Dispersion analysis of SPH for parabolic equations: High-order kernels against tensile instability

The Smoothed Particle Hydrodynamics (SPH) is a meshless particle-based method mainly used to solve dynamical problems for partial differential equations (PDE). By means of dispersion analysis we investigated four classical SPH-discretizations of parabolic PDE differing by the approximation of Laplacian.

We derived approximate dispersion relations (ADR) for considered SPH-approximations of the Burgers equation. We demonstrated how the analysis of the ADR allows both studying the approximation and stability of numerical scheme and explaining the features of the method that are known from practice, but are counter-intuitive from the theoretical point of view.

By means of the mathematical analysis of ADR, the phenomenon of conditional approximation of some schemes under consideration is shown. Moreover, we pioneered in obtaining the necessary condition for the stability of the SPH-approximation of parabolic equations in terms of the Fredholm integral operator applied to the function defined by the kernel of the SPH method. Using this condition, we revealed that passing from the classical second-order kernels to high-order kernels for some schemes leads to the appearance of tensile (short-wave) instability. Among the schemes under consideration, we found the one, for which the necessary condition for the stability of short waves is satisfied both for classical and high-order kernels. The fourth order of approximation in space of this scheme is shown theoretically and confirmed in practice.