Uncoupling Evolutionary Groundwater-surface Water Flows Using the Crank-nicolson Leapfrog Method

Abstract. Consider an incompressible fluid in a region Ωf flowing both ways across an interface, I, into a porous media domain Ωp saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a system with an exactly skew symmetric coupling. Spatial discretization by finite element method and time discretization by Crank-Nicolson LeapFrog gives a second order partitioned method requiring only one Stokes and one Darcy sub-physics and sub-domain solver per time step for the fully evolutionary Stokes-Darcy problem. Analysis of this method leads to a time-step condition sufficient for stability and convergence. Numerical tests verify predicted rates of convergence, however stability tests reveal the problem of growth of numerical noise in unstable modes in some cases. In such instances, the addition of time filters adds stability.