The stochastic scalar auxiliary variable approach for stochastic nonlinear Klein–Gordon equation

In this paper, we propose and analyze semi-implicit numerical schemes for the stochastic nonlinear Klein–Gordon equation (SNKGE) with multiplicative noise. These numerical schemes, called stochastic scalar auxiliary variable (SAV) schemes, are constructed by transforming the considered SNKGE into a higher dimensional stochastic system with a stochastic SAV. We prove that they can be solved explicitly, and preserve the modified energy evolution law and the regularity structure of the original system. These structure-preserving properties are the keys to overcoming the mutual effect of noise and nonlinearity. By providing new regularity estimates of the introduced SAV, we obtain the strong convergence rate of stochastic SAV schemes under Lipschitz conditions. Furthermore, based on the modified energy evolution laws, we derive the exponential moment bounds and sharp strong convergence rate of the proposed schemes for SNKGE with a non-globally Lipschitz nonlinearity in the additive noise case. To the best of our knowledge, this is the first result on the construction and strong convergence of semi-implicit schemes preserving averaged energy evolution law for SNKGEs.