Development of a particle interaction kernel for convection-diffusion scalar transport equation

In this study we derive a mathematically rigorous kernel function, which accounts for the interaction among particles, within the framework of the particle method, to predict a computationally more accurate solution for the convection-diffusion equation investigated at low as well as high Peclet numbers. Determination of the functional dependence of the kernel function on the distance vector between the particles is therefore a key to the success of the interaction model. The smoothed quantity for a scalar or for a vector at a particle location is mathematically identical to its collocated value provided that the kernel function is chosen as the delta function. Such a kernel is unfortunately not computable in a discrete context. Our guideline for developing the modified kernel function is therefore to make it closer to the delta function as much as possible in cases when diffusion dominates convection. To achieve this goal, we enforce five constraint conditions in a derivation of the kernel function for the pure diffusion equation. In addition, this kernel function has no effect on the particles outside of the disk, which has the user's specified radius re. To mimic the delta function we demand that the developed kernel function at r=re should smoothly approach zero. It is also desired to acquire the largest possible value for the kernel function near r=0. As flow convection prevailingly dominates its diffusion counterpart, particle interaction at the upstream side should be more favorably taken into account to avoid numerical oscillations due to convective instability. We present in this study a two-dimensional upwind kernel function to enhance numerical stability along the flow direction. The proposed upwind kernel function can render an exact solution for the investigated convection-diffusion equation in the limiting one-dimensional case. The proposed particle interaction model featuring the newly developed kernel function is validated through several problems that are amenable to analytical solutions or have available benchmark solutions. Analysis of the stability condition and spatial accuracy order of the proposed particle interaction model are also provided in details.