We solve the compressible Euler equations of gas dynamics with a fourth-order accurate entropy-stable discontinuous Galerkin (DG) method and combine it with a first-order accurate finite volume (FV) method at the node level to impose positivity of density and pressure [1,2].

The initial condition of this problem is given by:

\[

\begin{array}{rlrl}

\rho (t=0) &= \frac{1}{2}

+ \frac{3}{4} B,

&

p (t=0) &= 0.1, \\

v_1 (t=0) &= \frac{1}{2} \left( B-1 \right),

&

v_2 (t=0) &= \frac{1}{10} \sin(2 \pi x),

\end{array}

\]

with $B=\tanh \left( 15 y + 7.5 \right) – \tanh(15y-7.5)$, where $\rho$ is the density, $\vec{v}=(v_1,v_2)$ is the velocity, and $p$ is the pressure.

The video shows the evolution of the density in time using $1024^2$ and degrees of freedom. We use the entropy-conservative and kinetic energy preserving flux of Ranocha [3] for the volume fluxes and the Rusanov solver for the surface fluxes of the DG and FV methods. During this simulation, the FV method acts on average in 0.000086% of the computational domain, and never more than 0.002% of the computational domain at a specific time.

References

[1] A. M. Rueda-Ramírez, G. J. Gassner, A Subcell Finite Volume Positivity-Preserving Limiter for DGSEM Discretizations of the Euler Equations, WCCM-ECCOMAS2020, pp. 1–12.

[2] A. M. Rueda-Ramírez, W. Pazner, G. J. Gassner, Subcell limiting strategies for discontinuous galerkin spectral element methods, Computers & Fluids 247 (2022) 105627.

[3] H. Ranocha, Generalised summation-by-parts operators and entropy stability of numerical methods for hyperbolic balance laws, Cuvillier Verlag, 2018.