A conservative finite volume scheme with time-accurate local time stepping for scalar transport on unstructured grids

Abstract

In this article we propose a new conservative high resolution TVD (total variation diminishing) finite volume scheme with time-accurate local time stepping (LTS) on unstructured grids for the solution of scalar transport problems, which are typical in the context of water quality simulations. To keep the presentation of the new method as simple as possible, the algorithm is only derived in two space dimensions and for purely convective transport problems, hence neglecting diffusion and reaction terms. The new numerical method for the solution of the scalar transport is directly coupled to the hydrodynamic model of Casulli and Walters (2000) that provides the dynamics of the free surface and the velocity vector field based on a semi-implicit discretization of the shallow water equations. Wetting and drying is handled rigorously by the nonlinear algorithm proposed by Casulli (2009). The new time-accurate LTS algorithm allows a different time step size for each element of the unstructured grid, based on an element-local Courant-Friedrichs-Lewy (CFL) stability condition. The proposed method does not need any synchronization between different time steps of different elements and is by construction locally and globally conservative. The LTS scheme is based on a piecewise linear polynomial reconstruction in space–time using the MUSCL–Hancock method, to obtain second order of accuracy in both space and time.The new algorithm is first validated on some classical test cases for pure advection problems, for which exact solutions are known. In all cases we obtain a very good level of accuracy, showing also numerical convergence results; we furthermore confirm mass conservation up to machine precision and observe an improved computational efficiency compared to a standard second order TVD scheme for scalar transport with global time stepping (GTS). Then, the new LTS method is applied to some more complex problems, where the new scalar transport scheme has also been coupled to a semi-implicit model for the simulation of the free surface hydrodynamics, including nonlinear wetting and drying. The last application shown in this paper is carried out on a real geometry, for which we have taken available DTM data of the lake Guaiba in Brazil. Comparisons have been made in all cases with a second order TVD scheme based on GTS. For the new LTS algorithm we report a significant reduction in computational effort, with a savings of CPU time of the order of up to 95%.