High order pseudo arc-length method for strong discontinuity of detonation wave

In this paper, in order to improve the resolution of capturing discontinuities, we introduce the pseudo arc-length parameter to make the mesh move to the discontinuities adaptively. By combining the high-precision WENO scheme with the pseudo arc-length algorithm, the advantages of both schemes can be shown, on the one hand, the solution has a higher convergence rate, on the other hand, it has a higher resolution for the region solution with larger physical variation . Because the traditional high-order scheme is based on Cartesian grid, and the grid in the pseudo arc-length numerical calculation is deformed. In view of the non-uniform grid and non-orthogonal deformation grid caused by the grid moving, the original deformed physical space is mapped to the uniform orthogonal arc-length calculation space by introducing coordinate transformation, and then the classical higher order scheme is used to solve the governing equations in the computational coordinate system. Through the comparison of some numerical examples and the analysis of numerical errors, it can be found that the pseudo arc-length algorithm is better than the finite volume method with fixed mesh. The high-order pseudo arc-length algorithm has a very high resolution to capture discontinuities, and the density of the grid near the discontinuities is very high. The adaptive grid movement weakens the singularity of the governing equation near the discontinuity, so the whole solution is smooth and the numerical oscillation is not obvious. This shows that the pseudo arc-length algorithm can overcome the shortcomings of high-order schemes which easily cause numerical oscillations. Finally, the chemical reaction flow problem is calculated. The results show that the high-order pseudo arc-length numerical algorithm with less mesh number has faster convergence rate and higher discontinuous resolution. Therefore, the high-order pseudo arc-length algorithm has obvious advantages in dealing with the strong discontinuity problem of explosion and shock.