A New Stable Version of the SPH Method in Lagrangian Coordinates (Communications in Computational Physics, 2018)
The purpose of this paper is to solve some of the trouble spots of the classical SPH method by proposing an alternative approach. First, we focus on the problem of the stability for two different SPH schemes, one is based on the approach ofVila [25] and the other is proposed in this article which mimics the classical 1D Lax Wendroff scheme. In both approaches the classical SPH artificial viscosity term is removed preserving nevertheless the linear stability of the methods, demonstrated viathe von Neumann stability analysis. Moreover, the issue of the consistency for theequations of gas dynamics is analyzed. An alternative approach is proposed that consists of using Godunov-type SPH schemes in Lagrangian coordinates. This not onlyprovides an improvement in accuracy of the numerical solutions, but also assures thatthe consistency condition on the gradient of the kernel function is satisfied using anequidistant distribution of particles in Lagrangian mass coordinates. Three differentRiemann solvers are implemented for the first-order Godunov type SPH schemes inLagrangian coordinates, namely the Godunov flux based on the exact Riemann solver,the Rusanov flux and a new modified Roe flux, following the work of Munz [17]. Somewell-known numerical 1D shock tube test cases [22] are solved, comparing the numerical solutions of the Godunov-type SPH schemes in Lagrangian coordinates with thefirst-order Godunov finite volume method in Eulerian coordinates and the standardSPH scheme with Monaghan's viscosity term.