Accuracy and efficiency of time integration methods for 1D diffusive wave equation.

*(English)*Zbl 1392.76093Summary: The diffusive wave approximation of the Saint-Venant equations is commonly used in hydrological models to describe surface flow processes. Numerous numerical approaches can be used to solve this highly nonlinear equation. Nonlinear time integration schemes – also called methods of lines (MOL) – were proven very efficient to solve other nonlinear problems in geosciences but were never considered to deal with surface flow modeling with the diffusive wave equation. In this paper, we study the relative performance of different time and space integration schemes by comparing the results obtained with classical approaches and with nonlinear time integration approaches. The results show that (i) the integration method with a higher order in space shows high accuracy regarding an integrated indicator such as the global mass balance error but is less accurate regarding local indicators, and (ii) nonlinear time integration techniques perform better than classical ones. Overall, it seems that integration
techniques combining nonlinear time integration and a low spatial order need to be considered when developing hydrological modeling tools owing to their simplicity of implementation and very good performance.

##### MSC:

76S05 | Flows in porous media; filtration; seepage |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

76Rxx | Diffusion and convection |

35Q86 | PDEs in connection with geophysics |

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\textit{S. Weill} et al., Comput. Geosci. 18, No. 5, 697--709 (2014; Zbl 1392.76093)

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