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Uncoupling evolutionary groundwater‐surface water flows using the Crank–Nicolson Leapfrog method
Author(s) -
Kubacki Michaela
Publication year - 2013
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21751
Subject(s) - discretization , crank–nicolson method , mathematics , partial differential equation , convergence (economics) , solver , porous medium , stability (learning theory) , compressibility , stokes flow , darcy–weisbach equation , finite element method , domain (mathematical analysis) , darcy's law , mathematical analysis , mathematical optimization , mechanics , geometry , computer science , flow (mathematics) , porosity , physics , geology , geotechnical engineering , machine learning , economics , thermodynamics , economic growth
Consider an incompressible fluid in a region Ω f flowing both ways across an interface into a porous media domain Ω p 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 give a second‐order partitioned method requiring only one Stokes and one Darcy subphysics and subdomain 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. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013