A Comparison Between Relaxation and Kurganov–Tadmor Schemes

In this work we compare two semidiscrete schemes for the solution of hy-perbolic conservation laws, namely the relaxation [JX95] and the Kurganov Tadmor central scheme [KT00]. We are particularly interested in their behavior under small time steps, in view of future applications to convection diffusion problems. The schemes are tested on two benchmark problems, with one space variable. 1 Motivation We are interested in the solution of systems of equations of the form u t + f x (u) = D p xx (u), (1) where f (u) is hyperbolic, i.e., the Jacobian of f is provided with real eigen-values and a basis of eigenvectors for each u, while p(u) is a nondecreasing Lipschitz continuous function, with Lipschitz constant µ and D ≥ 0. We continue the study of convection diffusion equations with the aid of high order relaxation schemes started in [CNPS06] for the case of the purely parabolic problem. In many applications, such as multiphase flows in porous media, p(u) is nonlinear and possibly degenerate. In these conditions, an implicit solution of the diffusion term can be computationally very expensive: in fact it may be necessary to solve large nonlinear algebraic systems of equations which, moreover, can be singular at degenerate points, i.e., where p(u) = 0. For this reason, it is of interest to consider the explicit solution of (1). This in turn poses one more difficulty. An explicit solution of (1) requires a parabolic CFL condition, that is, stability will restrict the possible choice of the time step ∆t to ∆t ≤ C(∆x) 2 , where ∆x is the grid spacing. In other words, it may be necessary to choose very small time steps. But conventional solvers for convective operators typically work at their best for time steps close to a