Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities

Abstract

The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing nonlinearities. It remained an open question whether such a divergence phenomenon also holds in the case of stochastic partial differential equations with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. In this work we solve this problem by proving that full-discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations. This article also contains a short literature overview on existing numerical approximation results for stochastic differential equations with superlinearly growing nonlinearities.

65 pages

References in corpus (18)

• Strong Approximation of Stochastic Allen-Cahn Equation with White Noise
• Analysis of Some Splitting Schemes for the Stochastic Allen-Cahn Equation
• Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
• On strong $L^2$ convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations
• Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations
• Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients
• Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations
• Adaptive time-stepping for Stochastic Partial Differential Equations with non-Lipschitz drift
• Strong convergence rates of semi-discrete splitting approximations for stochastic Allen--Cahn equation
• Multilevel Monte Carlo Method for Ergodic SDEs without Contractivity
• A note on strong convergence of implicit scheme for SDEs under local one-sided Lipschitz conditions
• Convergence rates of truncated EM scheme for NSDDEs
• Weak convergence of Euler-Maruyama's approximation for SDEs under integrability condition
• A note on convergence and stability of the truncated Milstein method for stochastic differential equations
• Milstein-type Schemes of SDE Driven by Lévy Noise with Super-linear Diffusion Coefficients
• A New Efficient Explicit Scheme of Order $1.5$ for SDE with Super-linear Drift Coefficient
• Strong convergence for explicit space-time discrete numerical approximation for 2D stochastic Navier-Stokes equations
• Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations

Cited by in corpus (6)

• Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise
• $L^p$-Convergence Rate of Backward Euler Schemes for Monotone SDEs
• A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations
• Convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation
• Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus
• Temporal semi-discretizations of a backward semilinear stochastic evolution equation