Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations

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.