Preprint 1999-003

Operator Splitting Methods for Generalized Korteweg-de Vries Equations

Abstract: We apply the method of operator splitting on the generalized Korteweg--de Vries (KdV) equation $u_{t}+f(u)_{x} +\eps u_{xxx} = 0$, by solving the nonlinear conservation law $u_{t}+f(u)_{x}=0$ and the linear dispersive equation $u_{t}+\eps u_{xxx} = 0$ sequentially. We prove that if the approximation obtained by operator splitting converges, then the limit function is a weak solution of the generalized KdV equation. Convergence properties are analyzed numerically by studying the effect of combining different numerical methods for each of the simplified problems.

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Author(s):

Helge Holden, <holden@math.ntnu.no>

Kenneth Hvistendahl Karlsen, <kennethk@mi.uib.no>

Nils Henrik Risebro, <nilshr@math.uio.no>

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<holden@math.ntnu.no> March 5 1999.