SVI problem

Reference numerical solution

To obtain the reference numerical solution, the modified formulation of the SVI problem is applied. The flow is simulated using the HR-MC+ scheme [9] with the artificial viscosity method AV [8] on the uniform Cartesian grid consisted of square h×h cells, where h = 1/N = 1/7200, with the constant timestep corresponding to the CFL number 0.9. At the final time t₁ = 0.5⋅(1.4)^–0.5, coordinates of the grid nodes and the u_x, u_y, ρ, p variables are exported in binary format for the subdomain [–0.1, 1] × [0, 1]. Visualization of the reference numerical solution, download links for the corresponding data and the comparison recommendations are presented below. The evolution of the reference flow is illustrated and described for the time interval t ∈ [0, t₁] in the following presentation.

Numerical schlieren image	Vorticity ω = ∂uy/∂x – ∂ux/∂y

Reference data

Binary file containing the reference solution for the subdomain [–0.1, 1] × [0, 1] (the corresponding subgrid consists of 7920×7200 cells) is recorded using the standard Fortran functions.

The format of this binary file is described in detail by the source code of the supplementary Fortran program, which simplifies visualization of the reference solution by translating it to the coarser uniform Cartesian grid defined on the domain [⁠–⁠0.1, 0.9] × [0, 1] and consisted of square 3h×3h cells (2400×2400 cells in total). The translation procedure is consistent with the laws of conservation of mass, momentum and energy. The u_x variable is additionally adjusted by the value 3⋅(1.4)^0.5 to provide correspondence to the basic formulation of the problem. Output of the supplementary Fortran program is ASCII file in Tecplot format containing coordinates of the grid nodes and the u_x, u_y, ρ, p variables.

This ASCII file can be used as input to the simple Fortran program, which generates additional ASCII file in Tecplot format containing numerical schlieren and vorticity for the internal grid nodes (2399×2399 nodes in total).

Recommended parameters for assessing accuracy of numerical solutions

• Level of numerical artifacts behind the shock wave front (e.g., by comparing the density at the section x = 0.02 for the numerical and reference solutions);
• L² norm of the density difference between the numerical and reference solutions (e.g., for the subdomain Ω [0.24, 0.4] × [0.46, 0.62] using the function ε = { ∑_i∑_j(ρ_ij – ρ_ij,ref)² / [(i₂–i₁+1)(j₂–j₁+1)] }^0.5/ρ₂ ⋅ 100%, where ρ_ij and ρ_ij,ref are the average density values at ij cell for the numerical and reference solutions, i₁ and i₂ are the minimum and maximum indices along the x axis in the Ω subdomain, j₁ and j₂ are the minimum and maximum indices along the y axis in the Ω subdomain, ρ₂ is the density of the uniform background medium ahead of the shock wave);
• Minimum and maximum values of the vorticity ω (e.g., in the Ω subdomain) for the numerical solution in comparison with the corresponding values calculated for the reference solution;
• Integral values of the enstropy components E^– = 0.25⋅∫(ω–|ω|)²dxdy and E⁺ = 0.25⋅∫(ω+|ω|)²dxdy (integration is performed over a certain subdomain, e.g., Ω; E = ∫ω²dxdy = E^– + E⁺) in comparison with the corresponding values calculated for the reference solution.

The example of accuracy assessment for the numerical solutions obtained for the basic formulation of the SVI problem using some Godunov-type schemes on the uniform Cartesian grids consisting of square h×h cells, where h = 1/N and N = 200, 300, 400, 600, 800, 1200, 1600, is presented in [13].