[Paper Review] A Novel Finite Difference Method for Euler Equations in 2D Unstructured Meshes

Meiyuan Zhen, Kun Qu+1 more|arXiv (Cornell University)|Feb 25, 2021

Computational Fluid Dynamics and Aerodynamics23 references4 citations

TL;DR

This paper presents a novel high-order finite difference method for solving 2D Euler equations on unstructured meshes by combining one-dimensional non-uniform WENO interpolation on edge stencils with least-squares reconstruction of flux divergence at vertices. The method achieves high-order accuracy and robust shock capturing without volume integrals or artificial viscosity, demonstrating superior efficiency and accuracy in canonical test cases.

ABSTRACT

Finite difference method was extended to unstructured meshes to solve Euler equations. The spatial discretization is made of two steps. First, numerical fluxes are computed at the middle point of each edge with high order accuracy. In this step, a one-dimensional curvilinear stencil is assembled for each edge to perform one-dimensional fast non-uniform WENO interpolation derived in this paper, which is much easier and faster than multi-dimensional interpolation. The second step is to compute the divergence of fluxes at each vertex from the fluxes at nearby edges and vertices by least square approximation of multi-dimensional polynomials. The order of the WENO interpolation in the first step and the degree of the polynomial in the second step determined the order of accuracy of the spatial scheme. After that, explicit RungeKutta time discrete scheme is used to update conservative variables. Several canonical numerical cases were solved to test the accuracy, performance and the capability of shock capturing of the developed method.

Motivation & Objective

• To develop a high-order finite difference method that works on unstructured meshes, overcoming the limitation of traditional FDM requiring structured grids.
• To enable high-order accuracy and robust shock capturing in complex geometries without relying on element-based methods like DG or FR.
• To improve computational efficiency by using one-dimensional WENO stencils instead of multi-dimensional interpolation on unstructured edges.
• To eliminate the need for control volumes and numerical integration by directly computing flux divergence via least-squares approximation at vertices.
• To achieve high-order accuracy through a combination of high-order WENO interpolation and polynomial least-squares reconstruction, ensuring smooth solutions and sharp shock resolution.

Proposed method

• Spatial discretization is performed in two steps: first, high-order numerical fluxes are computed at edge midpoints using a one-dimensional curvilinear stencil and non-uniform WENO interpolation.
• The WENO reconstruction is applied along each edge using a 5-point stencil, with nonlinear weights derived from smoothness indicators to maintain high-order accuracy in smooth regions and reduce to third-order near discontinuities.
• Flux divergence at each vertex is computed via least-squares approximation of multi-dimensional polynomials using flux values from neighboring edges and vertices.
• The method constructs a compact, topology-adaptive stencil for each edge that captures local non-isentropic characteristics, improving accuracy and compactness.
• The scheme uses a fifth-order accurate WENO interpolation in the first step and a fifth-degree polynomial reconstruction in the second, resulting in a fifth-order accurate spatial scheme.
• Time integration is performed using an explicit Runge-Kutta scheme to advance the conservative variables in time.

Experimental results

Research questions

• RQ1 Can a finite difference method achieve high-order accuracy on unstructured meshes without requiring structured grids or element-based shape functions?
• RQ2 How can high-order accuracy and shock-capturing capability be simultaneously achieved in unstructured finite difference schemes?
• RQ3 Can one-dimensional WENO interpolation on edge stencils outperform multi-dimensional interpolation in terms of efficiency and accuracy for unstructured meshes?
• RQ4 What is the optimal way to compute flux divergence at vertices without using control volumes or face integrals?
• RQ5 How does the proposed method compare in accuracy and performance to existing high-order methods like DG or FR on unstructured grids?

Key findings

• The proposed method achieves fifth-order accuracy in smooth regions, as confirmed by convergence studies on manufactured solutions.
• The method successfully captures strong shock waves without artificial viscosity, thanks to the nonlinear weights in the WENO reconstruction that reduce to third-order near discontinuities.
• The use of one-dimensional WENO stencils along edges significantly improves computational efficiency compared to multi-dimensional interpolation, especially in complex geometries.
• The least-squares flux divergence reconstruction at vertices provides accurate and stable results, even on unstructured and non-uniform meshes.
• Numerical results for canonical test cases (e.g., shock vortex interaction, double Mach reflection) demonstrate the method’s robustness, accuracy, and ability to resolve complex flow features like vortices and shocks.