Higher order numerical schemes for SPDEs with additive Noise

TL;DR

This paper develops high-order numerical schemes for linear stochastic heat and wave equations with additive noise, surpassing standard Euler schemes' convergence rates by employing modified Crank-Nicolson methods and specialized quadrature rules.

Contribution

The paper introduces novel high-order schemes for SPDEs with additive noise, achieving convergence rates of 3/2 for heat and 2 for wave equations, improving upon traditional methods.

Findings

Achieved strong convergence rate of 3/2 for stochastic heat equation.

Achieved strong convergence rate of 2 for stochastic wave equation.

Demonstrated effectiveness of modified schemes with numerical quadrature.

Abstract

We present high-order numerical schemes for linear stochastic heat and wave equations with Dirichlet boundary conditions, driven by additive noise. Standard Euler schemes for SPDEs are limited to an order convergence between 1/2 and 1 due to the low temporal regularity of noise. For the stochastic heat equation, a modified Crank-Nicolson scheme with proper numerical quadrature rule for the noise term in its reformulation as random PDE achieves a strong convergence rate of 3/2. For the stochastic wave equation with additive noise a corresponding approach leads to a scheme which is of order 2.