Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL–condition associated with it, and it does not discriminate between stiff and non-stiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady–state solutions (or achieving them in the long time limit) with good accuracy. 1.

In the hydraulics industry, an accurate numerical approximation of the equations governing sediment transport in coastal regions has recently become a major topic of interest. These equations comprise the shallow water equations governing the water flow with the addition of a bed transport equation. It is common practice in industry to simplify the equations governing sediment transport by assuming the water flow is in an equilibrium state and the bed has a negligible effect on the water flow. However, this approach is limited as only steady flow and a slow moving bed can be approximated accurately in this manner. Thus, an unsteady approach is required that is considerably more robust and approximates the full system simultaneously. Until recently, the classic Lax-Wendroff scheme has been widely used in industry to obtain a numerical solution to the equations, but not surprisingly the numerical results obtained suffered from spurious oscillations resulting in the numerical scheme becoming unstable for long time periods. In industry, different measures have been applied to the use of classic Lax-Wendroff scheme to try to eliminate the spurious oscillations such as flux-limiter methods and using a small Courant number.

The morphodynamical equations can be used to model water flow and bed load sediment transport in rivers and coastal regions. They are a hyperbolic inhomogeneous system of conservation laws consisting of the standard shallow water equations and a bed-updating equation. Any numerical scheme that attempts to model these equations must satisfy the C-property. This ensures that the scheme settles to the correct steady state in the absence of flow. Work by Hudson [34] tested numerical schemes in a finite difference setting demonstrating the need for C-property satisfaction. However a model of a physical problem also usually contains a complicated domain that naturally suggests a finite element scheme should be employed. The Runge-Kutta Discontinuous Galerkin method is a finite element scheme, studied by Cockburn et al. [17, 16, 15, 13, 18], that combines a discontinuous spatial discretisation, finite volume style numerical fluxes, TVD Runge-Kutta time stepping and a TVD slope limiter to obtain a high order solution that retains ideal properties and explicitness. Schwanenberg et al. [68] showed, through comparisons with

finite-volume scheme for modeling shallow water flows over an