A finite element method for fully nonlinear water waves
Journal of Computational Physics
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
Performance of Discontinuous Galerkin Methods for Elliptic PDEs
SIAM Journal on Scientific Computing
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems
SIAM Journal on Numerical Analysis
The $\cal P^k+1-\cal S^k$ Local Discontinuous Galerkin Method for Elliptic Equations
SIAM Journal on Numerical Analysis
Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics)
Journal of Scientific Computing
Journal of Scientific Computing
Space-time discontinuous Galerkin method for nonlinear water waves
Journal of Computational Physics
hpGEM---A software framework for discontinuous Galerkin finite element methods
ACM Transactions on Mathematical Software (TOMS)
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In this paper, we discuss a discontinuous Galerkin finite (DG) element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the L2-norm is in both cases of optimal order and proportional to O(驴t2+hp+1), without the need for a separate velocity reconstruction, with p the polynomial order, h the mesh size and 驴t the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.