Technical Report 2011-009

Operator Splittings for Solving Nonlinear, Coupled Multiphysics Problems with an Application to the Numerical Solution of an Interface Problem

Owe Axelsson and Maya Neytcheva

April 2011


This work summarizes operator splitting methods to solve various kinds of coupled multiphysics problems. Such coupled problems are usually stiff. Furthermore, one is often interested in obtaining stationary solutions, which require integration over long time intervals. Therefore, an implicit and stable time-stepping method of at least second order of accuracy must be used, to allow for larger timesteps. To control the operator splitting errors for nonlinear problems, an approximate Newton solution method is proposed for each separate equation. After completion of some (normally few) Newton steps, the equations are updated with the current solution, thereby preparing for the next sequence of Newton steps.

An application for a nonlinear model of interface tracking problem arising in a multiphase flow is described. Hereby an inner-outer iterative solution method with a proper preconditioning for solving the arising linearized algebraic equations, which results in few iterations, is analyzed. There is no need to update the preconditioner during the iterations.

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