Complex Systems of Conservation Laws with Source Terms

Numerical Solution of Genuinely Multidimensional Conservation Laws with Source Terms

Multidimensional hyperbolic conservation or balance laws are characterised by the fact that information propagate in all possible directions. Classical finite volume methods that are based on dimensional splitting algorithms can typically prefer some directions. This might yield lost of accuracy. An alternative approach is to work with the so-called genuinely multidimensional finite volume methods; in particular we will use the finite volume evolution Galerkin (FVEG) method that belongs to this class.

The goal of the project is mathematical modelling and numerical simulation of complex multidimensional hyperbolic conservation laws with source terms using the FVEG method. Recent numerical experiments show that the FVEG method yields much better accuracy than classical dimensional splitting FVM; in fact it is typically 10 times more accurate under the same CPU time, see, e.g., [1],[2].

Numerical methods that will be developed within the project should be appropriate for various geophysical applications. For example, we  investigate Euler and Navier-Stokes equations with source terms or the shallow water equations that can be used in oceanographic or meteorological modelling.

Another interesting feature appearing in geophysical flows is their multiscale behaviour;  indeed the velocity of fluid flow is tyically 10-100 times smaller than the speed of sound or of the gravitational waves. In order to solve these types of flows we have proposed large time step IMEX type FVEG and discontinuous Galerkin EG schemes, see [3],[4],[5],[6]. Theoretically the question of asymtotic preserving property with respect to small Mach/Froude numbers have to be analysed.

Cooperations with S. Noelle (RWTH Aachen),  K.R. Arun (Trivandrum, India) ;  former collaborations G. Warnecke (University Magdeburg), K.W. Morton (Oxford) and Y. Zahaykah (Magdeburg, Jerusalem) .

Rising large warm air bubble with a small cold bubble on top,
simulated on GPU by the DG evolution Galerkin method (computed by L. Yelash and B.J. Block)