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].
\nThe initial condition of this problem is given by:
\n\\[
\n\\begin{array}{rlrl}
\n\\rho (t=0) &= \\frac{1}{2}
\n+ \\frac{3}{4} B,
\n&
\np (t=0) &= 0.1, \\\\
\nv_1 (t=0) &= \\frac{1}{2} \\left( B-1 \\right),
\n&
\nv_2 (t=0) &= \\frac{1}{10} \\sin(2 \\pi x),
\n\\end{array}
\n\\]
\nwith $B=\\tanh \\left( 15 y + 7.5 \\right) – \\tanh(15y-7.5)$, where $\\rho$ is the density, $\\vec{v}=(v_1,v_2)$ is the velocity, and $p$ is the pressure.<\/p>\n