  In this paper, we propose a stochastic conformal multi-symplectic method for
a class of damped stochastic Hamiltonian partial differential equations in
order to inherit the intrinsic properties, and apply the numerical method to
solve a kind of damped stochastic nonlinear Schrodinger equation with
multiplicative noise. It is shown that the stochastic conformal
multi-symplectic method preserves the discrete stochastic conformal
multi-symplectic conservation law, the discrete charge exponential dissipation
law almost surely, and we also deduce the recurrence relation of the discrete
global energy. Numerical experiments are preformed to verify the good
performance of the proposed stochastic conformal multi-symplectic method,
compared with a Crank-Nicolson type method. Finally, we present the mean square
convergence result of the proposed numerical method in temporal direction
numerically.
