Error estimates for a time discretization method for the Richards’ equation. (English) Zbl 1079.65542

Pop, Iuliu Sorin

Comput. Geosci. 6, No. 2, 141-160 (2002).

Summary: We present a numerical analysis of a time discretization method applied to Richards’ equation. Written in its saturation-based form, this nonlinear parabolic equation models water flow into unsaturated porous media. Depending on the soil parameters, the diffusion coefficient may vanish or explode, leading to degeneracy in the original parabolic equation. The numerical approach is based on an implicit Euler time discretization scheme and includes a regularization step, combined with the Kirchhoff transform. Convergence is shown by obtaining error estimates in terms of the time step and of the regularization parameter.

MSC:

65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76S05	Flows in porous media; filtration; seepage
35K65	Degenerate parabolic equations
35K55	Nonlinear parabolic equations

Keywords:

degenerate parabolic problem; error estimates; implicit scheme; regularization; Richards’ equation

Software:

UG