Computing rough solutions of the stochastic nonlinear wave equation
Jiachuan Cao, Buyang Li, Katharina Schratz
TL;DR
This work develops a novel low-regularity exponential integrator for the stochastic nonlinear wave equation with multiplicative Itô noise, enabling robust high-order convergence even when initial data lie in $H^{\gamma}\times H^{\gamma-1}$ for $\gamma$ near zero. By employing a filtered, frequency-localized scheme and a first-order exponential integrator based on the linear wave propagator, the authors prove convergence rates $O(\tau^{2\gamma-})$ in 1D/2D ($\gamma\in(0,1/2]$) and $O(\tau^{\max(\gamma,2\gamma-1/2-)})$ in 3D ($\gamma\in(1/2,3/4]$), surpassing existing methods under the same regularity. The paper also furnishes extensive 1D and 2D numerical experiments comparing against SEM and STM, showing improved accuracy and efficiency, particularly for rough initial data or discontinuities. The approach promises practical impact for simulating stochastic wave phenomena with challenging initial conditions and lays groundwork for higher-order methods in broader SPDE contexts.
Abstract
The regularity of solutions to the stochastic nonlinear wave equation plays a critical role in the accuracy and efficiency of numerical algorithms. Rough or discontinuous initial conditions pose significant challenges, often leading to a loss of accuracy and reduced computational efficiency in existing methods. In this study, we address these challenges by developing a novel and efficient numerical algorithm specifically designed for computing rough solutions of the stochastic nonlinear wave equation, while significantly relaxing the regularity requirements on the initial data. By leveraging the intrinsic structure of the stochastic nonlinear wave equation and employing advanced tools from harmonic analysis, we construct a time discretization method that achieves robust convergence for initial values \((u^{0}, v^{0}) \in H^γ \times H^{γ-1}\) for all \(γ> 0\). Notably, our method attains an improved error rate of \(O(τ^{2γ-})\) in one and two dimensions for \(γ\in (0, \frac{1}{2}]\), and \(O(τ^{\max(γ, 2γ- \frac{1}{2}-)})\) in three dimensions for \(γ\in (0, \frac{3}{4}]\), where \(τ\) denotes the time step size. These convergence rates surpass those of existing numerical methods under the same regularity conditions, underscoring the advantage of our approach. To validate the performance of our method, we present extensive numerical experiments that demonstrate its superior accuracy and computational efficiency compared to state-of-the-art methods. These results highlight the potential of our approach to enable accurate and efficient simulations of stochastic wave phenomena even in the presence of challenging initial conditions.