Theoretical modelling and numerical simulation of hyperbolic balance laws

In the first part, we formulate a one dimensional isothermal drift flux model of two phase flows which is fully hyperbolic and conservative. The exact solution of this system, consisting of elementary waves has been derived. Furthermore, using the singular surface theory the evolution of discontinuity is discussed. In the second part, we formulate a weakly compressible two-layer shallow water flow in channels with arbitrary cross sections. This model is a balance law with non-conservative products. Using the method of generalised Rankine-Hugoniot conditions we derive the Riemann invariants for the shock and contact wave. Furthermore, a high-resolution, non-oscillatory semi-discrete central-upwind scheme is presented. The scheme extends existing central-upwind semi-discrete numerical methods for hyperbolic balance laws. We validate the exact solution with the numerical approach.