Research On Radial Basis Function-finite Difference Method For Convection-diffusion-reaction Equations

In the calculations of partial differential equation,it is particularly important to construct stable and efficient numerical methods.Grid based methods are very popular,but they usually face difficulties in mesh generation and contradictions between accuracy and amount of computation.Radial basis function method as one of the mesh-less method has been widely used in data interpolation and partial differential equation solving,and gradually formed its own system.Considering the hyperbolic nature of convection-dominated diffusion-reaction equations,the method of characteristics combined with radial basis function method is proposed to numerically solve and simulate the equations.To numerically simulate convection-diffusion-reaction equations on surface,we proposed the characteristic radial basis function-finite difference method.By using the method of characteristics,the convection-diffusion-reaction equations can be transformed into diffusion equation,so that radial basis function-finite difference method only need to discrete diffusion and reaction terms.Moreover,we give the theoretical estimate of interpolation error bound on the surface,and demonstrate the convergence and effective of characteristic radial basis function-finite difference method for convection-diffusion-reaction equations on surface by some numerical tests.There are two questions when use above method to research convection-diffusionreaction equations:(1)how to overcome numerical oscillation caused by convectiondominate;(2)how to solve the problem quickly without reducing the accuracy.We start from plane and present the characteristic overlapped radial basis function-finite difference method to solve convection-dominated diffusion-reaction equations.This method combined the method of characteristics with the overlapped radial basis function-finite difference method to create the characteristic overlapped radial basis function-finite difference formula.The method of characteristics overcomes the instability problem caused by the dominance of convection.The overlapped radial basis function-finite difference method can save calculation cost.In numerical experiments,we verify the convergence of the method,and demonstrate the validity and accuracy of the method by solving boundary layer problem and three-dimensional problem.