Mesh-free Solution of ADR Equations: A Comparison Between the Application of Hybrid and Standard Radial Basis Functions

Three types of radial basis functions comprising two standard radial basis functions (RBF) and a hybrid Gaussian-cubic (HGC) kernel have been adopted in this paper to solve three-dimensional advection–dispersion-reaction (ADR) equations by an RBF-based mesh-free method. In this context, radial point interpolation method (RPIM) is implemented as a mesh-free discretization tool and the weak form of governing equation is attained by Galerkin weighted residual technique. Since the utilized RBFs contain shape parameter and weight coefficients (in the case of HGC kernel) as the influential parameters on the results accuracy, optimal values of these parameters are assessed via an optimization algorithm based on Particle Swarm Optimization technique. Accuracy/adequacy of the utilized RBFs as well as their stability have been elaborated by solving some examples at the end of the paper. Since the main aim of the present paper is the comparison between performance of different RBFs in ADR equation solution, simple examples with known analytical solutions have been utilized in a unitary cube domain. Accordingly, by solving time-dependent and time-independent problems, it has been concluded that all considered RBFs can be successfully used in RPIM to render results with acceptable accuracy. Besides, the hybrid RBF ended up with the highest stability in comparison with the other RBFs.