Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations
Journal of Computational Physics
Geometric integrators for the nonlinear Schrödinger equation
Journal of Computational Physics
Multisymplectic box schemes and the Korteweg-de Vries equation
Applied Numerical Mathematics - Special issue: Workshop on innovative time integrators for PDEs
On the preservation of phase space structure under multisymplectic discretization
Journal of Computational Physics
On Symplectic and Multisymplectic Schemes for the KdV Equation
Journal of Scientific Computing
Linear PDEs and Numerical Methods That Preserve a Multisymplectic Conservation Law
SIAM Journal on Scientific Computing
Dispersion, group velocity, and multisymplectic discretizations
Mathematics and Computers in Simulation
Symplectic wavelet collocation method for Hamiltonian wave equations
Journal of Computational Physics
Splitting multisymplectic integrators for Maxwell's equations
Journal of Computational Physics
Symplectic and multisymplectic numerical methods for Maxwell's equations
Journal of Computational Physics
Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations
Journal of Computational Physics
Journal of Computational Physics
Geometric numerical schemes for the KdV equation
Computational Mathematics and Mathematical Physics
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Multisymplectic (MS) integrators, i.e. numerical schemes which exactly preserve a discrete space-time symplectic structure, are a new class of structure preserving algorithms for solving Hamiltonian PDEs. In this paper we examine the dispersive properties of MS integrators for the linear wave and sine-Gordon equations. In particular a leapfrog in space and time scheme (a member of the Lobatto Runge-Kutta family of methods) and the Preissman box scheme are considered. We find the numerical dispersion relations are monotonic and that the sign of the group velocity is preserved. The group velocity dispersion (GVD) is found to provide significant information and succinctly explain the qualitative differences in the numerical solutions obtained with the different schemes. Further, the numerical dispersion relations for the linearized sine-Gordon equation provides information on the ability of the MS integrators to capture the sine-Gordon dynamics. We are able to link the numerical dispersion relations to the total energy of the various methods, thus providing information on the coarse grid behavior of MS integrators in the nonlinear regime.