Conservation laws in continuum physics are often coupled, for example the continuity equations for a reacting gas mixture or a plasma are coupled through multi-species diffusion and a complicated reaction mechanism. For space discretisation of these equations we employ the finite volume method. The purpose of this talk is to present novel flux vector approximation schemes that incorporate this coupling in the discretisation. More specifically, we consider as model problems linear advection-diffusion systems with a nonlinear source and linear diffusion-reaction systems, also with a nonlinear source.
The new flux approximation schemes are inspired by the complete flux scheme for scalar equations, see . An extension to systems of equations is presented in . The basic idea is to compute the numerical flux vector at a cell interface from a local inhomogeneous ODE-system, thus including the nonlinear source. As a consequence, the numerical flux vector is the superposition of a homogeneous flux, corresponding to the homogeneous ODE-system, and an inhomogeneous flux, taking into account the effect of the nonlinear source. The homogeneous ODE-system is either an advection-diffusion system or a diffusion-reaction system. In the first case, the homogeneous flux contains only real-valued exponentials, on the other hand, in the second case, also complex-valued components are possible, generating oscillatory solutions. The inclusion of the inhomogeneous flux makes that all schemes display second order convergence, uniformly in all parameters (Peclet and Damköhler numbers).
The performance of the novel schemes is demonstrated for several test cases, moreover, we investigate several limiting cases.
This is a joint work with J. van Dijk and R.A.M. van Gestel (Department of Applied Physics, Eindhoven University of Technology).
 J.H.M. ten Thije Boonkkamp and M.J.H. Anthonissen, The finite volume-complete flux scheme for advection-diffusion-reaction equations, J. Sci. Comput. 46 (2011) pp. 47-70.
 J.H.M. ten Thije Boonkkamp, J. van Dijk, L. Liu and K.S.C. Peerenboom, Extension of the complete flux scheme to systems of comservation laws, J. Sci. Comput. 53 (2012), pp. 552-568.