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SUMMARYIn this paper, a new family of high-order relaxation methods is constructed. These methods combine general higher-order reconstruction for spatial discretization and higher order implicit-explicit schemes or TVD Runge-Kutta schemes for time integration of relaxing systems. The new methods retain all the attractive features of classical relaxation schemes such as neither Riemann solvers nor characteristic decomposition are needed. Numerical experiments with the shallow-water equations in both one and two space dimensions on at and non-at topography demonstrate the high resolution and the ability of our relaxation schemes to better resolve the solution in the presence of shocks and dry areas without using either Riemann solvers or front tracking techniques.

We present a convergence and stability analysis of the finite element modified method of characteristics for the incompressible Navier-Stokes equations. The method consists of combining a second-order backward time discretization based on the characteristics method with a spatial discretization of finite element type. We obtain stability results and optimal error estimates in the L 2 -norm for velocity and pressure components under a time step restriction more relaxed than the standard Courant-Friedrichs-Levy condition. We also show some numerical results for two benchmark problems on the incompressible Navier-Stokes equations at different Reynolds numbers.

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractAn enriched partition-of-unity (PU) finite element method is developed to solve timedependent diffusion problems. In the present PU formulation, an exponential solution describing the spatial diffusion decay is embedded in the finite element shape function. It results in an enriched approximation, which is in the form of local asymptotic expansion. The temporal decay in the solution is embedded naturally in the PU expansion so that, unlike previous works in this area, the same system matrices may be used for every time step. In comparison with the traditional finite element analysis with p-version refinements, the present approach is much simpler, more robust and efficient, and yields more accurate solutions for a prescribed number of degrees of freedom. On the other hand, the notorious difficulty encountered in the meshless method in satisfying the essential boundary conditions is circumvented. Numerical results are presented for a transient diffusion equation with known analytical solution. The performance of the method is analysed on two applications: the transient heat equation with a single source and with multiple sources. The aim of such a method compared to the classical finite element method is to solve time-dependent diffusion applications efficiently and with an appropriate level of accuracy.

SUMMARYWe apply the lattice Boltzmann (LB) method for solving the shallow water equations with source terms such as the bed slope and bed friction. Our aim is to use a simple and accurate representation of the source terms in order to simulate practical shallow water flows without relying on upwind discretization or Riemann problem solvers. We validate the algorithm on problems where analytical solutions are available. The numerical results are in good agreement with analytical solutions. Furthermore, we test the method on a practical problem by simulating mean flow in the Strait of Gibraltar. The main focus is to examine the performance of the LB method for complex geometries with irregular bathymetry. The results demonstrate its ability to capture the main flow features.