Smoothed particle hydrodynamics with diffusive flux for advection-diffusion equation with discontinuities

The advection-diffusion equation (ADE) with variable diffusion coefficient can be written in a flux form to avoid rewriting the diffusion term with a drift. However, for solving the flux-form ADE using smoothed particle hydrodynamics (SPH), a double first-order derivative approximation has to be used to approximate the diffusion term, which creates non-physical oscillations at any discontinuities. This is one reason why the flux-form diffusion is rarely used in SPH. To prevent such oscillations, the conditions and causes of them are theoretically analyzed, and a new partial diffusive flux format is proposed. To improve the particle consistency, a corrective particle approximation is applied. The effectiveness of the proposed method is verified by solving four ADE cases with analytical solutions. The results shown that SPH in partial diffusive flux can fully eliminate the spurious oscillations, and achieve second-order accuracy and second-order uniform convergence for contaminant transport problems with discontinuities. Moreover, compared with the conventional diffusive flux format, the numerical error of the proposed method is reduced by at least one order of magnitude. An encouraging possibility for the application of the smoothed particle hydrodynamics with diffusive flux to the anisotropic dispersion is also provided.