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A velocity-based moving mesh Discontinuous Galerkin method for the advection-diffusion equation

Ezra Rozier, Jörn Behrens

TL;DR

The paper addresses robust numerical solution of unsteady advection-diffusion problems in convection-dominated regimes by introducing a velocity-based moving mesh DG method. It splits the flow into a smooth mean motion $\tilde{\mathbf{V}}$ and a residual $\mathbf{V}-\tilde{\mathbf{V}}$, uses an ALE-DG formulation on a fixed reference domain, and derives both a priori and robust a posteriori error estimates for the semi-discrete scheme. Theoretical results show second-order spatial convergence when the remaining advection is small and the mesh-velocity is sufficiently regular, while the a posteriori estimators enable local refinement that remains reliable under large mesh Peclet numbers. Numerical tests on a boundary-layer problem confirm that the moving mesh focuses resolution where needed, maintaining stability and improving accuracy compared to static meshes with DG."

Abstract

In convection-dominated flows, robustness of the spatial discretisation is a key property. While Interior Penalty Galerkin (IPG) methods already proved efficient in the situation of large mesh Peclet numbers, Arbitrary Lagrangian-Eulerian (ALE) methods are able to reduce the convection-dominance by moving the mesh. In this paper, we introduce and analyse a velocity-based moving mesh discontinuous Galerkin (DG) method for the solution of the linear advection-diffusion equation. By introducing a smooth parameterized velocity $\Tilde{V}$ that separates the flow into a mean flow, also called moving mesh velocity, and a remaining advection field $V-\Tilde{V}$, we made a convergence analysis based on the smoothness of the mesh velocity. Furthermore, the reduction of the advection speed improves the stability of an explicit time-stepping. Finally, by adapting the existing robust error criteria to this moving mesh situation, we derived robust \textit{a posteriori} error criteria that describe the potentially small deviation to the mean flow and include the information of a transition towards $V=\Tilde{V}$.

A velocity-based moving mesh Discontinuous Galerkin method for the advection-diffusion equation

TL;DR

The paper addresses robust numerical solution of unsteady advection-diffusion problems in convection-dominated regimes by introducing a velocity-based moving mesh DG method. It splits the flow into a smooth mean motion and a residual , uses an ALE-DG formulation on a fixed reference domain, and derives both a priori and robust a posteriori error estimates for the semi-discrete scheme. Theoretical results show second-order spatial convergence when the remaining advection is small and the mesh-velocity is sufficiently regular, while the a posteriori estimators enable local refinement that remains reliable under large mesh Peclet numbers. Numerical tests on a boundary-layer problem confirm that the moving mesh focuses resolution where needed, maintaining stability and improving accuracy compared to static meshes with DG."

Abstract

In convection-dominated flows, robustness of the spatial discretisation is a key property. While Interior Penalty Galerkin (IPG) methods already proved efficient in the situation of large mesh Peclet numbers, Arbitrary Lagrangian-Eulerian (ALE) methods are able to reduce the convection-dominance by moving the mesh. In this paper, we introduce and analyse a velocity-based moving mesh discontinuous Galerkin (DG) method for the solution of the linear advection-diffusion equation. By introducing a smooth parameterized velocity that separates the flow into a mean flow, also called moving mesh velocity, and a remaining advection field , we made a convergence analysis based on the smoothness of the mesh velocity. Furthermore, the reduction of the advection speed improves the stability of an explicit time-stepping. Finally, by adapting the existing robust error criteria to this moving mesh situation, we derived robust \textit{a posteriori} error criteria that describe the potentially small deviation to the mean flow and include the information of a transition towards .
Paper Structure (13 sections, 17 theorems, 121 equations, 2 figures)

This paper contains 13 sections, 17 theorems, 121 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Flow maps
  3. The semi-discrete interior penalty discretisation formulation for the unsteady advection-diffusion equation
  4. Notation and weak form
  5. Bilinear forms and function spaces for the semi-discretisation
  6. Properties of the bilinear operator
  7. Error estimates
  8. An a priori error estimate for the semi-discrete formulation
  9. An a posteriori error estimate for the semi-discrete formulation
  10. Proof of Theorem \ref{['thm:apststd']}
  11. A boundary layer test case
  12. Conclusion
  13. Proof of remark \ref{['rq:velocity']}

Key Result

Lemma 1

(Coercivity) For $\alpha$ large enough (depending on the value of the scalar $C_T$ in the inverse trace inequality ppt:ite) $a_h$ is $|||\cdot |||$-coercive. For all $\hat{v}_h \in U_h$

Figures (2)

  • Figure 1: Curves on which the particles are advected by the velocity $\Tilde{\textbf{V}}$
  • Figure 2: Plot of the criterion (upper row) and the error (lower row) after one and after twelve time steps for a static mesh (left) and a moving mesh (right). Note the differences in the scaling, which is left for better readability.

Theorems & Definitions (40)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 30 more