Comparison of the Crank Nicholson and generalized α methods applied to the groundwater flow equation

Abstract

This article is a revision and comparison of two numeric methods used to solve the transient part of the problem of flow through a saturated, porous medium. The transient part of the fundamental flow equation in a porous medium in one dimension is solved using two temporary integration schemas: the Crank Nicholson method and the generalized α-method. The spatial component of the equation is discretized using the finite element method. Even though both methods are traditionally stable, the precision of the methodology of the Crank Nicholson depends on the existing relationship between the Δx size of the discretized elements and the size Δt of the flow of time (albeit mostly dependent on the size of Δt), while in the α-generalized method it depends additionally of three parameters exclusive of this schema. The precision of both methods is compared for different values of Δx and Δt. Additionally, for the alpha method, the influence of two of the three parameters in the precision of the solution is analyzed.