A space-time backward substitution method for three-dimensional advection-diffusion equations

A meshless numerical scheme, named as space-time backward substitution method (STBSM), is proposed for 3D unsteady advection-diffusion equations (ADEs). In this method, the numerical solution is composed by approximation of boundary points and approximation of internal domain. Here, the space-time radial basis function (STRBF) is applied to construct internal approximation which satisfies homogeneous boundary condition. While, time dimension, as a new variable in radial coordinate, is incorporated into the global space-time basis function. Different from using other common global basis functions, the STBSM with STRBF does not require the temporal discretization, which makes it easy-to program and time saving. In addition, compactly supported space-time radial basis function (CS-STRBF) is introduced to reduce the original resultant full matrix to a sparse matrix, compared with global space-time radial basis function. It can overcome ill-conditioned matrix caused by an excessive condition number. Numerical results of two examples show that the proposed method provides accurate and stable solutions with low computational cost for three dimensional advection-diffusion equations. Moreover, comparison results indicate that the accuracy of the method is 2 - 4 orders of magnitude higher than the dimension splitting element-free Galerkin (DSEFG) [19] algorithm with less time expense.

Automated summary

A meshless numerical scheme, STBSM, is proposed for solving 3D unsteady advection-diffusion equations using space-time backward substitution method and space-time radial basis function (STRBF). The method introduces compactly supported space-time radial basis function (CS-STRBF) to improve computational efficiency and reduce matrix condition number.