I implemented the filtering technique as described in [1] (also used in bamps, see [2]). It seems like this does not cure the general problem of violation of positivity of energy and density.
I decided to take a break from the problem of getting the EV to work with shock tube tests.
I prepared an email to Jean-Luc Guermond to ask about details of the EV method and how their implementation deals with the shock tube problems.
I studied penalty terms in [3]. They are needed for implementation in BAMPS.
I picked up my advection equation project in BAMPS and started to implement penalty terms for it.
Last week’s meeting was cancelled.
Implemented smooth test problem from [4] - works.
The 1D shock tube test from [4] already fails in the first (few) time steps (sub steps in RK). Looks like I am stuck at the same problem which I have with the ordinary Euler equations.
Studied derivation of boundary conditions and how one counts does. Keyword: Kreiss’ condition.
I went back to the literature and started questioning my DG implementation from the beginning. A few questions came about, see diary entry <2020-07-09>. Most important question: Do I handle non-linearities correctly?
Started formulating questions for Jean-Luc Guermond which we would like to contact soon if I can’t get it to work.
Discussion with Yorgos about blast wave problems (this are the problems where my program fails). His experiences are that he cannot evolve those tests with an ordinary FD stencil and the EV method alone. Also Radice told him that the only reason why they can pass those tests in [12] is that they use a positivity preserving filter that is inbuilt into the THC program.
Because of some numerical investigations I did and the discussion I had with Yorgos, I started to read up on a positivity preserving filter for the Euler equations, [11].
I implemented the EV method for the DG approximation. First tests show that it does its work and allows to evolve discontinuous solutions with reduced Gibbs-phenomena.
I forwarded my results to Yorgos and I also conducted the results in the group meeting. I have then contacted Bernd, Yorgos and Sebastiano and suggested that we should meet next week and discuss the current status of Yorgos’ and my projects and on how to continue.
I have implemented the convection-diffusion equation. This is needed to implement the EV method for a DG code.
I discussed with Bernd how to proceed further with the EV study: At the moment, I should continue experimenting with Julia, because I am familiar with the language and it is rather simple to fast to prototype things. However, the long term goal is to implement the EV method into BAMPS.
Yorgos and I should agree on certain milestones for which we should make comparisons of our projects (or better results). The ultimate test of our methods should be evolution of a TOV star.
Get EV + DG to work in Julia. Conduct results to Yorgos, Bernd and Sebastiano. Agree with Yorgos about the next benchmark/milestone problem for which we should test the methods we are studying.
I have written a program that solves the advection equation with a DG method. The program can use LG and LGL grid. The current implementation suffers from a stability issue that renders a time evolution useless.
I read [17] (suggested by Bernd). This paper discusses an alternative construction of an artificial viscosity. They posse a very interesting questions that may be worth to study (also suggested by Bernd): Do all artificial viscosity approaches that try to regularize non-smooth solutions destroy the convergence rate of spectral methods?
Asked Yorgos about his project: The current status is that EV does not bring any improvements in his code if used in conjunction with a WENO method and it performs worse if used with an ordinary FD stencil compared to the WENO method without EV. He then tested the ELH approach [12] and he sent me plots that give very nice results for the advection and Burgers’ equations.
Fix the stability issues with the DG method, implement the Burgers’ equation and start working on the EV implementation.
I implemented the EV method for the Burgers’ equation. The method is working and improves the approximation of the shock solution of the Burgers’ equation considerably.
I could not reproduce the results of [9] exactly, my results show remaining oscillations (v1.0). This may be a minor fix if one knows where the problem sits, but the concept was proven and I am looking forward to present these results in the next group meeting.
Yorgos tried to incorporate the EVM method in a FD code. The current status of his calculations is that the provides an improvement compared to an ordinary centered FD code. However, the WENO methods that are available in BAM already outperform the FD + EVM method by a large margin. Adding EVM to a WENO FD code did not show any major improves, besides an (insignificant?) increase in the convergence order.
I made a julia program to solve the advection and Burgers’ equation. This was a rather simple exercise, once I got the FFT derivative going. The program can be found at https://git.uni-jena.de/fatteneder/evm.
I started to implement the entropy viscosity method as described in [9] for the advection equation with a pseudospectral method.
In this week’s journal club, Yorgos spoke about [6], [9], [12]. I think neither Yorgos, Bernd nor Sebastiano are yet really convinced about this method and further investigation is definitely needed.
This week I studied the book by LeVeque, which provided me much insight about nonlinear conservation laws and the viscosity method. As a first project, I will now try to reproduce the simplest results of the first of the papers that Yorgos discussed.