The introduction of the surfing scheme for shock capturing with high-stability and high-speed convergence

Abstract

In this paper, a novel shock capturing scheme with high stability and speed of solution is presented. The scheme divides the dependent variables into two parts, i.e. Surf and Surfer, like surfing sport. The Surf part is calculated by applying the concept of Sobolev gradient on the dependent variables. Furthermore, it has the least difference with these variables, along with the lowest discretization error and the same equation. Hence, it is expected that this part can be solved at high stability conditions and large time steps. In addition, the Surfer part is obtained from the difference between the Surf part and the dependent variables. Therefore, only the error-maker operators (for example, a discontinuity like a shock) are in this part and accordingly it has a local nature. Due to this feature, it can be solved with fewer points compared to the Surf part. Therefore, its equation is solved as quickly as Surf equation, despite its limited stability conditions. Finally, the summation of the Surf and Surfer solutions makes the main solution. The scheme has been applied to the inviscid Burgers’ equation with an initial value of the step function, and one-dimensional inviscid Euler flow through a nozzle with a shock wave. The stability condition for the Burgers’ equation has been increased to a Courant number of 1010 and the solution time for the Euler flow case has been decreased by one-fifth. Comparison of the results with traditional methods indicates the ultra-high stability and ultra-speed convergence of this scheme.