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

Antonio Baeza, Pep Mulet, David Zorío

TL;DR

This paper presents a technique for the extrapolation of information from the interior of the computational domain to ghost cells designed for structured Cartesian meshes (which, as opposed to non-structured meshes, cannot be adapted to the morphology of the domain boundary).

Abstract

The application of suitable numerical boundary conditions for hyperbolic conservation laws on domains with complex geometry has become a problem with certain difficulty that has been tackled in different ways according to the nature of the numerical methods and mesh type. In this paper we present a technique for the extrapolation of information from the interior of the computational domain to ghost cells designed for structured Cartesian meshes (which, as opposed to non-structured meshes, cannot be adapted to the morphology of the domain boundary). This technique is based on the application of Lagrange interpolation with a filter for the detection of discontinuities that permits a data dependent extrapolation, with higher order at smooth regions and essentially non oscillatory properties near discontinuities.

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

TL;DR

This paper presents a technique for the extrapolation of information from the interior of the computational domain to ghost cells designed for structured Cartesian meshes (which, as opposed to non-structured meshes, cannot be adapted to the morphology of the domain boundary).

Abstract

The application of suitable numerical boundary conditions for hyperbolic conservation laws on domains with complex geometry has become a problem with certain difficulty that has been tackled in different ways according to the nature of the numerical methods and mesh type. In this paper we present a technique for the extrapolation of information from the interior of the computational domain to ghost cells designed for structured Cartesian meshes (which, as opposed to non-structured meshes, cannot be adapted to the morphology of the domain boundary). This technique is based on the application of Lagrange interpolation with a filter for the detection of discontinuities that permits a data dependent extrapolation, with higher order at smooth regions and essentially non oscillatory properties near discontinuities.

Paper Structure

This paper contains 24 sections, 1 theorem, 95 equations, 12 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Numerical schemes
  3. Finite difference WENO schemes
  4. Meshing procedure
  5. Ghost cells
  6. Normal lines
  7. Choice of nodes on normal lines
  8. Extrapolation
  9. Motivation
  10. Stencil selection by thresholding
  11. Numerical experiments
  12. One-dimensional experiments
  13. Linear advection, $\mathcal{C}^{\infty}$ solution.
  14. Linear advection, discontinuous solution.
  15. Burgers equation.
  16. ...and 9 more sections

Key Result

lemma thmcounterlemma

For any $\lambda \in(0,1)$ and $s \in\mathbb N$,

Figures (12)

  • Figure 1: Examples of choice of stencil: (a) $C_y=2$, $N_{q,i}\in\mathcal{S}_q$; (b) $C_x=1$, $N_{q,i}\in\mathcal{S}_q$.
  • Figure 2: Stability.
  • Figure 3: (a)$\Delta t=0.9h_x$, $t=0.147$. Oscillations appear; (b) $\Delta t=0.9\frac{h_x}{8}$, $t=1$. No oscillations
  • Figure 4: Comparison of different extrapolations for the linear advection test with discontinuous solution.
  • Figure 5: Shock in Burgers equation, $n=80$, $\delta=0.75$, $\delta'=0.5$.
  • ...and 7 more figures

Theorems & Definitions (2)

  • lemma thmcounterlemma
  • proof