Ethan J. Kubatko

A Basin- to Channel-Scale Unstructured Grid Hurricane Storm Surge Model Applied to Southern Louisiana

Abstract Southern Louisiana is characterized by low-lying topography and an extensive network of sounds, bays, marshes, lakes, rivers, and inlets that permit widespread inundation during hurricanes. A basin- to channel-scale implementation of the Advanced Circulation (ADCIRC) unstructured grid hydrodynamic model has been developed that accurately simulates hurricane storm surge, tides, and river flow in this complex region. This is accomplished by defining a domain and computational resolution appropriate for the relevant processes, specifying realistic boundary conditions, and implementing accurate, robust, and highly parallel unstructured grid numerical algorithms. The model domain incorporates the western North Atlantic, the Gulf of Mexico, and the Caribbean Sea so that interactions between basins and the shelf are explicitly modeled and the boundary condition specification of tidal and hurricane processes can be readily defined at the deep water open boundary. The unstructured grid enables highly refi...

A Performance Comparison of Continuous and Discontinuous Finite Element Shallow Water Models

We present a comparative study of two finite element shallow water equation (SWE) models: a generalized wave continuity equation based continuous Galerkin (CG) model—an approach used by several existing SWE models—and a recently developed discontinuous Galerkin (DG) model. While DG methods are known to possess a number of favorable properties, such as local mass conservation, one commonly cited disadvantage is the larger number of degrees of freedom associated with the methods, which naturally translates into a greater computational cost compared to CG methods. However, in a series of numerical tests, we demonstrate that the DG SWE model is generally more efficient than the CG model (i) in terms of achieving a specified error level for a given computational cost and (ii) on large-scale parallel machines because of the inherently local structure of the method. Both models are verified on a series of analytic test cases and validated on a field-scale application.

Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge-Kutta time discretizations

This paper investigates the use of a special class of strong-stability-preserving (SSP) Runge–Kutta time discretization methods in conjunction with discontinuous Galerkin (DG) finite element spatial discretizatons. The class of SSP methods investigated here is defined by the property that the number of stages s is greater than the order k of the method. From analysis, CFL conditions for the linear (L 2 ) stability of the methods defined using the s > k SSP schemes are obtained that are less restrictive than those of the ‘‘standard’’ so-called RKDG methods that use s = k SSP Runge–Kutta schemes. The improvement in the CFL conditions for linear stability of the methods more than offsets the additional work introduced by the increased number of stages. Given that the CFL conditions for linear stability are what must be respected in practice in order to maintain high-order accuracy, the use of the s > k SSP schemes results in RKDG methods that are more efficient than those previously defined. Furthermore, with the application of a slope limiter, the nonlinear stability properties of the forward Euler method and the DG spatial discretization, which have been previously proven, are preserved with these methods under less restrictive CFL conditions than those required for linear stability. Thus, more efficient RKDG methods that possess the same favorable accuracy and stability properties of the ‘‘standard’’ RKDG methods are obtained. Numerical results verify the CFL conditions for stability obtained from analysis and demonstrate the efficiency advantages of these new RKDG methods.

Adaptive hierarchic transformations for dynamically p-enriched slope-limiting over discontinuous Galerkin systems of generalized equations

We study a family of generalized slope limiters in two dimensions for Runge-Kutta discontinuous Galerkin (RKDG) solutions of advection-diffusion systems. We analyze the numerical behavior of these limiters applied to a pair of model problems, comparing the error of the approximate solutions, and discuss each limiters advantages and disadvantages. We then introduce a series of coupled p-enrichment schemes that may be used as standalone dynamic p-enrichment strategies, or may be augmented via any in the family of variable-in-p slope limiters presented.

Optimal Strong-Stability-Preserving Runge---Kutta Time Discretizations for Discontinuous Galerkin Methods

Discontinuous Galerkin (DG) spatial discretizations are often used in a method-of-lines approach with explicit strong-stability-preserving (SSP) Runge–Kutta (RK) time steppers for the numerical solution of hyperbolic conservation laws. The time steps that are employed in this type of approach must satisfy Courant–Friedrichs–Lewy stability constraints that are dependent on both the region of absolute stability and the SSP coefficient of the RK method. While existing SSPRK methods have been optimized with respect to the latter, it is in fact the former that gives rise to stricter constraints on the time step in the case of RKDG stability. Therefore, in this work, we present the development of new “DG-optimized” SSPRK methods with stability regions that have been specifically designed to maximize the stable time step size for RKDG methods of a given order in one space dimension. These new methods represent the best available RKDG methods in terms of computational efficiency, with significant improvements over methods using existing SSPRK time steppers that have been optimized with respect to SSP coefficients. Second-, third-, and fourth-order methods with up to eight stages are presented, and their stability properties are verified through application to numerical test cases.

ADMESH: An advanced, automatic unstructured mesh generator for shallow water models

In this paper, we present the development and application of a two-dimensional, automatic unstructured mesh generator for shallow water models called Admesh. Starting with only target minimum and maximum element sizes and points defining the boundary and bathymetry/ topography of the domain, the goal of the mesh generator is to automatically produce a high-quality mesh from this minimal set of input. From the geometry provided, properties such as local features, curvature of the boundary, bathymetric/topographic gradients, and approximate flow characteristics can be extracted, which are then used to determine local element sizes. The result is a high-quality mesh, with the correct amount of refinement where it is needed to resolve all the geometry and flow characteristics of the domain. Techniques incorporated include the use of the so-called signed distance function, which is used to determine critical geometric properties, the approximation of piecewise linear coastline data by smooth cubic splines, a so-called mesh function used to determine element sizes and control the size ratio of neighboring elements, and a spring-based force equilibrium approach used to improve the element quality of an initial mesh obtained from a simple Delaunay triangulation. Several meshes of shallow water domains created by the new mesh generator are presented.

A Comparison of Artificial Viscosity, Limiters, and Filters, for High Order Discontinuous Galerkin Solutions in Nonlinear Settings

Nonlinear systems of equations demonstrate complicated regularity features that are often obfuscated by overly diffuse numerical methods. Using a discontinuous Galerkin finite element method, we study a nonlinear system of advection–diffusion–reaction equations and aspects of its regularity. For numerical regularization, we present a family of solutions consisting of: (1) a sharp, computationally efficient slope limiter, known as the BDS limiter, (2) a standard spectral filter, and (3) a novel artificial diffusion algorithm with a solution-dependent entropy sensor. We analyze these three numerical regularization methods on a classical test in order to test the strengths and weaknesses of each, and then benchmark the methods against a large application model.

Dynamic p-enrichment schemes for multicomponent reactive flows

We present a family of p-enrichment schemes. These schemes may be separated into two basic classes: the first, called fixed tolerance schemes, rely on setting global scalar tolerances on the local regularity of the solution, and the second, called dioristic schemes, rely on time-evolving bounds on the local variation in the solution. Each class of p-enrichment scheme is further divided into two basic types. The first type (the Type I schemes) enrich along lines of maximal variation, striving to enhance stable solutions in ‘‘areas of highest interest.’’ The second type (the Type II schemes) enrich along lines of maximal regularity in order to maximize the stability of the enrichment process. Each of these schemes are tested on three model systems. The first is an academic exact system where basic analysis is easily performed. Then we discuss a pair of application model problems arising in coastal hydrology. The first being a contaminant transport model, which addresses a declinature problem for a contaminant plume with respect to a bay inlet setting. And the second, a multicomponent chemically reactive flow model of estuary eutrophication arising in the Gulf of Mexico.

A Framework for Running the ADCIRC Discontinuous Galerkin Storm Surge Model on a GPU

Abstract Hybrid architectures utilizing GPUs provide a unique opportunity in a high performance computing environment. However, there are many legacy codes, particularly written in Fortran, that can not take immediate advantage of GPUs. Furthermore, many of these codes are under active development and so completely rewriting the code may not be an option. The advanced circulation and storm surge finite element model (ADCIRC) is one such code base. In this paper we present our semi-automatic methodology for porting portions of ADCIRC to run on the GPU and some preliminary scaling results of these subroutines. We have implemented a C++ array class and pre-processor macros to create a type of application framework to simplify the conversion and maintenance tasks. This allows the C++ syntax to be similar to Fortran, to provide for a more straight forward syntactical conversion from the original Fortran to C++ and simplified calling conventions between the two. After the necessary subroutines are converted to the C++ framework, the CUDA library can be easily used and also we are able to provide a simplified abstraction layer for accessing basic GPU functionality. For example, the problem of transferring the correct data on/o_ the GPU is addressed by our framework by a one time code change and a script to resolve data dependencies. Although it is currently specific to ADCIRC, our framework provides a starting point for utilizing GPUs with legacy Fortran codes, from which more specific GPU optimizations can be implemented.

hp discontinuous Galerkin methods for the vertical extent of the water column in coastal settings part I

In this article, we present novel, high-order, discontinuous Galerkin (DG) methods for the vertical extent of the water column in coastal settings. We examine the shallow water equations (SWE) in the context of DG spatial discretizations coupled with explicit Runge-Kutta (RK) time stepping. All the primary variables, including the free surface elevation, are discretized using discontinuous polynomial spaces of arbitrary order. The difficulty of mismatched lateral boundary faces that accompanies the use of a discontinuous free surface is overcome through the use of a so-called sigma-coordinate system in the vertical, which transforms the bottom boundary and free surface into coordinate surfaces. We develop high-order methods for the SWE that exhibit optimal orders of convergence for all the primary variables via two distinct paths: the first involves the use of a convolution kernel made up of B-splines to filter out errors in the DG discretization of the surface elevation and the corresponding pressure flux. The second involves a method that evaluates the discrete depth-integrated velocity exactly, eliminating the need to solve the depth-integrated momentum equation altogether. The result is a simple and efficient high-order scheme that can be extended to the full three-dimensional SWE.