Stochastic conformal schemes for damped stochastic Klein-Gordon equation with additive noise

In this article, stochastic conformal schemes of the damped stochastic Klein-Gordon equation with additive noise are studied. It is shown that this equation possesses the stochastic conformal multi-symplectic conservation law. Under appropriate boundary conditions, the global momentum evolution law and the global energy evolution law are proposed. We chiefly develop the stochastic conformal Preissman scheme, the stochastic conformal discrete gradient scheme and the stochastic conformal Euler box scheme to preserve the geometric structures of the original system. Specifically, we make theoretical discussions on the three proposed schemes to obtain corresponding discrete conservation laws or discrete evolution laws. Then the damped stochastic linear Klein-Gordon equation and the damped stochastic nonlinear Klein-Gordon equation with cubic nonlinearity are taken as examples to demonstrate the validity of the proposed schemes. Through numerical experiments and comparisons, the superiorities of the proposed schemes are fully shown, which are consistent with our theoretical analysis. Moreover, the mean square convergence orders of the three stochastic conformal schemes in time direction and space direction are tested numerically.