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Entropy stable non-oscillatory fluxes: An optimized wedding of entropy conservative flux with non-oscillatory flux

Ritesh Kumar Dubey

TL;DR

This novel approach paves a way to construct non-oscillatory entropy stable flux as a simple combination of $(F^*$ and $F^s)$ which inherently optimize the numerical diffusion in the entropystable flux ($\hat{F}$) such that it reduces to the underlying non-oscopeillatory flux ($F ^s$) in the flux sign stable region.

Abstract

This work frames the problem of constructing non-oscillatory entropy stable fluxes as a least square optimization problem. A flux sign stability condition is defined for a pair of entropy conservative flux ($F^*$) and a non-oscillatory flux ($F^s$). This novel approach paves a way to construct non-oscillatory entropy stable flux ($\hat{F}$) as a simple combination of $(F^*$ and $F^s)$ which inherently optimize the numerical diffusion in the entropy stable flux ($\hat{F}$) such that it reduces to the underlying non-oscillatory flux ($F^s$) in the flux sign stable region. This robust approach is (i) agnostic to the choice of flux pair $(F^*,F^s)$, (ii) does not require the computation of costly dissipation operator and high order reconstruction of scaled entropy variable to construct the diffusion term. Various non-oscillatory entropy stable fluxes are constructed and exhaustive computational results for standard test problems are given which show that these entropy stable schemes completely remove spurious oscillations in approximating the discontinuities compared to the non-oscillatory schemes using underlying fluxes ($F^s$) only. Moreover, these entropy stable schemes maintain the formal order of accuracy of the lower order flux in the pair.

Entropy stable non-oscillatory fluxes: An optimized wedding of entropy conservative flux with non-oscillatory flux

TL;DR

This novel approach paves a way to construct non-oscillatory entropy stable flux as a simple combination of and which inherently optimize the numerical diffusion in the entropystable flux () such that it reduces to the underlying non-oscopeillatory flux () in the flux sign stable region.

Abstract

This work frames the problem of constructing non-oscillatory entropy stable fluxes as a least square optimization problem. A flux sign stability condition is defined for a pair of entropy conservative flux () and a non-oscillatory flux (). This novel approach paves a way to construct non-oscillatory entropy stable flux () as a simple combination of and which inherently optimize the numerical diffusion in the entropy stable flux () such that it reduces to the underlying non-oscillatory flux () in the flux sign stable region. This robust approach is (i) agnostic to the choice of flux pair , (ii) does not require the computation of costly dissipation operator and high order reconstruction of scaled entropy variable to construct the diffusion term. Various non-oscillatory entropy stable fluxes are constructed and exhaustive computational results for standard test problems are given which show that these entropy stable schemes completely remove spurious oscillations in approximating the discontinuities compared to the non-oscillatory schemes using underlying fluxes () only. Moreover, these entropy stable schemes maintain the formal order of accuracy of the lower order flux in the pair.

Paper Structure

This paper contains 20 sections, 6 theorems, 98 equations, 14 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Entropy stable scheme
  3. Entropy Stable fluxes and numerical diffusion
  4. Quest for suitable dissipation operator
  5. Need for sign stable high order reconstruction
  6. Non-Oscillatory Schemes
  7. High order non-oscillatory fluxes
  8. Least square optimization Problem
  9. Non-oscillatory Entropy stable flux
  10. First order Entropy stable flux
  11. High order entropy stable TeCNO flux
  12. Example of a flux sign stable pair $(\mathbf{F}^{*,4}, \mathbf{F}^{s,3})$
  13. Numerical Result
  14. Scalar conservation law
  15. Linear transport equation:
  16. ...and 5 more sections

Key Result

Theorem 1

Let a consistent numerical flux ${\mathbf{F}}_{i+\frac{1}{2}}={\mathbf{F}}_{i+\frac{1}{2}}^*$ satisfies where $\psi$ is entropy potential given by Then the semi-discrete scheme semi_scheme with numerical flux $\mathbf{F}^*$ is second order accurate and entropy conservative i.e., solution computed by scheme satisfies the discrete entropy equality disEC with numerical entropy flux

Figures (14)

  • Figure 1: Linear transport equation for non-oscillatory property: Each sub figure corresponds to the fixed EC flux $F^*$ with different $F^s$ fluxes.
  • Figure 2: Linear transport equation for non-oscillatory property: Each sub figure corresponds to different EC flux with fixed $F^s$ flux.
  • Figure 3: Linear transport equation for non-oscillatory property: Different EC fluxes with WENOJS and WENOZ fluxes.
  • Figure 4: Non-oscillatory solution of Burgers equation by EC-m-$F^w$-n schemes corresponding to \ref{['BurgerIC2']} at $T_f=0.5,\; CFL = 0.8,\; N=80$. Small spurious oscillations can be observed in solution by the TECNO Fjordholm2012 and TeC-WENOJS3 BbRk schemes.
  • Figure 5: Entropy decay corresponding to the computed solution in Figure \ref{['fig:burgertest6eceno']} by EC-m-$F^w$-n schemes.
  • ...and 9 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • Theorem 6