An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
Mathematics of Computation
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Journal of Computational Physics
Journal of Computational Physics
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Proceedings of the third ARO workshop on Adaptive methods for partial differential equations
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The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
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Journal of Scientific Computing
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SIAM Journal on Numerical Analysis
Enhanced accuracy by post-processing for finite element methods for hyperbolic equations
Mathematics of Computation
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Journal of Computational Physics
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Journal of Scientific Computing
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Journal of Computational Physics
Discontinuous Galerkin spectral/hpelement modelling of dispersive shallow water systems
Journal of Scientific Computing
A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect
Journal of Computational Physics
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Journal of Computational Physics
Discontinuous Hamiltonian Finite Element Method for Linear Hyperbolic Systems
Journal of Scientific Computing
Polymorphic nodal elements and their application in discontinuous Galerkin methods
Journal of Computational Physics
Journal of Scientific Computing
Journal of Scientific Computing
Local discontinuous Galerkin methods for the generalized Zakharov system
Journal of Computational Physics
The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs
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On the Negative-Order Norm Accuracy of a Local-Structure-Preserving LDG Method
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SIAM Journal on Numerical Analysis
Computers & Mathematics with Applications
Journal of Computational Physics
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In this paper we review the existing and develop new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L2 stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.