Convergence of a method based on the exponential integrator and Fourier spectral discretization for stiff stochastic PDEs

Abstract

In this article, we propose and analyze some efficient numerical methods for the solution of a class of stiff stochastic partial differential equations (SPDEs) with additive noise. First, we apply the polynomial chaos (PC) expansion to treat the randomness, which results to a set of deterministic stiff PDEs. Then this system of PDEs is discretized in space by the Fourier pseudo‐spectral method, enabling us to use the FFT algorithm to reduce computational cost efficiently. To overcome the instability in discretizing time component of resulting system of ordinary differential equations (ODEs), we apply the exponential time differencing (ETD), integrating factor Runge‐Kutta (IFRK) and modified exponential time differencing Runge‐Kutta (ETDRKB) methods. We also analyze the strong error bound of numerical approximation in a framework of Lipschitz nonlinearities for the method based on the ETDRKB scheme. Some numerical experiments are included to demonstrate the performance and convergence of methods. Numerical results show that the method based on the ETDRKB scheme is more efficient for stiff SPDEs.

A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

“…In the past few years, there has been an emerging trend for solving forward and inverse problems using machine learning strategies due to their capability to handle various types of model problems in many disciplines, especially in partial differential equations (PDEs) [15][16][17]. Because of the successful outcome of neural networks in a broad range of problems such as image processing [18], predicting disease [19,20], and finance [21,22], some mathematicians tried to use these techniques to solve challenging problems [23][24][25]. Among many neural network-based methods in the literature, the two state-of-the-art researches in solving PDE methods are the Gaussian processes regression (GPR) for PDEs [16] and the physics-informed neural networks (PINNs) [17].…”