| We study the application of high-order semi-discrete central-upwind schemes to multidimensional systems of balance laws. These schemes are Godunov-type methods -- a piecewise polynomial interpolant, reconstructed from the cell averages computed at time $t$, is evolved to the next time level, $t+\Delta t$, using a particularly simple spatial discretization combined with a stable ODE solver of an appropriate order. The numerical fluxes are obtained by including the Riemann fans into the control volumes of a (varying) size, determined by one-sided local speeds of propagation, and by passing to the limit as $\Delta t\to 0$.
The main advantages of central schemes is their simplicity, since no (approximate) Riemann problem solver, characteristic decomposition, operator or dimensional splitting is required. The main challenge in application of central-upwind schemes to balance laws is how to preserve the balance between the numerical fluxes and the source terms.
In the case of Saint-Venant system of non-homogeneous shallow water equations, the goal is achieved by using an appropriate quadrature for the source average. At the same time, a special treatment is required to accurately capture ``dry'' states, where the water height is (almost) zero.
The application of the central-upwind schemes to balance laws with a stiff source term will be also discussed.
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