Unconditionally stable splitting methods for the shallow water equations.

*(English)*Zbl 0945.76059Summary: The front-tracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multi-dimensions and shock tracking in one dimension. The front-tracking method is unconditionally stable, but for practical computations feasible CFL numbers are moderately above unity (typically between 1 and 5). The method resolves shocks sharply and is efficient.

The numerical technique is applied to four test cases: an expanding bore with rotational symmetry; the time development of two constant water levels separated by a dam that breaks instantaneously; comparison with an explicit analytic solution of water waves rotating over a parabolic bottom profile; the flow over an obstacle in one dimension.

The numerical technique is applied to four test cases: an expanding bore with rotational symmetry; the time development of two constant water levels separated by a dam that breaks instantaneously; comparison with an explicit analytic solution of water waves rotating over a parabolic bottom profile; the flow over an obstacle in one dimension.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

35L65 | Hyperbolic conservation laws |