Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
A first course in the numerical analysis of differential equations
A first course in the numerical analysis of differential equations
Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations
Journal of Computational Physics
Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations
SIAM Journal on Numerical Analysis
Multisymplectic box schemes and the Korteweg-de Vries equation
Applied Numerical Mathematics - Special issue: Workshop on innovative time integrators for PDEs
Numerical Solution of Partial Differential Equations: An Introduction
Numerical Solution of Partial Differential Equations: An Introduction
Linear PDEs and Numerical Methods That Preserve a Multisymplectic Conservation Law
SIAM Journal on Scientific Computing
Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations
Journal of Computational Physics
On multi-symplectic partitioned Runge-Kutta methods for Hamiltonian wave equations
Applied Mathematics and Computation
Dispersive properties of multisymplectic integrators
Journal of Computational Physics
Trigonometrically-fitted ARKN methods for perturbed oscillators
Applied Numerical Mathematics
On Multisymplecticity of Partitioned Runge-Kutta Methods
SIAM Journal on Scientific Computing
New modified Runge-Kutta-Nyström methods for the numerical integration of the Schrödinger equation
Computers & Mathematics with Applications
Efficient energy-preserving integrators for oscillatory Hamiltonian systems
Journal of Computational Physics
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In this paper, we study the integration of Hamiltonian wave equations whose solutions have oscillatory behaviors in time and/or space. We are mainly concerned with the research for multi-symplectic extended Runge-Kutta-Nystrom (ERKN) discretizations and the corresponding discrete conservation laws. We first show that the discretizations to the Hamiltonian wave equations using two symplectic ERKN methods in space and time respectively lead to an explicit multi-symplectic integrator (Eleap-frogI). Then we derive another multi-symplectic discretization using a symplectic ERKN method in time and a symplectic partitioned Runge-Kutta method, which is equivalent to the well-known Stormer-Verlet method in space (Eleap-frogII). These two new multi-symplectic schemes are extensions of the leap-frog method. The numerical stability and dispersive properties of the new schemes are analyzed. Numerical experiments with comparisons are presented, where the two new explicit multi-symplectic methods and the leap-frog method are applied to the linear wave equation and the Sine-Gordon equation. The numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation.