Improving approximation accuracy in Godunov-type smoothed particle hydrodynamics methods

The study examines the origin of errors resulting from the approximation of the right hand sides of the Euler equations using the Godunov type contact method of smoothed particle hydrodynamics (CSPH). The analytical expression for the numerical shear viscosity in CSPH method is obtained. In our recent study the numerical viscosity was determined by comparing the numerical solution of momentum diffusion in the shear flow with theoretical one. In this study we deduce the analytical expression for the numerical viscosity which is found to be similar to numerical one, confirming the obtained results. To reduce numerical diffusion, diffusion limiters are typically applied to expressions for contact values of velocity and pressure, as well as higher-order reconstruction schemes. Based on the performed theoretical analysis, we propose a new method for correcting quantities at interparticle contacts in CSPH method, which can be easily extended to the MUSCL-type (Monotonic Upstream-centered Scheme for Conservation Laws) method. Original CSPH and MUSCL-SPH approaches and ones with aforementioned correction are compared. •The sources of approximation errors in the contact Godunov-type SPH method (CSPH) are identified.•A simple approximate analytical formula for the numerical shear viscosity of CSPH method is derived.•A new kernel gradient correction technique for Godunov-type SPH schemes is proposed.