Unconditionally Stable Space-Time Finite Element Method for the Shallow Water Equations

We introduce the automatic variationally stable finite element (AVS-FE) method [1, 3] for the shallow water equations (SWE). The AVS-FE method uses a first order system integral formulation of the under- lying partial differential equations (PDEs) and, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2], employs the concept of optimal test functions to ensure discrete stability. The AVS-FE method distinguishes itself by using globally conforming FE trial spaces, e.g., H1(Ω) and H(div,Ω) and their broken counterparts for the test spaces. The broken topology of the test spaces allows us to compute numerical approximations of the local restrictions of the optimal test functions in a completely decoupled fashion, i.e., element-by-element. The test functions can be com- puted with sufficient numerical accuracy by using the same local p-level as applied for the trial space. The unconditional discrete stability of the method allows for straightforward implementation of transient problems in existing FE solvers such as FEniCS. Furthermore, the AVS-FE method comes with a built-in a posteriori error estimate as well as element-wise error indicators allowing us to perform mesh adaptive refinements in both space and time. The application of this method to complex physical domains requires large FE meshes leading to signif- icant computational costs. However, since the computation of optimal test functions as well as element- wise error indicators are all local, the method is an excellent candidate for parallel processing. We show numerical verifications for the SWE utilizing the built-in error indicators to drive mesh adaptive refine- ments

A study of the potential effects of deepening the Corpus Christi Ship Channel on hurricane storm surge

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Cross-mode Stabilized Stochastic Shallow Water Systems Using Stochastic Finite Element Methods

The development of surrogate models to study uncertainties in hydrologic systems requires significant effort in the development of sampling strategies and forward model simulations. Furthermore, in applications where prediction time is critical, such as prediction of hurricane storm surge, the predictions of system response and uncertainties can be required within short time frames. Here, we develop an efficient stochastic shallow water model to address these issues. To discretize the physical and probability spaces we use a Stochastic Galerkin method and a Incremental Pressure Correction scheme to advance the solution in time. To overcome discrete stability issues, we propose cross-mode stabilization methods which employs existing stabilization methods in the probability space by adding stabilization terms to every stochastic mode in a modes-coupled way. We extensively verify the developed method for both idealized shallow water test cases and hindcasting of past hurricanes. We subsequently use the developed and verified method to perform a comprehensive statistical analysis of the established shallow water surrogate models. Finally, we propose a predictor for hurricane storm surge under uncertain wind drag coefficients and demonstrate its effectivity for Hurricanes Ike and Harvey