Exact solutions of the two dimensional nonlinear shallow water wave equations for flow involving linear bottom friction and with no forcing are found for flow above parabolic bottom topography. These solutions also involve moving shorelines. The motion decays over time. In the solution of the three simultaneous nonlinear partial differential shallow water wave equations it is assumed that the velocity is a function of time only and along one axis. This assumption reduces C374 the three simultaneous nonlinear partial differential equations to two simultaneous linear ordinary differential equations . The solutions found are useful for testing numerical solutions of the nonlinear shallow water wave equations which include bottom friction and whose flow involves moving shorelines.
Exact solutions of the nonlinear shallow water wave equations for forced flow involving linear bottom friction in a region with quadratic bathymetry have been found. These solutions also involve moving shorelines. The motion decays over time. In the solution of the three simultaneous nonlinear partial differential shallow water wave equations it is assumed that the velocity is a function of time only and along one axis. This assumption reduces the three simultaneous nonlinear partial differential equations to two simultaneous linear ordinary differential equations. The analytical model has been tested against a numerical solution with good agreement between the numerical and analytical solutions. The analytical model is useful for testing the accuracy of a moving boundary shallow water numerical model.
A new moving boundary shallow water wave equation numerical model is presented. The model is adapted from the Selective Lumped Mass (slm) numerical model. The wetting and drying scheme used is different to that in the slm model. The slm model is finite element in space, using fixed triangular elements, finite difference in time and is explicit. The numerical model has been tested against an analytical solution with good agreement between the numerical and analytical solutions. The numerical model proposed is useful for comparing against analytical moving boundary solutions. The analytical solution is presented for the first time, being for the case of frictionless one dimensional moving boundary nonlinear shallow water wave flow with cosine forcing in a bed with quadratically varying depth. The analytical solution, which is explicit, is useful for testing numerical models.
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