SVI problem

Modified formulation

In the basic formulation of the SVI problem, where the coordinate system moves along with the initial shock wave, this shock wave remains flat and stationary until it begins to interact with the vortex; afterwards its front bends and slowly moves through the computational domain. It is known that shock-capturing methods demonstrate an increased level of numerical artifacts for such flows – appearance of artificial low-frequency disturbances behind the shock wave (the problem of simulating a slowly moving shock wave; see, for example, [11, 12] and [9, sec. 3.4]). To eliminate these effects, the SVI problem can be solved in a coordinate system fixed to the stationary medium ahead of the shock wave. In this modified formulation, the problem is stated as follows. 

In the computational domain [–0.5, 1.5] × [0, 1], superposition of the stationary medium (u_x1 = u_y1 = 0) with parameters ρ₁ = p₁ = 1 and the vortex centered at (x_V, y_V) = (1, 0.5) is defined; the vortex parameters are described in the basic formulaion of the problem. Flow symmetry condition is set at the upper, lower, and left boundaries of the domain; supersonic inflow with the parameters

u_x2 = –20⋅(1.4)^0.5/9,   u_y2 = 0,   ρ₂ = 27/7,   p₂ = 31/3

is set at the right boundary.

The flow is simulated until the time t = t₁ = 0.5⋅(1.4)^–0.5, when the shock wave is nominally (without accounting for the influence of the vortex) located at the cross section x = 0.

Similarly to the basic formulaion of the problem, the start-up errors can be eliminated by setting the parameters of the supersonic inflow in the region x > 1.28 at the time t = t₁/6.

The exact solution of the SVI problem in the modified formulation corresponds to its solution in the basic formulation, except for of the longitudinal velocity component u_x, the values of which differ by 3⋅(1.4)^0.5. Hence, if we obtain the solution for the modified formulation of the SVI problem and apply the aforementioned correction for the velocity component to it, the resulting solution will correspond to the basic formulation of the problem.