A time splitting algorithm for numerical solution of Richard’s equation

Abstract

► Advection and diffusion forming the Richard’s equation are studied separately. ► A non-iterative numerical algorithm is proposed based on time splitting technique. ► A combined finite volume and characteristic methods are proposed for advection. ► To preserve the efficiency, an implicit procedure is proposed for characteristic part. ► A 4th order scheme is designed for diffusing without noticeable change in efficiency. The Richard’s equation mathematically expressing the infiltration problem is a transport type of equation describing two different phenomena namely advection and diffusion being hyperbolic and parabolic type of differential equation, respectively each having their own physical properties thus a single numerical scheme would not essentially be suitable for both. In this paper a numerical algorithm based on time splitting or fractional step method has been proposed and described. This has allowed designing and applying some different numerical schemes more compatible with the mathematical and physical properties of the corresponding phenomenon. In advection part an implicit characteristic based method has been employed to calculate the cell face fluxes and then the new unknowns have been obtained using finite volume method (FVM) explicitly. Two different speed type quantities namely wave celerity and mass velocity have been distinguished in the advection of infiltrated water to the soil, the magnitude of the first being several times larger than the other. The implicit characteristic based method has been designed specifically to cope with the mentioned high wave celerity. An implicit numerical scheme has been employed in which the space derivative term contributed in definition of cell face flux has been discretized in fourth order accurate manner. The extra cells getting involved due to the higher order discretization have been taken in account in a way not to change the tri-diagonal matrix coefficient shape preserving the efficiency of the algorithm. The performance, accuracy and efficiency of the out coming non-iterative numerical algorithm have been successfully examined by some test cases. The relative importance of the advection and diffusion terms in the concerned transport equation and the ration of their contribution in the overall infiltrated flux also have been discussed in detail.