Non-asymptotic uniform in time error bounds for new and old numerical schemes for SPDEs

Abstract

We study numerical schemes for Stochastic Partial Differential Equations (SPDEs). We introduce a general method of proof of non-asymptotic uniform in time error bounds on numerical integrators for SPDEs, ensuring the schemes capture both the transient and the long term dynamics faithfully. We then consider SPDEs with non-globally Lipshitz nonlinearities, which include for example the stochastic Allen-Cahn equation and some stochastic advection-diffusion equations. For the case of Allen-Cahn type SPDEs we show that the classic semi-implicit Euler time-discretization can exhibit finite time blow up. This motivates analysing other schemes which do not suffer from this blow-up problem. We consider three numerical schemes for SPDEs with non globally Lipshitz nonlinearity: a fully implicit scheme and two tamed schemes. For these schemes we prove non-asymptotic uniform in time error bounds by leveraging our general criterion, and provide numerical comparisons. While the main emphasis in this paper is on the properties of the time-discretization, the schemes we consider are full space-time discretization of the SPDE.

Figures (11)

• Figure 1: Time series of FEM solutions for the first moment with initial condition $u_0(x) = 0$. All schemes appear quite comparable. Shaded regions reflect one standard deviation.
• Figure 2: Time series of FEM solutions for the first moment for initial condition $u_0(x) = 1$. All schemes appear quite comparable. Shaded regions reflect one standard deviation.
• Figure 3: Time series of FEM solutions for the first moment for initial condition $u_0(x) = 100$. For all computed values of $\tau$, SIE experiences blowup. Otherwise, the schemes appear quite comparable. Shaded regions reflect one standard deviation.
• Figure 4: Time series of FEM solutions for the first moment for initial condition $u_0(x) = 10\sin(10\pi x)$. All methods appear quite comparable. Shaded regions reflect one standard deviation.
• Figure 5: Time series of spectral Galerkin solutions for the first moment for initial condition $u_0(x) = 0$. All methods are quite comparable. Shaded regions reflect one standard deviation.

Theorems & Definitions (28)

• Theorem 2.1
• proof : Proof of Theorem \ref{['thm: UiT theorem in our case']}
• Theorem 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Corollary 3.4
• proof
• Proposition 4.7: Proposition 3.3 in brehier2022approximation and Proposition 6.2.2 in Cerrai01
• Lemma 4.8: Proposition 3.3 in brehier2022approximation