Jennifer Proft

Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations

We consider the approximation of a simplified model of the depth-averaged two-dimensional shallow water equations by two approaches. In both approaches, a discontinuous Galerkin (DG) method is used to approximate the continuity equation. In the first approach, a continuous Galerkin method is used for the momentum equations. In the second approach a particular DG method, the nonsymmetric interior penalty Galerkin method, is used to approximate momentum. A priori error estimates are derived and numerical results are presented for both approaches.

A posteriori discontinuous Galerkin error estimates for transient convection–diffusion equations

A posteriori error estimates are derived for unsteady convection-diusion equations discretized with the non-symmetric interior penalty and the local discontinuous Galerkin methods. First, an error representation formula in a user specied output functional is derived using duality techniques. Then, an L 2 (L 2) a posteriori estimate consisting of elementwise residual-based error indicators is obtained by eliminating the dual solution. Numerical experiments are performed to assess the convergence rate of the various error indicators on a model problem.

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations

We develop a high-order positivity-preserving discontinuous Galerkin (DG) scheme for linear Vlasov-Boltzmann transport equations (Vlasov-BTE) under the action of quadratically confined electrostatic potentials. The solutions of such BTEs are positive probability distribution functions and it is very challenging to have a mass-conservative, high-order accurate scheme that preserves positivity of the numerical solutions in high dimensions. Our work extends the maximum-principle-satisfying scheme for scalar conservation laws in a recent work by X. Zhang and C.-W. Shu to include the linear Boltzmann collision term. The DG schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy. A discussion of the standard semi-discrete DG schemes for the BTE are included as a foundation for the stability and error estimates for this new scheme. Numerical results of the relaxation models are provided to validate the method.

Coupling of continuous and discontinuous Galerkin methods for transport problems

In this paper, we formulate a coupled discontinuous/continuous Galerkin method for the numerical solution of convection–diffusion (transport) equations, where convection may be dominant. One motivation for this approach is to use a discontinuous method where the solution is rough, e.g., in regions of high gradients, and use a continuous method where the solution is smooth. In this approach, the domain is decomposed into two regions, and appropriate transmission conditions are applied at the interface between regions. In one region, a local discontinuous Galerkin method is applied, and in the other region a standard continuous Galerkin method is employed. Stability and a priori error estimates for the coupled method are derived, and numerical results in one space dimension are given for smooth problems and problems with sharp fronts. 2002 Elsevier Science B.V. All rights reserved.

A three-dimensional discontinuous Galerkin model applied to the baroclinic simulation of Corpus Christi Bay

In this work, we present results of a numerical study of Corpus Christi Bay, Texas and surrounding regions and compare simulated model results to recorded data. The validation data for the year 2000 include the water elevation, velocity, and salinity at selected locations. The baroclinic computations were performed using the University of Texas Bays and Estuaries 3D (UTBEST3D) simulator based on a discontinuous Galerkin finite element method for unstructured prismatic meshes. We also detail some recent advances in the modeling capabilities of UTBEST3D, such as a novel turbulence scheme and the support for local vertical discretization on parts of the computational domain. All runs were conducted on parallel clusters; an evaluation of parallel performance of UTBEST3D is included.

Fidelity of the integrated kinetic energy factor as an indicator of storm surge impacts

After the devastating hurricane season of 2005, shortcomings with the Saffir–Simpson Hurricane Scale’s (SSHS) ability to characterize a tropical cyclone’s potential to generate storm surge became widely apparent. As a result, several alternative surge indices were proposed to replace the SSHS, including Powell and Reinhold’s integrated kinetic energy (IKE) factor, Kantha’s Hurricane Surge Index (HSI), and Irish and Resio’s Surge Scale (SS). Of the previous, the IKE factor is the only surge index to date that truly captures a tropical cyclone’s integrated intensity, size, and wind field distribution. However, since the IKE factor was proposed in 2007, an accurate assessment of this surge index has not been performed. This study provides the first quantitative evaluation of the IKEs ability to serve as a predictor of a tropical cyclone’s potential surge impacts as compared to other alternative surge indices. Using the tightly coupled Advanced Circulation and Simulating Waves Nearshore models, the surge and wave responses of Hurricane Ike in 2008 and 78 synthetic tropical cyclones were evaluated against the SSHS, IKE, HSI, and SS. Results along the northwestern Gulf of Mexico coastline demonstrate that the HSI performs best in capturing the peak surge response of a tropical cyclone, while the IKE accounting for winds greater than tropical storm intensity (IKETS) provides the most accurate estimate of a tropical cyclone’s regional surge impacts. These results demonstrate that the appropriate selection of a surge index ultimately depends on what information is of interest to be conveyed to the public and/or scientific community.

Shallow water modeling using discontinuous and coupled finite element methods

Publisher Summary This chapter discusses two approaches for shallow water flow modeling based on discontinuous and continuous approximating spaces. In the first approach, it discretises the primitive continuity equation using a discontinuous Galerkin (DG) method, coupled to a continuous finite element approximation of the momentum equations. This approach is useful when local conservation is important and it uses discontinuous approximations for the hyperbolic continuity equation while allowing the momentum equation to be approximated through more traditional and continuous functions. In the second approach, it discretises both equations using DG methods. In both approaches, a DG method is used to approximate the continuity equation. This DG approach has several appealing features, particularly the ability to incorporate upwinding and stability postprocessing into the solution to model highly advective flows, the ability to use different polynomial orders of approximation in different parts of the domain (and for different variables, if so desired), and the ability to easily use nonconforming meshes. The shallow water equations (SWE) model flow in the domains whose characteristic wavelength in the horizontal is much larger than the water depth. The SWEs consist of a first-order hyperbolic continuity equation for the water elevation, coupled to momentum equations for the horizontal depth-averaged velocities.