An alternative SPH formulation: ADER-WENO-SPH

We present a new class of fully-discrete one-step SPH schemes based on a mesh-free ADER (Arbitrary DERivatives in space and time) reconstruction on moving particles in multiple space dimensions. In particular, the new SPH scheme computes mesh-free and local high order accurate polynomials in space and time to evaluate numerical fluxes at the midpoint between two interaction particles with a proper Riemann solver within the general SPH framework of Vila (1999) for nonlinear systems of hyperbolic conservation laws. The new scheme has been carefully tested against reference solutions for both the compressible Euler and the magneto-hydrodynamics (MHD) equations. The capability of the proposed scheme to accurately capture shocks and rarefaction waves for 1D and 2D problems with minimal amount of diffusion has been demonstrated. Via numerical evidence it has been shown that the new fully-discrete one-step ADER-WENO-SPH method is computationally more efficient than WENO-SPH schemes based on classical Runge–Kutta time-stepping. This is mainly due to the fact that with ADER timestepping the expensive stencil and neighbor search needs to be done only once per time step, while with Runge–Kutta time integrators the neighbor and stencil search is needed in each Runge–Kutta stage again.