Eulerian-Lagrangian time-stepping methods for convection-dominated problems
International Journal of Computer Mathematics - Recent Advances in Computational and Applied Mathematics in Science and Engineering
Finite element P1 solution of unsteady thermal flow past a circular cylinder with radiation
International Journal of Computer Mathematics - Recent Advances in Computational and Applied Mathematics in Science and Engineering
An Eulerian--Lagrangian method for coupled parabolic-hyperbolic equations
Applied Numerical Mathematics
Solving Wick-stochastic water waves using a Galerkin finite element method
Mathematics and Computers in Simulation
Applied Numerical Mathematics
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We present a new numerical technique to approximate solutions to unsteady free surface flows modelled by the two-dimensional shallow water equations. The method we propose in this paper consists of an Eulerian–Lagrangian splitting of the equations along the characteristic curves. The Lagrangian stage of the splitting is treated by a non-oscillatory modified method of characteristics, while the Eulerian stage is approximated by an implicit time integration scheme using finite element method for spatial discretization. The combined two stages lead to a Lagrange–Galerkin method which is robust, second order accurate, and simple to implement for problems on complex geometry. Numerical results are shown for several test problems with different ranges of difficulty.