Method of Order Reduction for the High-Dimensional Convection-Diffusion-Reaction Equation with Robin Boundary Conditions Based on MQ RBF-FD

Abstract

A novel method of order reduction is proposed to the high-dimensional convection-diffusion-reaction equation with Robin boundary condition based on the multiquadric radial basis function-generated finite difference method (MQ RBF-FD). The main motivation is to get not only a second-order accurate solution but also a second-order accurate gradient. Key to the proposed method is introducing the intermediate variables representing the first-order derivatives to reduce the original second-order problem into an equivalent system of first-order partial differential equations. Then a discrete scheme for the latter is constructed, in which MQ RBF-FD method is applied to approximate the first-order derivatives of the original variable at the center point with decoupled method. Moreover, we can obtain an equivalent discrete scheme about the original variable and intermediate variables which can be proven all second-order convergent, that is, the convergence rate of the gradient of solution is also second-order. Finally numerical examples are presented to show the efficiency and accuracy of the proposed method.