In this work, we study a kinetic model that contains stiff relaxation terms in the source. This model can be applied to any non linear hyperbolic problem without source term, that we will call macroscopic problem, to obtain a larger system of equations with linear fluxes and non–linear source terms, the microscopic problem, that converges asymptotically to the original hyperbolic system.
We want to apply this kinetic model to shallow water equations with a source term that depends on the profile of the bathymetry. Thus, the model must be extended in order to include this term and to gain the asymptotic convergence to the macroscopic problem.
To solve the equations with high order methods, we use an IMEX (implicit–explicit) discretization in time to stabilize the relaxation term, with DeC (deferred correction) time integration, a high order iterative time integration technique, and RD (residual distribution) space discretization, a finite– element based method that can generalize well–known discontinuous Galerkin, finite volume and finite difference schemes.
The scheme proposed must verify many essential physical and numerical properties in order to guarantee the quality of the simulations. First of all, the scheme should be AP (asymptotic preserving), which means that in the relaxation limit, we will recast the macroscopic model of the shallow water equations. Then, we should guarantee the well balancedness of the solution in the lake at rest case, where no oscillations should occur when the surface level of the water is constant and the speed is zero. Moreover, we want our scheme to guarantee positivity of water height everywhere in the domain, also close to wet and dry area.
We show some numerical tests to prove the quality of the scheme.
- Aregba-Driollet, D. and Natalini, R., Discrete Kinetic Schemes for Systems of Conservation Laws. Birkhauser Basel, Basel, 1999.
- R. Abgrall, and D. Torlo. Asymptotic preserving Deferred Correction Residual Distribution schemes. arXiv e-prints, page arXiv:1811.09284, Nov 2018.
- A. Dutt, L. Greengard, and V. Rokhlin. Spectral Deferred Correction Methods for Ordinary Differential Equations. BIT Numerical Mathematics, 40(2):241–266, 2000.
- M. Ricchiuto and R. Abgrall. Explicit Runge-Kutta residual distribution schemes for time dependent problems: Second order case. Journal of Computational Physics, 229(16):5653–5691, 2010.