On the convergence of the solution to the integral SPH advection-diffusion equation with rotating transport velocity field

The convergence of the integral smoothed particle hydrodynamics (SPH) solution to the advection-diffusion equation in two dimensions in the particular case of a rigid rotation transport velocity field, is established in this paper. The approximation to the Laplacian operator is considered (Morris et al. 1997). The convergence of the SPH solution to the exact one is established in Fourier space. It is shown that convergence is guaranteed if a certain condition on the Fourier transform of the kernel, that we call positivity condition, is fulfilled. This condition is that the first moment of the radial profile of the kernel's Fourier transform is positive. The analytical result is illustrated with a numerical verification. The numerical solutions obtained with different kernels, including those more commonly used by the SPH community (namely, the Wendland kernels and the cubic and linear spline kernels), are compared in terms of both the L-2 norm and the norm of the maximum. In addition to that, a pathological kernel, for which the positivity condition is not satisfied, is presented and tested, leading to non convergent results. This fact suggests that the Positivity condition is not only a sufficient but a necessary condition for convergence, in the case of the advection-diffusion equation. Moreover, the derivation of the exact solution for one of the numerical examples discussed in the paper is presented in detail in an appendix. This is an interesting test case for the advection-diffusion equation due to its discontinuous nature, therefore, the authors consider it useful to thoroughly present the obtaining of its exact solution. The paper is closed with conclusions and future work.