Upwind Biased Local RBF Scheme with PDE Centres for the Steady Convection Diffusion Equations with Continuous and Discontinuous Boundary Conditions (Communications in Computational Physics, 2019)
RBF based grid-free scheme with PDE centres is experimented in this workfor solving Convection-Diffusion Equations (CDE), a simplified model of the Navier-Stokes equations. For convection dominated problems, very few integration schemescan give converged solutions for the entire range of diffusivity wherein sharp layers areexpected in the solutions and accurate computation of these layers is a big challengefor most of the numerical schemes. Radial Basis Function (RBF) based Local HermitianInterpolation (LHI) with PDE centres is one such integration scheme which has somebuilt in upwind effect and hence may be a good solver for the convection dominatedproblems. In the present work, to get convergent solutions consistently for small diffusion parameters, an explicit upwinding is also introduced in to the RBF based schemewith PDE centres, which was initially used to solve some time dependent problemsin [10]. RBF based numerical schemes are one type of grid free numerical schemesbased on the radial distances and hence very easy to use in high dimensional problems. In this work, the RBF scheme, with different upwind biasing, is used to a varietyof steady benchmark problems with continuous and discontinuous boundary data andvalidated against the corresponding exact solutions. Comparisons of the solutions ofthe convective dominant benchmark problems show that the upwind biasing eitherin source centres or PDE centres gives convergent solutions consistently and is stablewithout any oscillations especially for problems with discontinuities in the boundaryconditions. It is observed that the accuracy of the solutions is better than the solutionsof other standard integration schemes particularly when the computations are carriedout with fewer centers.
Keywords: Grid-free scheme; radial basis function; convection-diffusion; multi-quadric; PDE points; upwind.