Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
A deterministic approximation of diffusion equations using particles
SIAM Journal on Scientific and Statistical Computing
A finite difference scheme for the K(2, 2) compacton equation
Journal of Computational Physics
Journal of Computational Physics
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
A numerical study of compactons
Mathematics and Computers in Simulation
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
Particle methods for dispersive equations
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
A Local Discontinuous Galerkin Method for KdV Type Equations
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
A Particle Method for the KdV Equation
Journal of Scientific Computing
Padé numerical method for the Rosenau-Hyman compacton equation
Mathematics and Computers in Simulation
Self-similar radiation from numerical Rosenau-Hyman compactons
Journal of Computational Physics
Local discontinuous Galerkin methods for the Cahn-Hilliard type equations
Journal of Computational Physics
Radiation in numerical compactons from finite element methods
MATH'05 Proceedings of the 8th WSEAS International Conference on Applied Mathematics
ADER finite volume schemes for nonlinear reaction--diffusion equations
Applied Numerical Mathematics
Journal of Scientific Computing
Journal of Computational and Applied Mathematics
Journal of Scientific Computing
Local discontinuous Galerkin methods for the generalized Zakharov system
Journal of Computational Physics
A local discontinuous Galerkin method for directly solving Hamilton-Jacobi equations
Journal of Computational Physics
An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations
Journal of Scientific Computing
Finite volume schemes for dispersive wave propagation and runup
Journal of Computational Physics
The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs
Journal of Computational and Applied Mathematics
Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems
Journal of Scientific Computing
Journal of Scientific Computing
Analysis for one-dimensional time-fractional Tricomi-type equations by LDG methods
Numerical Algorithms
Geometric numerical schemes for the KdV equation
Computational Mathematics and Mathematical Physics
Hi-index | 31.48 |
We develop local discontinuous Galerkin (DG) methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions, the so-called "compactons". The schemes we present extend the previous works of Yan and Shu on approximating solutions for linear dispersive equations and for certain KdV-type equations. We present two classes of DG methods for approximating solutions of such PDEs. First, we generate nonlinearly stable numerical schemes with a stability condition that is induced from a conservation law of the PDE. An alternative approach is based on constructing linearly stable schemes, i.e., schemes that are linearly stable to small perturbations. The numerical simulations we present verify the desired properties of the methods including their expected order of accuracy. In particular, we demonstrate the potential advantages of using DG methods over pseudospectral methods in situations where discontinuous fronts and rapid oscillations co-exist in a solution.