Positivity Preserving Implicit and Partially Implicit Time Integration Methods in the Context of the DG Scheme Applied to Shallow Water Flows

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This contribution is concerned with the development of unconditionally positive implicit time integration schemes in the context of shallow water flows discretized by the DG scheme. For explicit time integration—which is mostly applied in combination with wetting and drying shallow water flows—both linear stability and positivity preservation require very small time steps. Also for implicit Runge-Kutta schemes, positivity preservation generally leads to additional time step restrictions. In this work, we discuss two possible extensions to implicit time integration schemes that guarantee non-negativity of the water height for any time step size while still preserving the conservativity of the space discretization.