Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
A first course in the numerical analysis of differential equations
A first course in the numerical analysis of differential equations
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations
Journal of Computational Physics
Square-Conservative Schemes for a Class of Evolution Equations Using Lie-Group Methods
SIAM Journal on Numerical Analysis
On Symplectic and Multisymplectic Schemes for the KdV Equation
Journal of Scientific Computing
Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation
Journal of Computational Physics
Dispersive properties of multisymplectic integrators
Journal of Computational Physics
On the multisymplecticity of partitioned Runge-Kutta and splitting methods
International Journal of Computer Mathematics - Splitting Methods for Differential Equations
Explicit multi-symplectic methods for Klein-Gordon-Schrödinger equations
Journal of Computational Physics
Multi-symplectic integration of coupled non-linear Schrodinger system with soliton solutions
International Journal of Computer Mathematics
Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations
Mathematics and Computers in Simulation
Dispersion, group velocity, and multisymplectic discretizations
Mathematics and Computers in Simulation
A multisymplectic explicit scheme for the modified regularized long-wave equation
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
High-order compact splitting multisymplectic method for the coupled nonlinear Schrödinger equations
Computers & Mathematics with Applications
Symplectic and multisymplectic numerical methods for Maxwell's equations
Journal of Computational Physics
Numerical wave propagation on non-uniform one-dimensional staggered grids
Journal of Computational Physics
The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs
Journal of Computational and Applied Mathematics
Linear Stability of Partitioned Runge-Kutta Methods
SIAM Journal on Numerical Analysis
Numerical integration of the Ostrovsky equation based on its geometric structures
Journal of Computational Physics
Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations
Journal of Computational Physics
Conformal conservation laws and geometric integration for damped Hamiltonian PDEs
Journal of Computational Physics
Journal of Computational Physics
Local structure-preserving algorithms for the "good" Boussinesq equation
Journal of Computational Physics
Geometric numerical schemes for the KdV equation
Computational Mathematics and Mathematical Physics
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We develop and compare some geometric integrators for the Korteweg--de Vries equation, especially with regard to their robustness for large steps in space and time, Δx and Δt, and over long times. A standard, semi-explicit, symplectic finite difference scheme is found to be fast and robust. However, in some parameter regimes such schemes are susceptible to developing small wiggles. At the same instances the fully implicit and multisymplectic Preissmann scheme, written as a 12-point box scheme, stays smooth. This is accounted for by the ability of the box scheme to preserve the shape of the dispersion relation of any hyperbolic system for all Δx and Δt. We also develop a simplified 8-point version of this box scheme which maintains its advantageous features.