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High Order Weighted Extrapolation for Boundary Conditions for Finite Difference Methods on Complex Domains with Cartesian Meshes

Antonio Baeza, Pep Mulet, David Zorío

TL;DR

The paper tackles the challenge of imposing accurate, high-order boundary conditions for hyperbolic conservation laws on Cartesian meshes with complex geometry. It develops a fully weighted boundary extrapolation framework, incorporating dimensionless weights and a least-squares stabilization option to handle discontinuities and intricate boundaries. A family of weight designs (SW, IW, UW, GAW) and a WLS variant are analyzed through comprehensive 1D and 2D tests, demonstrating robustness and high-order accuracy near boundaries. Among the approaches, Weighted Least Squares with Global Average Weight (WLS-GAW) consistently offers the best balance of accuracy and stability, enabling high-order boundary treatment in challenging domains and suggesting directions for parallelization and 3D extensions.

Abstract

The design of numerical boundary conditions is a challenging problem that has been tackled in different ways depending on the nature of the problem and the numerical scheme used to solve it. In this paper we present a new weighted extrapolation technique which entails an improvement with respect to the technique that was developed in [1]. This technique is based on the application of a variant of the Lagrange extrapolation through the computation of weights capable of detecting regions with discontinuities. We also present a combination of the above technique with a least squares approach in order to stabilize the scheme in some cases where Lagrange extrapolation can turn the scheme mildly unstable. We show that this combined extrapolation technique can tackle discontinuities more robustly than the procedure introduced in [1].

High Order Weighted Extrapolation for Boundary Conditions for Finite Difference Methods on Complex Domains with Cartesian Meshes

TL;DR

The paper tackles the challenge of imposing accurate, high-order boundary conditions for hyperbolic conservation laws on Cartesian meshes with complex geometry. It develops a fully weighted boundary extrapolation framework, incorporating dimensionless weights and a least-squares stabilization option to handle discontinuities and intricate boundaries. A family of weight designs (SW, IW, UW, GAW) and a WLS variant are analyzed through comprehensive 1D and 2D tests, demonstrating robustness and high-order accuracy near boundaries. Among the approaches, Weighted Least Squares with Global Average Weight (WLS-GAW) consistently offers the best balance of accuracy and stability, enabling high-order boundary treatment in challenging domains and suggesting directions for parallelization and 3D extensions.

Abstract

The design of numerical boundary conditions is a challenging problem that has been tackled in different ways depending on the nature of the problem and the numerical scheme used to solve it. In this paper we present a new weighted extrapolation technique which entails an improvement with respect to the technique that was developed in [1]. This technique is based on the application of a variant of the Lagrange extrapolation through the computation of weights capable of detecting regions with discontinuities. We also present a combination of the above technique with a least squares approach in order to stabilize the scheme in some cases where Lagrange extrapolation can turn the scheme mildly unstable. We show that this combined extrapolation technique can tackle discontinuities more robustly than the procedure introduced in [1].

Paper Structure

This paper contains 24 sections, 83 equations, 12 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Numerical scheme
  3. Meshing procedure
  4. Extrapolation
  5. Weighted extrapolation
  6. Simple weights (SW)
  7. Improved weights (IW)
  8. Examples
  9. Unique weight extrapolation (UW)
  10. Tuned weight ($\lambda$-UW)
  11. Global average weight (GAW)
  12. Weighted least squares extrapolation (WLS)
  13. Numerical experiments
  14. One-dimensional experiments
  15. Linear advection, $\mathcal{C}^{\infty}$ solution.
  16. ...and 9 more sections

Figures (12)

  • Figure 1: Examples of choice of stencil. We use the stencil $\mathcal{S}(P)=\{N_1, N_2, N_3, N_4\}$ in case of outflow boundary and conditions and the stencil $\mathcal{S}(P)=\{P_0, P_1, P_2, P_3\}$ in case of Dirichlet boundary conditions.
  • Figure 2: Comparison of the SW extrapolation results against the expected optimal result.
  • Figure 3: Illustration of the grid configuration in Example 1.
  • Figure 4: Illustration of the grid configuration in Example 2.
  • Figure 5: Plot of the mapping $\omega\rightarrow\widetilde{\omega}$ for $\lambda=14$.
  • ...and 7 more figures

Theorems & Definitions (2)

  • proof
  • proof