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We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weak convergence rate of the exponential Euler scheme when the linear operator is self adjoint and also provide the full weak convergence rate for non-self-adjoint linear operator with additive noise. Key part of the proof does not rely on Malliavin calculus. For non-self-adjoint operators, we analyse the optimal strong error for spatially semi-discrete approximations for both multiplicative and additive noise with truncated and non-truncated noise. Depending on the regularity of the noise and the initial solution, we found that in some cases the rate of weak convergence is twice the rate of the strong convergence. Our convergence rate is in agreement with some numerical results in two dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we give conditions for stochastic Runge–Kutta (SRK) methods to preserve quadratic invariants. It is shown that SRK methods preserving quadratic invariants are symplectic. Based on both convergence order conditions and quadratic invariant-preserving conditions, we construct some SRK schemes preserving quadratic invariants with strong and weak convergence order with the help of computer algebra, respectively. Numerical experiments are executed to verify our theoretical analysis and show the superiority of these schemes. [ABSTRACT FROM AUTHOR]