Uniform-in-time weak error estimates of explicit full-discretization schemes for SPDEs with non-globally Lipschitz coefficients

This article is devoted to long-time weak approximations of stochastic partial differential equations (SPDEs) evolving in a bounded domain D Rᵈ, d 3, with non-globally Lipschitz and possibly non-contractive coefficients. Both the space-time white noise (d=1) and the trace-class noise in multiple dimensions d=2, 3 are examined for the considered SPDEs. Based on a spectral Galerkin spatial semi-discretization, we propose a class of novel full-discretization schemes of exponential type, which are explicit, easily implementable and preserve the ergodicity of the original dissipative SPDEs with possibly non-contractive coefficients. The uniform-in-time weak approximation errors are carefully analyzed in a low regularity and non-contractive setting, with uniform-in-time weak convergence rates obtained. A key ingredient is to establish the uniform-in-time moment bounds (in L^4q-2-norm, q 1) for the proposed fully discrete schemes in a super-linear setting. This is highly non-trivial for the explicit full-discretization schemes and new arguments are elaborated by fully exploiting a contractive property of the semi-group in L^4q-2, the dissipativity of the nonlinearity and the particular benefit of the taming strategy. Numerical experiments are finally reported to verify the theoretical findings.