Backward Error Analysis for Numerical Integrators
SIAM Journal on Numerical Analysis
Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations
Journal of Computational Physics
Multi-symplectic integration methods for Hamiltonian PDEs
Future Generation Computer Systems - Special issue: Geometric numerical algorithms
Multisymplectic box schemes and the Korteweg-de Vries equation
Applied Numerical Mathematics - Special issue: Workshop on innovative time integrators for PDEs
Linear PDEs and Numerical Methods That Preserve a Multisymplectic Conservation Law
SIAM Journal on Scientific Computing
Conformal conservation laws and geometric integration for damped Hamiltonian PDEs
Journal of Computational Physics
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Conformal symplecticity is generalized to forced-damped multi-symplectic PDEs in 1+1 dimensions. Since a conformal multi-symplectic property has a concise form for these equations, numerical algorithms that preserve this property, from a modified equations point of view, are available. In effect, the modified equations for standard multi-symplectic methods and for space-time splitting methods satisfy a conformal multi-symplectic property, and the splitting schemes exactly preserve global symplecticity in a special case. It is also shown that the splitting schemes yield incorrect rates of energy/momentum dissipation, but this is not the case for standard multi-symplectic schemes. These methods work best for problems where the dissipation coefficients are small, and a forced-damped semi-linear wave equation is considered as an example.