Simulation of Nondifferentiable Models for Groundwater Flow and Transport
NORTH CAROLINA STATE UNIV AT RALEIGH CENTER FOR RESEARCH IN SCIENTIFIC COMPUTATION
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Non-Lipschitz continuous nonlinearities arise frequently in models for groundwater flow and species transport. The van Genuchten and Mualem PSK relations for unsaturated flow and the Freundlich equilibrium expressions in reactive transport are examples. Numerical methods such as nonlinear solvers based on Newtons method, error estimators for differential equations, and stepsize and order control methods for temporal integration, are designed for differentiable problems and may fail when applied to nonsmooth nonlinear problems. In this paper we consider two approaches to this problem 1 adding new equations to smooth the nonlinearity and 2 approximating the nonlinearity with a smoother function, such as a spline. In both cases, we replace the non-Lipschitz continuous functions with Lipschitz continuous, but sometimes non-differentiable, nonlinearities. The mathematical properties of Lipschitz continuous nonlinear equations enable standard solvers to work well. We will describe some recent theoretical advances that explain this success and use those results to justify a stepsize and error control strategy for temporal integration. We illustrate the results with two computational examples.
- Hydrology, Limnology and Potamology
- Numerical Mathematics
- Fluid Mechanics