The semi-implicit Euler scheme is the default time discretization for stochastic PDEs. For equations with globally Lipschitz nonlinearities, it works fine. But many physically important SPDEs — the stochastic Allen-Cahn equation, stochastic advection-diffusion — have non-globally Lipschitz nonlinearities. The cubic term in Allen-Cahn grows without bound.
This paper proves that the classic semi-implicit Euler scheme can blow up in finite time for Allen-Cahn type SPDEs. The standard numerical method for these equations is provably unsafe. The blow-up is not a rare event or a numerical artifact — it's a structural consequence of applying a scheme designed for bounded nonlinearities to an unbounded one.
The fix is taming: modified schemes that prevent the nonlinearity from driving the solution to infinity. The paper analyzes a fully implicit scheme and two tamed schemes, proving non-asymptotic uniform-in-time error bounds for all three. “Uniform in time” means the error bound doesn't grow with the simulation horizon — the scheme remains faithful to both transient dynamics and long-time behavior. The general criterion they develop applies beyond Allen-Cahn: any SPDE where the nonlinearity can overwhelm the discretization needs taming, and the paper provides the framework to verify when a tamed scheme succeeds.