On the convergence of the solutions to the integral SPH heat and advection–diffusion equations: Theoretical analysis and numerical verification

Abstract

The convergence of the continuous Smoothed Particle Hydrodynamics (SPH) solution to the advection–diffusion equation is established in this paper. As a by-product of this result, the convergence of the SPH heat equation is also established. The most prevalent SPH approximations to diffusion operators are considered. The convergence is demonstrated using Fourier Analysis. First, the Fourier representation of the SPH approximation to the Hessian matrix is obtained. Next, the Fourier representations of the SPH diffusion operators are derived. Finally, the convergence of the SPH solution to the advection–diffusion equation (which includes, in particular, the case of the heat equation) is established in the Fourier space. The proof of convergence requires a certain condition over the Fourier transform of the SPH kernel to be satisfied. An example of kernel not satisfying this condition is numerically tested, leading to non-convergent solutions. To the authors’ knowledge, this is the first proof of convergence of an SPH equation involving the Laplacian operator. These analytical results are illustrated with a variety of numerical verification cases. In particular, the influence of the choice of the kernel on the nature of convergence is thoroughly studied.