[Paper Review] On entropy stable temporal fluxes

Ayoub Gouasmi, Karthik Duraisamy+1 more|arXiv (Cornell University)|Jul 10, 2018

Computational Fluid Dynamics and Aerodynamics16 references4 citations

TL;DR

This paper extends entropy stability to the temporal dimension in space-time discontinuous Galerkin schemes by deriving entropy-conservative and entropy-stable fluxes for time interfaces, analogous to spatial fluxes. It proves that upwinding in time is inherently entropy-stable and can be decomposed into an entropy-conservative flux plus a positive-definite dissipation term, offering a foundation for improved stability in high-order simulations of turbulent flows.

ABSTRACT

Entropy-stable (ES) schemes have gained considerable attention over the last decade, especially in the context of turbulent flow simulations using high-order methods. While promising because of their nonlinear stability properties, ES schemes have to address a number of issues to become practical. One of them is how much entropy should be produced by the scheme at a certain level of under-resolution. This problem has been so far studied by considering different ES interfaces fluxes in the spatial discretization only because they can be tuned to generate a certain amount of entropy. In this note, we point out that, in the context of space-time discretizations, the same applies to ES interface fluxes in the temporal direction.

Motivation & Objective

• Address the lack of entropy stability analysis in temporal discretizations of high-order schemes for turbulent flow simulations.
• Investigate whether entropy production in time can reduce the need for spatial dissipation in under-resolved simulations.
• Develop entropy-conservative and entropy-stable temporal fluxes for hyperbolic conservation laws using the same framework as spatial fluxes.
• Demonstrate that upwinding in time is inherently entropy-stable by decomposing it into an entropy-conservative flux and a positive-definite dissipation term.
• Explore the trade-off between causality and entropy stability in space-time DG formulations, particularly in the context of sub-grid scale modeling.

Proposed method

• Adapt Tadmor's entropy stability framework to the temporal direction by treating time as a coordinate analogous to space.
• Derive the entropy-conservative temporal flux using the integral of the temporal flux Jacobian along a straight-line path in entropy variable space.
• Introduce a dissipation term of the form $ T^{n+rac{1}{2}} \Delta v^{n+rac{1}{2}} $, where $ T^{n+rac{1}{2}} $ is symmetric positive definite, to achieve entropy stability in time.
• Prove that upwinding in time ($ u^{n+rac{1}{2}} = u^n $) is equivalent to an entropy-conservative flux plus a positive-definite dissipation matrix.
• Use the viscosity form of the flux to rewrite upwinding as a combination of an entropy-conservative flux and a dissipation term, with the dissipation matrix derived from the integral of the temporal Jacobian.
• Show that the temporal jacobian $ H(v) $ is positive definite, ensuring entropy production under jumps, and that the resulting dissipation term acts as a temporal implicit sub-grid scale model.

Experimental results

Research questions

• RQ1 Can entropy-stable fluxes be constructed for the temporal direction in space-time discontinuous Galerkin schemes, analogous to spatial fluxes?
• RQ2 To what extent does entropy production in time reduce the need for spatial dissipation in under-resolved turbulent flow simulations?
• RQ3 Is upwinding in time inherently entropy-stable, and if so, how can it be decomposed into entropy-conservative and dissipative components?
• RQ4 What are the implications of using non-causal entropy-conservative temporal fluxes in terms of coupling between time slabs and computational cost?
• RQ5 How does temporal entropy stability interact with spatial entropy stability in the context of large eddy simulation and sub-grid scale modeling?

Key findings

• Entropy-conservative temporal fluxes can be derived by integrating the temporal flux Jacobian along a straight-line path in entropy variable space, analogous to the spatial case.
• Entropy-stable temporal fluxes are achieved by adding a dissipation term $ T^{n+rac{1}{2}} \Delta v^{n+rac{1}{2}} $ with a symmetric positive definite matrix $ T^{n+rac{1}{2}} $, ensuring non-negative entropy production.
• Upwinding in time is proven to be entropy-stable by expressing it as the sum of an entropy-conservative flux and a positive-definite dissipation term, with the dissipation matrix derived from the integral of the temporal Jacobian.
• The dissipation term in the upwind flux decomposition is shown to be symmetric positive definite, confirming the entropy stability of time-upwinding.
• The temporal dissipation term can be interpreted as a time-implicit sub-grid scale model, suggesting a physical analogy to spatial sub-grid scale modeling.
• Non-causal entropy-conservative temporal fluxes can be used selectively at time interfaces to enable block-implicit time integration, at the cost of increased memory and arithmetic intensity, while preserving entropy stability.