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On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws

Hendrik Ranocha, Andrew R. Winters, Michael Schlottke-Lakemper, Philipp Öffner, Jan Glaubitz, Gregor J. Gassner

Abstract

We use the framework of upwind summation-by-parts (SBP) operators developed by Mattsson (2017, doi:10.1016/j.jcp.2017.01.042) and study different flux vector splittings in this context. To do so, we introduce discontinuous-Galerkin-like interface terms for multi-block upwind SBP methods applied to nonlinear conservation laws. We investigate the behavior of the upwind SBP methods for flux vector splittings of varying complexity on Cartesian as well as unstructured curvilinear multi-block meshes. Moreover, we analyze the local linear/energy stability of these methods following Gassner, Svärd, and Hindenlang (2022, doi:10.1007/s10915-021-01720-8). Finally, we investigate the robustness of upwind SBP methods for challenging examples of shock-free flows of the compressible Euler equations such as a Kelvin-Helmholtz instability and the inviscid Taylor-Green vortex.

On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws

Abstract

We use the framework of upwind summation-by-parts (SBP) operators developed by Mattsson (2017, doi:10.1016/j.jcp.2017.01.042) and study different flux vector splittings in this context. To do so, we introduce discontinuous-Galerkin-like interface terms for multi-block upwind SBP methods applied to nonlinear conservation laws. We investigate the behavior of the upwind SBP methods for flux vector splittings of varying complexity on Cartesian as well as unstructured curvilinear multi-block meshes. Moreover, we analyze the local linear/energy stability of these methods following Gassner, Svärd, and Hindenlang (2022, doi:10.1007/s10915-021-01720-8). Finally, we investigate the robustness of upwind SBP methods for challenging examples of shock-free flows of the compressible Euler equations such as a Kelvin-Helmholtz instability and the inviscid Taylor-Green vortex.
Paper Structure (26 sections, 6 theorems, 121 equations, 8 figures, 21 tables)

This paper contains 26 sections, 6 theorems, 121 equations, 8 figures, 21 tables.

Table of Contents

  1. Introduction
  2. Review of upwind SBP operators and flux vector splitting
  3. Flux vector splitting
  4. Upwind SBP operators
  5. Some useful properties of upwind SBP operators
  6. Upwind SBP operators in periodic domains
  7. Formulation of upwind SBP methods for nonlinear problems
  8. Local upwind SBP formulation with SATs and numerical fluxes
  9. Global upwind SBP formulation
  10. Classical flux vector splitting using upwind SBP operators
  11. Formulation in two-dimensional curvilinear coordinates
  12. Analysis of local linear/energy stability
  13. A special choice of entropy
  14. Discussion
  15. Numerical experiments
  16. ...and 11 more sections

Key Result

theorem 3.3

Consider two upwind SBP operators $D_{\pm, l/r}$ on the grids $\pmb{x}_{l/r}$ with $\pmb{x}_{N_l,l} = \pmb{x}_{1,r}$. Then, yield upwind SBP operators on the joint grid $\pmb{x} = (\pmb{x}_{1,l}, \dots, \pmb{x}_{N_l,l}, \pmb{x}_{1,r}, \dots, \pmb{x}_{N_r,r})^T$ with $N = N_l + N_r$ nodes. These global operators have the same order of accuracy as the less accurate one of the given local operators.

Figures (8)

  • Figure 1: Non-overlapping quadrilateral meshes used for the convergence testing on unstructured meshes. The local coordinate axes on each element denoted with $\xi$ and $\eta$ demonstrate that several elements have flipped local coordinate systems with respect to their neighbor elements.
  • Figure 2: Spectra of semidiscretizations of the 1D linear scalar advection equation with periodic boundary conditions. The maximum real part of all eigenvalues is around machine precision.
  • Figure 3: Non-overlapping quadrilateral meshes used for the free-stream preservation testing.
  • Figure 4: Discrete $L^2$ of the density for long-time simulations of the isentropic vortex for the 2D compressible Euler equations.
  • Figure 5: Isentropic vortex evolution up to final time $t=20$ on a heavily distorted quadrilateral mesh of 64 elements. All curvilinear interior boundaries of the warping \ref{['eq:warping']} are approximated with quadratic polynomials. This result used the curvilinear van Leer-Hänel splitting, Example \ref{['ex:van-Leer-Hanel_curved']}, with 17 nodes in each spatial direction and the 8th order interior, 4th order boundary accurate upwind SBP operators.
  • ...and 3 more figures

Theorems & Definitions (34)

  • example 2.1
  • example 2.2
  • example 2.3
  • definition 2.4
  • definition 2.5
  • example 2.6
  • definition 2.7
  • definition 2.8
  • example 2.9
  • remark 3.1
  • ...and 24 more