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510
Multigrid Solution for HighOrder Discontinuous Galerkin . . .
, 2004
"... A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. Thi ..."
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Cited by 45 (16 self)
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A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. This limited coupling maximizes the effectiveness of the pmultigrid solver, which relies on an elementline Jacobi smoother. The elementline Jacobi smoother solves implicitly on lines of elements formed based on the coupling between elements in a p = 0 discretization of the scalar transport equation. Fourier analysis of 2D scalar convectiondiffusion shows that the elementline Jacobi smoother as well as the simpler element Jacobi smoother are stable independent of p and flow condition. Mesh refinement studies for simple problems with analytic solutions demonstrate that the discretization achieves optimal order of accuracy of O(h p+1). A subsonic, airfoil test case shows that the multigrid convergence rate is independent of p but weakly dependent on h. Finally, higherorder is shown to outperform grid refinement in terms of the time required to reach a desired accuracy level.
Adaptive mesh methods for one and twodimensional hyperbolic conservation laws
 SIAM J. Numer. Anal
"... Abstract. We develop efficient moving mesh algorithms for one and twodimensional hyperbolic systems of conservation laws. The algorithms are formed by two independent parts: PDE evolution and meshredistribution. The first part can be any appropriate highresolution scheme, and the second part is ..."
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Cited by 44 (8 self)
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Abstract. We develop efficient moving mesh algorithms for one and twodimensional hyperbolic systems of conservation laws. The algorithms are formed by two independent parts: PDE evolution and meshredistribution. The first part can be any appropriate highresolution scheme, and the second part is based on an iterative procedure. In each iteration, meshes are first redistributed by an equidistribution principle, and then on the resulting new grids the underlying numerical solutions are updated by a conservativeinterpolation formula proposed in this work. The iteration for the meshredistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state. The main idea of the proposed method is to keep the massconservation of the underlying numerical solution at each redistribution step. In one dimension, we can show that the underlying numerical approximation obtained in the meshredistribution part satisfies the desired TVD property, which guarantees that the numerical solution at any time level is TVD, provided that the PDE solver in the first part satisfies such a property. Several test problems in one and two dimensions are computed using the proposed moving mesh algorithm. The computations demonstrate that our methods are efficient for solving problems with shock discontinuities, obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach.
Nodal highorder discontinuous Galerkin methods for the spherical shallow water equations.
 J. Comput. Phys.,
, 2002
"... We present a highorder discontinuous Galerkin method for the solution of the shallow water equations on the sphere. To overcome wellknown problems with polar singularities, we consider the shallow water equations in Cartesian coordinates, augmented with a Lagrange multiplier to ensure that fluid ..."
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Cited by 40 (5 self)
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We present a highorder discontinuous Galerkin method for the solution of the shallow water equations on the sphere. To overcome wellknown problems with polar singularities, we consider the shallow water equations in Cartesian coordinates, augmented with a Lagrange multiplier to ensure that fluid particles are constrained to the spherical surface. The global solutions are represented by a collection of curvilinear quadrilaterals from an icosahedral grid. On each of these elements the local solutions are assumed to be well approximated by a highorder nodal Lagrange polynomial, constructed from a tensorproduct of the LegendreGaussLobatto points, which also supplies a highorder quadrature. The shallow water equations are satisfied in a local discontinuous element fashion with solution continuity being enforced weakly. The numerical experiments, involving a comparison of weak and strong conservation forms and the impact of overintegration and filtering, confirm the expected highorder accuracy and the potential for using such highly parallel formulations in numerical weather prediction. c 2002 Elsevier Science (USA)
A DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR TIME DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS WITH HIGHER ORDER DERIVATIVES
"... Abstract. In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applie ..."
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Cited by 39 (10 self)
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Abstract. In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal (k + 1)th order of accuracy when using piecewise kth degree polynomials, under the condition that k + 1 is greater than or equal to the order of the equation. 1.
Optimal a priori error estimates for the hpversion of the local discontinuous Galerkin method for convectiondiffusion problems
 Math. Comp
"... Abstract. We study the convergence properties of the hpversion of the local discontinuous Galerkin finite element method for convectiondiffusion problems; we consider a model problem in a onedimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper b ..."
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Cited by 35 (7 self)
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Abstract. We study the convergence properties of the hpversion of the local discontinuous Galerkin finite element method for convectiondiffusion problems; we consider a model problem in a onedimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the meshwidth h, in the polynomial degree p, and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both h and p. The theoretical results are confirmed in a series of numerical examples. 1.
A Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems,” submitted
 SIAM J. for Numerical Analaysis
, 2006
"... Abstract. We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is the ..."
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Cited by 34 (14 self)
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Abstract. We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is then applied to the resulting system of equations. The numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level, thus resulting in a global system that involves only the original unknown variable. The proposed method is closely related to the local discontinuous Galerkin (LDG) method [B. Cockburn and C.W. Shu, SIAM J. Numer. Anal., 35 (1998), pp. 2440–2463], but, unlike the LDG method, the sparsity pattern of the CDG method involves only nearest neighbors. Also, unlike the LDG method, the CDG method works without stabilization for an arbitrary orientation of the element interfaces. The computation of the numerical interface fluxes for the CDG method is slightly more involved than for the LDG method, but this additional complication is clearly offset by increased compactness and flexibility.
A triangular cutcell adaptive method for highorder discretizations of the compressible Navier–Stokes equations
, 2007
"... ..."
ADER schemes for threedimensional nonlinear hyperbolic systems
 J. Comput. Phys
, 2005
"... In this paper we carry out the extension of the ADER approach to multidimensional nonlinear systems of conservation laws. We implement nonlinear schemes of up to fourth order of accuracy in both time and space. Numerical results for the compressible Euler equations illustrate the very high order o ..."
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Cited by 30 (6 self)
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In this paper we carry out the extension of the ADER approach to multidimensional nonlinear systems of conservation laws. We implement nonlinear schemes of up to fourth order of accuracy in both time and space. Numerical results for the compressible Euler equations illustrate the very high order of accuracy and nonoscillatory properties of the new schemes. Compared to the stateofart finitevolume WENO schemes the ADER schemes are faster, more accurate and need less computer memory. Key words: highorder schemes, weighted essentially nonoscillatory, ADER, generalized Riemann problem, three space dimensions. 1 1
A posteriori error estimates for general numerical methods for scalar conservation laws
 Mat.Aplic.Comp.14 (1995), 37–45. CMP 95:15
"... Abstract. A new upper bound is provided for the L ∞norm of the difference between the viscosity solution of a model steady state HamiltonJacobi equation, u, and any given approximation, v. This upper bound is independent of the method used to compute the approximation v; it depends solely on the v ..."
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Cited by 29 (4 self)
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Abstract. A new upper bound is provided for the L ∞norm of the difference between the viscosity solution of a model steady state HamiltonJacobi equation, u, and any given approximation, v. This upper bound is independent of the method used to compute the approximation v; it depends solely on the values that the residual takes on a subset of the domain which can be easily computed in terms of v. Numerical experiments investigating the sharpness of the a posteriori error estimate are given. 1.
Enhanced accuracy by postprocessing for finite element methods for hyperbolic equations
 MATH. COMPUT
, 2003
"... We consider the enhancement of accuracy, by means of a simple postprocessing technique, for finite element approximations to transient hyperbolic equations. The postprocessing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if th ..."
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Cited by 28 (4 self)
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We consider the enhancement of accuracy, by means of a simple postprocessing technique, for finite element approximations to transient hyperbolic equations. The postprocessing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whoseedgesareofsizeoftheorderof∆x only. For example, when polynomials of degree k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k +1/2 inthe L 2norm, whereas the postprocessed approximation is of order 2k +1;ifthe exact solution is in L 2 only, in which case no order of convergence is available for the DG method, the postprocessed approximation converges with order k +1/2 inL 2 (Ω0), where Ω0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.