On the maximum time step in weakly compressible SPH

In the SPH method for viscous fluids, the time step is subject to empirical stability criteria. We proceed to a stability analysis of the Weakly Compressible SPH equations using the von Neumann approach in arbitrary space dimension for unbounded flow. Considering the continuous SPH interpolant based on integrals, we obtain a theoretical stability criterion for the time step, depending on the kernel standard deviation, the speed of sound and the viscosity. The stability domain appears to be almost independent of the kernel choice for a given space discretisation. Numerical tests show that the theory is very accurate, despite the approximations made. We then extend the theory in order to study the influence of the method used to compute the density, of the gradient and divergence SPH operators, of background pressure, of the model used for viscous forces and of a constant velocity gradient. The influence of time integration scheme is also studied, and proved to be prominent. All of the above theoretical developments give excellent agreement against numerical results. It is found that velocity gradients almost do not affect stability, provided some background pressure is used. Finally, the case of bounded flows is briefly addressed from numerical tests in three cases: a laminar Poiseuille flow in a pipe, a lid-driven cavity and the collapse of a water column on a wedge.