Comparison implementation characteristics

V. Zingan et al. (2013), V. N. Zingan (2012) Yu and Hesthaven (2017) Nazarov and Larcher (2017)
DG method FEM SEM FEM
implementation deal.II library (for 2D) n/a FEniCS project
viscous flux Navier-Stokes simple Navier-Stokes & Guermond-Popov
definition EV cell wise constant cell wise constant point wise
EV parameters ce, cmax, P_T, P_rho ce, cmax ce, cmax, P, (lambda ?)
numerical flux Lax-Friedrich (or Jean-Luc Guermond) Lax-Friedrich n/a
numerical flux diffusion term Lax-Friedrich central n/a
time derivative entropy Crank-Nicolson, BDF2 n/a 1st or 2nd order (BDF, ?)
smoother n/a C1 n/a

Algorithm details

V. N. Zingan (2012)

Divide problem domain \(\Omega\) into set of \(K\) disjoint elements. Define piecewise constant mesh function as \(h_K = L/K\) in 1D, where \(L\) domain length.

Define some reference domain and (linear) mappings between the reference and subdomains. Then define the finite element space approximation of a function \(v\) as an expansion in terms of linearly mapped basis functions from the reference element to the subdomain, e.g. \[ v(x) \approx v_h(x) = \sum_i v_{h,i} \Phi_i(x), \] where \(\Phi_i(x)\) are the linearly mapped basis functions and \(v_{h,i} := v_h(x_i)\).

Notation: The inner (outer) trace of \(v_h\) onto the subcell’s boundary is denoted by \(v_h^+\) (\(v_h^-\)).

The regularized initial boundary value problem describing the conservation law is given by \[ \partial_t u + \partial_x \cdot f(u) = - \partial_x \cdot q, \qquad (x,t) \in \Omega \times (0,T], \] \[ u(x,0) = u_0(x), \qquad x \in \Omega, \] where \(q := q(\mu, \partial_x \mu)\) is the viscous flux and \(\mu = \mu(x)\) is the nonlinear entropiy viscosity. The associated entropy condition is \[ \partial_t \eta(u) + \partial_x \psi(u) \leq 0, \] where \((\eta(u), \psi(u))\) is the entropy pair.

Note
This definition of \(h_K\) differs from the one given in V. Zingan et al. (2013) where \(h_K = L/(KN)\) and \(N\) is the highest polynomial order of the approximation.
Note
I use the exact opposite definition which I have adapted from Hesthaven and Warburton (2007) .

My experiences with DG+EV for the Euler equations

  • Time integration: Usage of RK4 scheme with adaptive stepping yields (most of the time) unstable results. This can be (partially) resolved by switching to a SSPRK3 scheme (with custom adaptive stepping). A bug in the backwards difference approximation of the entropy caused problems during evolutions with too big time steps. After fixing this, tests show that the RK4 and sSPRK3 perform (qualitatvely) equally well.

  • Evolution of different test problems with numerical fluxes similar to Yu (2017) showed that certain tests fail already in the first time step, because the initial discontinuity yields unphysical values (negative temperatures). This also happens when using either the Navier-Stokes or Guermond-Popov regularization, cf. Nazarov (2017). The reason for this is unclear, because Zingan (2013) present results that do no suffer from this issue.

  • Study of the blast wave problem revealed that already in the first time steps (compare previous item) unphysical values for the solution cause the program to crash. As a potential improvement, positivity preserving schemes implemented via a limiter could be imposed. However, the only limiter considered so far by me only preserves positivity of the cell average value, which is not enough.

Open questions

  • How to initialize entropy viscosity?

  • How to deal with blast wave problems? Or problems with big jumps in the pressure?

Bibliography

Hesthaven, Jan S, and Tim Warburton. 2007. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Science & Business Media.
Nazarov, Murtazo, and Aurélien Larcher. 2017. “Numerical Investigation of a Viscous Regularization of the Euler Equations by Entropy Viscosity.” Computer Methods in Applied Mechanics and Engineering 317: 128–52.
Yu, Jian, and Jan S Hesthaven. 2017. “A Comparative Study of Shock Capturing Models for the Discontinuous Galerkin Method.” Elsevier.
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.