SPH Entropy Errors and the Pressure Blip

Abstract: The spurious pressure jump at a contact discontinuity, in SPH simulations of the compressible Euler equations is investigated. From the spatiotemporal behaviour of the error, the SPH pressure jump is likened to entropy errors observed for artificial viscosity based finite difference/volume schemes. The error is observed to be generated at start-up and dissipation is the only recourse to mitigate it's effect. We show that similar errors are generated for the Lagrangian plus remap version of the Piecewise Parabolic Method (PPM) finite volume code (PPMLR). Through a comparison with the direct Eulerian version of the PPM code (PPMDE), we argue that a lack of diffusion across the material wave (contact discontinuity) is responsible for the error in PPMLR. We verify this hypothesis by constructing a more dissipative version of the remap code using a piecewise constant reconstruction. As an application to SPH, we propose a hybrid GSPH scheme that adds the requisite dissipation by utilizing a more dissipative Riemann solver for the energy equation. The proposed modification to the GSPH scheme, and it's improved treatment of the anomaly is verified for flows with strong shocks in one and two dimensions. The result that dissipation must act across the density and energy equations provides a consistent explanation for many of the hitherto proposed "cures" or "fixes" for the problem.