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A fully implicit, mass‐conserving, semi‐Lagrangian scheme for the f ‐plane shallow‐water equations
Author(s) -
Thuburn J.
Publication year - 2007
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1697
Subject(s) - jacobian matrix and determinant , newton's method , discretization , rate of convergence , mathematics , nonlinear system , convergence (economics) , augmented lagrangian method , newton fractal , plane (geometry) , local convergence , scheme (mathematics) , space (punctuation) , iterative method , mathematical analysis , mathematical optimization , geometry , computer science , physics , computer network , channel (broadcasting) , quantum mechanics , economics , economic growth , operating system
A fully implicit, mass‐conserving, semi‐Lagrangian discretization of the shallow‐water equations on a doubly periodic f ‐plane is proposed. The scheme requires the solution of a nonlinear and nonlocal system of equations at each time step. When a Newton method is used to solve this system of equations, a partial elimination of the unknowns can be carried out to leave a near standard elliptic problem at each Newton iteration, for which efficient solution methods are well known. Moreover, the nonlinearity is, in fact, rather weak so that a small number of Newton iterations (1–3) should be sufficient for practical application. Setting up the Newton method and solving the resulting elliptic problem can be made cheaper by making certain approximations to the terms in the Jacobian matrix; the price to pay is a slowing of the Newton convergence rate. Numerical experiments are carried out to quantify the effects on convergence rate of such approximations to help decide the optimum trade‐off between cost per Newton iteration and iteration count. Copyright © 2007 John Wiley & Sons, Ltd.