Moving Mesh with Streamline Upwind Petrov-Galerkin (MM-SUPG) Method for Time-dependent Convection-Dominated Convection-Diffusion Problems

TL;DR

This work addresses numerical stabilization of time-dependent, convection-dominated convection–diffusion problems with sharp layers. It introduces MM-SUPG, which combines streamline upwind Petrov–Galerkin stabilization with moving mesh adaptation via an MMPDE-driven metric, enabling effective handling of both isotropic and anisotropic diffusion. The study derives conditions under which the discrete solution satisfies the discrete maximum principle for anisotropic diffusion and demonstrates through isotropic tests that MM-SUPG yields smoother, more accurate solutions than fixed-mesh or non-SUPG moving-mesh variants. The results highlight the practical impact of integrated stabilization and adaptive meshing for reducing nonphysical oscillations and improving accuracy in convection-diffusion simulations.

Abstract

Time-dependent convection-diffusion problems is considered, particularly when the diffusivity is very small and sharp layers exist in the solutions. Nonphysical oscillations may occur in the numerical solutions when using regular mesh with standard computational methods. In this work, we develop a moving mesh SUPG (MM-SUPG) method, which integrates the streamline upwind Petrov-Galerkin (SUPG) method with the moving mesh partial differential equation (MMPDE) approach. The proposed method is designed to handle both isotropic and anisotropic diffusivity tensors. For the isotropic case, we focus on improving the stability of the numerical solution by utilizing both artificial diffusion from SUPG and mesh adaptation from MMPDE. And for the anisotropic case, we focus on the positivity of the numerical solution. We introduce a weighted diffusion tensor and develop a new metric tensor to control the mesh movement. We also develop conditions for time step size so that the numerical solution satisfies the discrete maximum principle (DMP). Numerical results demonstrate that the proposed MM-SUPG method provides results better than SUPG with fixed mesh or moving mesh without SUPG.

Figures (14)

• Figure 1: Example \ref{['example1']} -- side view of four different methods with $\Delta t = 0.01$, ${\boldsymbol b}$ is given by (\ref{['example1flow']}), $N = 32,768$ at $T=5.0$.
• Figure 2: Example \ref{['example1']} -- side view of MM-FEM and MM-SUPG methods with $\Delta t = 0.005$, ${\boldsymbol b}$ is given by (\ref{['example1flow']}), $N = 32,768$ at $T=5.0$.
• Figure 3: Example \ref{['example2']} -- top view with four different methods with $\Delta t = 0.001$ for ${\boldsymbol b} = (2,3)$, $N = 32768$ at $T=0.5$.
• Figure 4: Example \ref{['example2']} -- side view with the four different methods with $\Delta t = 0.001$, ${\boldsymbol b} = (2,3)$, $N = 32768$ at $T=0.5$.
• Figure 5: Example \ref{['example2']} -- cap view moving mesh with $\Delta t = 0.001$, ${\boldsymbol b} = (2,3)$, $N = 32768$ at $T=0.5$.

Theorems & Definitions (18)

• Lemma 4.1
• Lemma 4.2
• Theorem 4.3
• proof
• Corollary 4.3.1
• proof
• Lemma 4.4
• proof
• Lemma 4.5
• proof