Computing rough solutions of the stochastic nonlinear wave equation

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 harmonic analysis tools (particularly low/high-frequency decomposition) to enable convergence analysis under low regularity, we construct a time discretization method that achieves robust convergence for initial values ( u 0 , v 0 ) ∈ H γ × H γ − 1 (u⁰, v⁰) ∈ Hγ × Hγ -1 for all γ > 0 γ > 0 . Notably, our method attains an improved error rate of O ( τ 2 γ − ) O(τ 2γ -) in one and two dimensions for γ ∈ ( 0 , 1 2 ] γ ∈ (0, 1/2] , and O ( τ max ( γ , 2 γ − 1 2 − ) ) O(τ max (γ , 2γ - 1/2-)) in three dimensions for γ