexity of constructing accurate and robust reconstructions on dynamically varying Lagrangian meshes. This reconstruction process is needed for flux evaluations on time-dependent upstream quadrilaterals and time integrations along moving characteristics. To address this, we propose a strategy that utilizes a WENO reconstruction on a fixed Eulerian mesh for spatial reconstruction, and updates intermediate solutions on the Eulerian background mesh for implicit-explicit RK temporal integration. This strategy leverages efficient reconstruction and remapping algorithms to manage the complexities of polynomial reconstructions on time-dependent quadrilaterals, while ensuring local mass conservation. The proposed scheme ensures mass conservation due to the flux-form semi-discretization and the mass-conservative reconstruction on both background and upstream cells. Extensive numerical tests have been performed to verify the effectiveness of the proposed scheme.
Paper Structure (13 sections, 2 theorems, 45 equations, 15 figures, 5 tables)

This paper contains 13 sections, 2 theorems, 45 equations, 15 figures, 5 tables.

Table of Contents

  1. Introduction
  2. EL-RK-FV-WENO schemes
  3. First-order EL-RK-FV scheme
  4. High-order EL-RK-FV-WENO schemes
  5. Flux approximation
  6. Approximation of the diffusion term
  7. High-order IMEX RK temporal discretization
  8. EL-RK-FV-WENO scheme for the nonlinear convection-diffusion equation
  9. Numerical tests
  10. Linear models
  11. Nonlinear models
  12. Conclusion
  13. Third-order WENO-ZQ reconstruction method for Eulerian mesh

Key Result

Proposition 2.2

(Mass conservation for the remapping method) Local integrals of $\widetilde{u}^{\text{WENO}}$ over $\{\widetilde{I}_{i,j}(t)\}$ are consistent with the integrals of $u^{\text{WENO}}$.

Figures (15)

  • Figure 1: Schematic illustration for the dynamic region $\widetilde{I}_{i,j}(t)$.
  • Figure 1: (Swirling deformation flow) Log-log plot of CFL numbers versus $L^2$ errors with fixed meshes $160\times160$ and $320\times320$ at $t = 1.5$ of the EL-RK-FV-WENO scheme.
  • Figure 1: Stencil for the 3rd-order WENO-ZQ reconstruction.
  • Figure 2: (Swirling deformation flow) Mesh plot and contour plot of a discontinuous initial condition.
  • Figure 3: Schematic illustration of full discretization with IMEX(1,2,2).
  • ...and 10 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Proposition 2.2
  • Proof 1
  • Proposition 2.3
  • Proof 2
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Remark A.1