Notes about literature


https://doi.org/10.1016/j.cma.2016.12.010

Numerical investigation of a viscous regularization of the Euler equations by entropy viscosity - Nazarov, Larcher, 2017

They discuss that Navier-Stokes fluxes are not capable of stabilizing the Euler equation with discontinuities, however, the Guermond-Popov flux can. They present and discuss both fluxes. Because the Navier-Stokes flux does not respect a minimum entropy principle in certain cases, the Guermond-Popov flux was constructed to account for this.

They also present the details of the computation of the entropy viscosity flux. Their definition differs from the Zingan et al. 2013 paper in a way that they use a nodal based viscosity, e.g. point-wise viscosity instead of cell-wise constant viscosity.

They suggest two ways to compute the point-wise entropy residual 1. via an integral over subcell elements (???) 2. interpolation of the residual onto the reference basis element (I think this is what I do)

They use a SSPRK method and give an explicit formula for time stepping.

In section 3.6 a comparison between computations that use the Navier-Stokes, Navier-Stokes + additional viscosity for the mass equation and the Guermond-Popov flux is given. In the presented 1D test case the Guermond-Popov flux performs best.

They state that implementing boundary conditions for the Euler equations is not straightforward and they provide references to two books - look this up. They also discuss so-called slip-boundary conditions (kind of reflective BCs?) and how to impose them in the strong and the weak form.

In section 5.1 they present numerical results for the smooth test problem for the Euler equations in 2D and in 5.2 they present results for the non smooth Sod shock tube test problem in 2D.

https://www.math.purdue.edu/~zhan1966/research/paper/euler.pdf

On positivity preserving high order discontinuous Galerking schemes for compressible Euler equations on rectangular meshes - Zhang, Shu, 2010

TODO Look up: - total variation diminishing (TVD) property, - total variation bounded (TVB) limiter. see notes

The paper discusses how to enforce positivity of energy density and pressure for the Euler equations in numerical simulations that use either finite volume or DG methods. Positivity of these variables is needed, because otherwise the system of Euler equations is not hyperbolic and, thus, not well defined anymore. In the introduction they mention that it is well known that when solving hyperbolic conservation law system with high order numerical schemes, the positivity of those quantities is generally not automatically guaranteed.

They give a simplified (compared to previous work) condition that has to be imposed onto the state vector during the numerical evolution. The proof of this condition is based on concave and convex properties of functions and sets. TODO Look up convexity and concavity of sets and functions.

The positivity is enforced by limiting the density first and the pressure afterwards. The limiting is done with a linear scaling factor. In Section 2 they give a recipe for their algorithm.

They also test their scheme numerical with different 1D and 2D smooth and shock tube problems. To pass the shock tube test they use a TVB limiter before they apply their positivity preserving limiter. TODO Figure out if a TVB limiter is necessary or if the EV method serves as a substitute for this. The results look good and show that their method does the job, e.g. conserves positivity. However, they do not provide results for calculations where this is needed (maybe because in those cases the evolution is not possible like with my program and the blast wave tests?).


https://www.math.tamu.edu/~bojan.popov/paper_CMAME_revised_2015_accepted.pdfp

Entropy viscosity method for the single material euler equations in Lagrangian frame

The paper discusses the Euler equations expressed in the Lagrangian frame (coordinates move with the fluid, e.g. time-dependent coordinate transformation to reference element). They discuss this transformation explicitly.

They discuss entropy principles and they build their artificial viscosity upon a generalized entropy density (which differs from the physical one). They use the Guermond-Popov regularization to incorporate artificial diffusion.

They discuss compatibility of the viscosity with general requirements for artificial viscosity tensors (as discussed somewhere else). Those requirements include - invariance of the viscous terms wrt orthogonal transformations, - minimum principle of the specific entropy, - Galilean invariance of the regularized system, - preservation of radial symmetry, - no artificial dissipation in the momentum equation for linear velocities and no artificial viscosity diffusion in any equation for regions with expansion Those requirements include - invariance of the viscous terms wrt orthogonal transformations, - minimum principle of the specific entropy, - Galilean invariance of the regularized system, - preservation of radial symmetry, - no artificial dissipation in the momentum equation for linear velocities and no artificial viscosity diffusion in any equation for regions with expansion.

They propose a time step control algorithm based on a heuristic technique, but they do not mention why this algorithm is needed.

They solve the equations using a continuous FEM method with some external FEM library. No details about the used grids are given, so I assume they use equidistant ones. I have not studied all the results, I only looked at the 1D shock tube test - the results look ok, but worse than the ones presented in V. N. Zingan (2012), V. Zingan et al. (2013).

In the conclusion they mention a follow up work in which they try to replace the notion of artificial viscosity based on differential operators by a graph Laplacian approach.

My thoughts

The discussion of compatibility properties may be useful for extending the artificial viscosity method to relativistic problems (Galilean -> Lorentz invariance). The time step control algorithm is worth to implement and test to see whether it allows to over the problems of blast wave tests. Also the follow up work may be of interest if it removes the necessity of computing second order derivatives.


Entropy stable discontinuous Galerkin schemes for the Relativistic Euler equations

https://arxiv.org/abs/1911.07488

The work builds upon entropy conservative and entropy stable fluxes which are defined in (23), (24). The names stem from the fact, that they guarantee local conservation of entropy in single elements. In (29) they give the entropy conservation flux (taken over from [36]).

Importantly: In Remark 1 they note that they use a bounds preserving limiter to keep the solution in physical space!

They provide several tests for their 1D & 2D implementation. Perhaps I can also use them to test my program.

Thoughts

The results look good. The method is not yet clear to me, it looks like that they just have used a suitable flux together with a limiter. The presented results for the shock test problem 7 requires more than 2000 cells (for k=1 and k=2 methods) to deliver acceptable results.


arXiv:1609.06792


doi:10.1016/j.jcp.2016.02.079


arxiv:2005.02317

Zingan, Valentin Nikolaevich. 2012. Discontinuous Galerkin Finite Element Method for the Nonlinear Hyperbolic Problems with Entropy-Based Artificial Viscosity Stabilization. Texas A&M University.
Zingan, Valentin, Jean-Luc Guermond, Jim Morel, and Bojan Popov. 2013. “Implementation of the Entropy Viscosity Method with the Discontinuous Galerkin Method.” Computer Methods in Applied Mechanics and Engineering 253: 479–90.