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35
hpDISCONTINUOUS GALERKIN METHODS FOR THE HELMHOLTZ EQUATION WITH LARGE WAVE NUMBER
, 2009
"... This paper develops some interior penalty hpdiscontinuous Galerkin (hpDG) methods for the Helmholtz equation in two and three dimensions. The proposed hpDG methods are defined using a sesquilinear form which is not only meshdependent but also degreedependent. In addition, the sesquilinear form ..."
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Cited by 20 (3 self)
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This paper develops some interior penalty hpdiscontinuous Galerkin (hpDG) methods for the Helmholtz equation in two and three dimensions. The proposed hpDG methods are defined using a sesquilinear form which is not only meshdependent but also degreedependent. In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order p. Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts. It is proved that the proposed hpdiscontinuous Galerkin methods are absolutely stable (hence, wellposed). For each fixed wave number k, suboptimal order error estimates in the broken H 1norm and the L 2norm are derived without any mesh constraint. The error estimates and the stability estimates are improved to optimal order under the mesh condition k 3 h 2 p −1 ≤ C0 by utilizing these stability and error estimates and using a stabilityerror iterative procedure To overcome the difficulty caused by strong indefiniteness of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [19, 20, 33], which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size h, the polynomial degree p, the wave number k, as well as all the penalty parameters for the numerical solutions.
Discontinuous galerkin methods for modeling hurricane storm surge
 Advances in Water Resources
, 2011
"... This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. ..."
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Cited by 18 (6 self)
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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
New Interior Penalty Discontinuous Galerkin Methods for the KellerSegel Chemotaxis Model
 SIAM J. NUMER. ANAL
, 2008
"... We develop a family of new interior penalty discontinuous Galerkin methods for the KellerSegel chemotaxis model. This model is described by a system of two nonlinear PDEs: a convectiondiffusion equation for the cell density coupled with a reactiondiffusion equation for the chemoattractant concent ..."
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Cited by 17 (3 self)
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We develop a family of new interior penalty discontinuous Galerkin methods for the KellerSegel chemotaxis model. This model is described by a system of two nonlinear PDEs: a convectiondiffusion equation for the cell density coupled with a reactiondiffusion equation for the chemoattractant concentration. It has been recently shown that the convective part of this system is of a mixed hyperbolicelliptic type, which may cause severe instabilities when the studied system is solved by straightforward numerical methods. Therefore, the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original KellerSegel model in the form of a convectiondiffusionreaction system with a hyperbolic convective part. We then design interior penalty discontinuous Galerkin methods for the rewritten KellerSegel system. Our methods employ the centralupwind numerical fluxes, originally developed in the context of finitevolume methods for hyperbolic systems of conservation laws. In this paper, we consider Cartesian grids and prove error estimates for the proposed highorder discontinuous Galerkin methods. Our proof is valid for preblowup times since we assume boundedness of the exact solution. We also show that the blowup time of the exact solution is bounded from above by the blowup time of our numerical solution. In the numerical tests presented below, we demonstrate that the obtained numerical solutions have no negative values and are oscillationfree, even though no slope limiting technique has been implemented.
A performance comparison of nodal discontinuous Galerkin methods on triangles and quadrilaterals
"... This work presents a study on the performance of nodal bases on triangles and on quadrilaterals for discontinuous Galerkin solutions of hyperbolic conservation laws. A nodal basis on triangles and two tensor product nodal bases on quadrilaterals are considered. The quadrilateral element bases are co ..."
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Cited by 7 (1 self)
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This work presents a study on the performance of nodal bases on triangles and on quadrilaterals for discontinuous Galerkin solutions of hyperbolic conservation laws. A nodal basis on triangles and two tensor product nodal bases on quadrilaterals are considered. The quadrilateral element bases are constructed from the Lagrange interpolating polynomials associated with the Legendre–Gauss–Lobatto points and from those associated with the classical Legendre–Gauss points. Settings of interest concern the situation in which a mesh of triangular elements is obtained by dividing each quadrilateral element into two triangular elements or vice versa, the mesh of quadrilateral elements is obtained by merging two adjacent triangular elements. To assess performance, we use a linear advecting rotating plume transport problem as a test case. For cases where the order of the basis is low to moderate, the computing time used to reach a given final time for the quadrilateral elements is shorter than that for the triangular elements. The numerical results also show that the quadrilateral elements yield higher computational efficiency in terms of cost to
Highorder semiimplicit timeintegrators for a triangular discontinuous Galerkin oceanic shallow water model
 Internat. J. Numer. Methods Fluids
"... SUMMARY We extend the explicit in time highorder triangular discontinuous Galerkin (DG) method to semiimplicit (SI) and then apply the algorithm to the twodimensional oceanic shallow water equations; we implement highorder SI timeintegrators using the backward difference formulas from orders o ..."
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SUMMARY We extend the explicit in time highorder triangular discontinuous Galerkin (DG) method to semiimplicit (SI) and then apply the algorithm to the twodimensional oceanic shallow water equations; we implement highorder SI timeintegrators using the backward difference formulas from orders one to six. The reason for changing the timeintegration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for highorder DG methods. Changing the timeintegration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite elementtype area integrals, but also the finite volumetype boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI timeintegrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: highorder accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of timeexplicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new highorder SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new highorder SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use highorder (HO) timeintegrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with loworder time discretizations. Published in
Discontinuous Galerkin Method for 1D Shallow Water Flow with Water Surface Slope Limiter
"... Abstract—A water surface slope limiting scheme is tested and compared with the water depth slope limiter for the solution of one dimensional shallow water equations with bottom slope source term. Numerical schemes based on the total variation diminishing RungeKutta discontinuous Galerkin finite ele ..."
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Cited by 5 (0 self)
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Abstract—A water surface slope limiting scheme is tested and compared with the water depth slope limiter for the solution of one dimensional shallow water equations with bottom slope source term. Numerical schemes based on the total variation diminishing RungeKutta discontinuous Galerkin finite element method with slope limiter schemes based on water surface slope and water depth are used to solve onedimensional shallow water equations. For each slope limiter, three different Riemann solvers based on HLL, LF, and Roe flux functions are used. The proposed water surface based slope limiter scheme is easy to implement and shows better conservation property compared to the slope limiter based on water depth. Of the three flux functions, the Roe approximation provides the best results while the LF function proves to be least suitable when used with either slope limiter scheme. Keywords—Discontinuous finite element, TVD RungeKutta scheme, slope limiters, Riemann solvers, shallow water flow. O I.
Pressure forcing and dispersion analysis for discontinuous Galerkin approximations to oceanic fluid flows,
 J. Comput. Phys.
, 2013
"... Abstract This paper is part of an effort to examine the application of discontinuous Galerkin (DG) methods to the numerical modeling of the general circulation of the ocean. One step performed here is to develop an integral weak formulation of the lateral pressure forcing that is suitable for usage ..."
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Abstract This paper is part of an effort to examine the application of discontinuous Galerkin (DG) methods to the numerical modeling of the general circulation of the ocean. One step performed here is to develop an integral weak formulation of the lateral pressure forcing that is suitable for usage with a DG method and with a generalized vertical coordinate that includes level, terrainfitted, isopycnic, and hybrid coordinates as examples. This formulation is then tested, in special cases, with analyses of dispersion relations and numerical stability and with some computational experiments. These results suggest that the advantages of DG methods may significantly outweigh their disadvantages, in the settings tested here. This paper also outlines some other issues that need to be addressed in future work.
model
, 2010
"... Abstract This paper evaluates the parallel performance and scalability of an unstructured grid Shallow Water Equation (SWE) hurricane storm surge model. We use the ADCIRC model, which is based on the generalized wave continuity equation continuous Galerkin method, within a parallel computational fra ..."
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Abstract This paper evaluates the parallel performance and scalability of an unstructured grid Shallow Water Equation (SWE) hurricane storm surge model. We use the ADCIRC model, which is based on the generalized wave continuity equation continuous Galerkin method, within a parallel computational framework based on domain decomposition and the MPI (Message Passing Interface) library. We measure the performance of the model run implicitly and explicitly on various grids. We analyze the performance as well as accuracy with various spatial and temporal discretizations. We improve the output writing performance by introducing sets of dedicated writer cores. Performance is measured on the Texas Advanced Computing Center Ranger machine. A high resolution 9,314,706 finite element node grid with 1 s time steps can complete a day of real time hurricane storm surge simulation in less than 20 min of computer wall clock time, using 16,384 cores with sets of dedicated writer cores.
GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations
"... We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of ts ..."
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We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multirate AdamsBashforth scheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. We compare the performance of the kernels expressed in a portable threading language OCCA, when cross compiled with OpenCL, CUDA, and OpenMP at runtime.
Accepted by:
, 2012
"... ii Numerical models for one and twodimensional shallow water flows are developed using discontinuous Galerkin method. Formulation and characteristics of shallow water equations are discussed. The wellbalanced property and wetting/drying treatment are provided in the numerical models. The shockca ..."
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ii Numerical models for one and twodimensional shallow water flows are developed using discontinuous Galerkin method. Formulation and characteristics of shallow water equations are discussed. The wellbalanced property and wetting/drying treatment are provided in the numerical models. The shockcapturing property is achieved by the approximate Riemann solvers in the schemes. Effects of different approximate Riemann solvers are also investigated. The Total Variation Diminishing property is achieved by adoption of slope limiters. Different slope limiters and their effects are compared through numerical tests. Numerical tests are performed to validate the models. These tests include dambreak flows, hydraulic jump and shocks in channels, and flows in natural rivers. Results show that the numerical models developed in present work are robust, accurate, and efficient for modeling shallow water flows. The onedimensional model shows that the area based slope limiter provided the best solution in natural channels. The slope limiter based on the water depth or water