Operator splitting scheme based on barycentric Lagrange interpolation collocation method for the Allen-Cahn equation

2021 ◽  
Author(s):  
Yangfang Deng ◽  
Zhifeng Weng
Keyword(s):  
Collocation Method ◽  
Lagrange Interpolation ◽  
Operator Splitting ◽  
Splitting Scheme ◽  
Cahn Equation

scholarly journals Operator splitting around Euler-Maruyama scheme and high order discretization of heat kernels

2020 ◽  
Author(s):  
Toshihiro Yamada ◽  
Yuga Iguchi
Keyword(s):  
Commutative Algebra ◽  
Computational Algorithm ◽  
Diffusion Processes ◽  
Operator Splitting ◽  
Heat Kernels ◽  
Heat Equations ◽  
Discretization Method ◽  
Splitting Scheme ◽  
Order Operator ◽  
Numerical Examples

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker-Campbell-Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler-Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

scholarly journals Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

2018 ◽  
Vol 39 (4) ◽  
pp. 2096-2134 ◽  
Author(s):  
Charles-Edouard Bréhier ◽  
Jianbo Cui ◽  
Jialin Hong
Keyword(s):  
Convergence Rate ◽  
Strong Convergence ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Splitting Scheme ◽  
Cahn Equation ◽  
Strong Convergence Rate

Abstract This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space–time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.

Sign in / Sign up

Export Citation Format

Share Document