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47
Optimal discontinuous Galerkin methods for wave propagation
 SIAM J. Numer. Anal
"... Abstract. We have developed and analyzed a new class of discontinuous Galerkin methods (DG) which can be seen as a compromise between standard DG and the finite element (FE) method in the way that it is explicit like standard DG and energy conserving like FE. In the literature there are many methods ..."
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Cited by 18 (7 self)
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Abstract. We have developed and analyzed a new class of discontinuous Galerkin methods (DG) which can be seen as a compromise between standard DG and the finite element (FE) method in the way that it is explicit like standard DG and energy conserving like FE. In the literature there are many methods that achieve some of the goals of explicit time marching, unstructured grid, energy conservation, and optimal higher order accuracy, but as far as we know only our new algorithms satisfy all the conditions. We propose a new stability requirement for our DG. The stability analysis is based on the careful selection of the two FE spaces which verify the new stability condition. The convergence rate is optimal with respect to the order of the polynomials in the FE spaces. Moreover, the convergence is described by a series of numerical experiments.
The VOLNA code for the numerical modeling of tsunami waves: Generation, propagation and inundation
 Eur. J. Mech. B/Fluids
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A Central Discontinuous Galerkin Method for HamiltonJacobi Equations
 J SCI COMPUT
, 2010
"... In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of HamiltonJacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin m ..."
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Cited by 15 (1 self)
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In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of HamiltonJacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since HamiltonJacobi equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to directly solve such equations. By recognizing and following a “weightedresidual” or “stabilizationbased” formulation of central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method for HamiltonJacobi equations. The L2 stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance of the method in approximating the viscosity solutions of general HamiltonJacobi equations are demonstrated through extensive numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.
Discontinuous Galerkin timedomain solution of Maxwell’s equations on locallyrefined nonconforming cartesian grids
 COMPEL
"... methods, nonconforming grid The use of the prominent FDTD method for the time domain solution of electromagnetic wave propagation past devices with small geometrical details can require very ne grids and can lead to very important computational time and storage. Our purpose is to develop a numeric ..."
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Cited by 9 (0 self)
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methods, nonconforming grid The use of the prominent FDTD method for the time domain solution of electromagnetic wave propagation past devices with small geometrical details can require very ne grids and can lead to very important computational time and storage. Our purpose is to develop a numerical method able to handle possibly nonconforming locally rened grids, based on portions of Cartesian grids in order to use existing preand postprocessing tools. We present a Discontinuous Galerkin method to solve the threedimensional timedomain Maxwell's equations on conforming or nonconforming orthogonal grids. The method is based on a centered mean approximation for surface integrals and a secondorder leapfrog scheme for advancing in time. The dispersion error is partially analyzed in the cubic uniform mesh case. We show that the divergence of elds are weakly preserved on conforming grids. In the most general case, we prove that the resulting scheme is stable and that it conserves a discrete analog of the electromagnetic energy. For nonconforming grids, the local sets of basis functions are enriched at subgrid interfaces in order to get rid of possible spurious wave re
ections. Numerical simulation on realistic congurations are promising: the numerical method proposed here makes it possible to handle devices with extremely small
A Discontinuous Galerkin Method for Ideal Two Fluid Plasma Equations
"... Abstract. A discontinuous Galerkin method for the ideal 5 moment twofluid plasma system is presented. The method uses a second or third order discontinuous Galerkin spatial discretization and a third order TVD RungeKutta time stepping scheme. The method is benchmarked against an analytic solution ..."
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Cited by 6 (1 self)
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Abstract. A discontinuous Galerkin method for the ideal 5 moment twofluid plasma system is presented. The method uses a second or third order discontinuous Galerkin spatial discretization and a third order TVD RungeKutta time stepping scheme. The method is benchmarked against an analytic solution of a dispersive electron acoustic square pulse as well as the twofluid electromagnetic shock [1] and existing numerical solutions to the GEM challenge magnetic reconnection problem [2]. The algorithm can be generalized to arbitrary geometries and three dimensions. An approach to maintaining small gauge errors based on error propagation is suggested.
Highorder central ENO finitevolume scheme for ideal MHD on . . .
, 2012
"... A highorder central essentially nonoscillatory (CENO) finitevolume scheme is developed for the compressible ideal magnetohydrodynamics (MHD) equations solved on threedimensional (3D) cubedsphere grids. The proposed formulation is an extension to 3D geometries of a recent highorder MHD CENO sc ..."
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Cited by 5 (3 self)
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A highorder central essentially nonoscillatory (CENO) finitevolume scheme is developed for the compressible ideal magnetohydrodynamics (MHD) equations solved on threedimensional (3D) cubedsphere grids. The proposed formulation is an extension to 3D geometries of a recent highorder MHD CENO scheme developed on twodimensional (2D) grids. The main technical challenge in extending the 2D method to 3D cubedsphere grids is to properly handle the nonplanar cell faces that arise in cubedsphere grids. This difficulty is solved by considering general hexahedral cells with trilinear faces, which allow us to compute fluxes, areas and volumes with highorder accuracy by transforming to a reference cubic cell. The 3D CENO scheme is implemented within a flexible multiblock cubedsphere grid framework to fourthorder accuracy, resulting in a highorder solution method for cubedsphere grids with unique capabilities in terms of adaptive refinement and parallel scalability. The highorder method is applicable to the solution of general hyperbolic conservation laws with, in principle, arbitrary order. The CENO scheme is based on a hybrid solution reconstruction procedure that provides highorder accuracy in smooth regions, even for smooth extrema, and nonoscillatory transitions at discontinuities. The scheme is applied herein to MHD in combination with a GLM divergence correction technique to control
Moving mesh discontinuous Galerkin method for hyperbolic conservation laws
 J. Sci. Comput
"... In this paper, a moving mesh discontinuous Galerkin (DG) method is developed to solve the nonlinear conservation laws. In the mesh adaptation part, two issues have received much attention. One is about the construction of the monitor function which is used to guide the mesh redistribution. In this ..."
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Cited by 4 (0 self)
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In this paper, a moving mesh discontinuous Galerkin (DG) method is developed to solve the nonlinear conservation laws. In the mesh adaptation part, two issues have received much attention. One is about the construction of the monitor function which is used to guide the mesh redistribution. In this study, a heuristic posteriori error estimator is used in constructing the monitor function. The second issue is concerned with the solution interpolation which is used to interpolates the numerical solution from the old mesh to the updated mesh. This is done by using a scheme that mimics the DG method for linear conservation laws. Appropriate limiters are used on seriously distorted meshes generated by the moving mesh approach to suppress the numerical oscillations. Numerical results are provided to show the efficiency of the proposed moving mesh DG method. KEY WORDS: Moving mesh method; discontinuous Galerkin method; nonlinear conservation laws; monitor function.
Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an arti cial compressibility ux
 Int. J. Numer. Methods Fluids
, 2007
"... Stokes problem based on an artificial compressibility flux ..."
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Cited by 3 (2 self)
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Stokes problem based on an artificial compressibility flux
Convergence of locally divergencefree discontinuousGalerkin methods for the induction equations of the MHD system ∗
, 2004
"... We present the convergence analysis of locally divergencefree discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition ∆t ∼ h 4/3, we obtain error estimates in ..."
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Cited by 3 (0 self)
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We present the convergence analysis of locally divergencefree discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition ∆t ∼ h 4/3, we obtain error estimates in L 2 of order O(∆t 2 + h m+1/2) where m is the degree of the local polynomials. 1