Abstract
We consider the finite element (FE) approximation of the two dimensional shallow water equations (SWE) by considering discretizations in which both space and time are established using a stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to establish stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow us to employ mesh adaptive strategies and perform space-time mesh refinements, i.e., local time stepping. We establish a priori error estimates for the AVS-FE method and linearized SWE and perform numerical verifications to confirm corresponding asymptotic convergence behavior. In an effort to keep the computational cost low, we consider an alternative space-time approach in which the space-time domain is partitioned into finite sized space-time slices. Hence, we can perform adaptive mesh refinements on each individual slice to preset error tolerances as needed for a particular application. Numerical verifications comparing the two alternatives indicate the space-time slices are superior for simulations over long times, whereas the solutions are indistinguishable for short times. Multiple numerical verifications show the adaptive mesh refinement capabilities of the AVS-FE method, as well the application of the method to some commonly applied benchmarks for the SWE.
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The datasets and computer codes generated in the current study are available from the corresponding author on reasonable request.
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This work has been supported by the United States National Science Foundation - NSF PREEVENTS Track 2 Program, under NSF Grant Number 1855047.
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This work has been supported by the United States National Science Foundation - NSF PREEVENTS Track 2 Program, under NSF Grant Number 1855047.
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Valseth, E., Dawson, C. A stable space-time FE method for the shallow water equations. Comput Geosci 26, 53–70 (2022). https://doi.org/10.1007/s10596-021-10108-4
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DOI: https://doi.org/10.1007/s10596-021-10108-4
Keywords
- Shallow water equations
- Discontinuous Petrov-Galerkin
- Adaptivity
- Space-time FE method
- Local time stepping