Numerical Ergodicity and Optimal Strong Error Estimates for a Class of Novel Tamed Schemes to Superlinear SPDEs

We construct a class of novel tamed schemes for superlinear stochastic partial differential equations (SPDEs), including the stochastic Allen–Cahn equation driven by either multiplicative or additive noise. The schemes preserve the same Lyapunov structure as the original system, and we rigorously establish their longtime unconditional stability. Furthermore, we prove that the corresponding Galerkin-based fully discrete tamed schemes inherit the unique ergodicity of the underlying SPDEs and achieve optimal strong convergence rates in both the multiplicative- and additive-noise cases.