A generalised finite difference scheme based on compact integrated radial basis function for flow in heterogeneous soils

In the present paper, we develop a generalised finite difference approach based on compact integrated radial basis function (CIRBF) stencils for solving highly nonlinear Richards equation governing fluid movement in heterogeneous soils. The proposed CIRBF scheme enjoys a high level of accuracy and a fast convergence rate with grid refinement owing to the combination of the integrated RBF approximation and compact approximation where the spatial derivatives are discretised in terms of the information of neighbouring nodes in a stencil. The CIRBF method is first verified through the solution of ordinary differential equations, 2-D Poisson equations and a Taylor-Green vortex. Numerical comparisons show that the CIRBF method outperforms some other methods in the literature. The CIRBF method in conjunction with a rational function transformation method and an adaptive time-stepping scheme is then applied to simulate 1-D and 2-D soil infiltrations effectively. The proposed solutions are more accurate and converge faster than those of the finite different method used with a second-order central difference scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the 1D-IRBF and higher order compact schemes.