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 \((u^{0}, v^{0}) \in H^{\gamma} \times H^{\gamma-1}\) for all \(\gamma > 0\). Notably, our method attains an improved error rate of \(O(\tau^{2\gamma-})\) in one and two dimensions for \(\gamma \in (0, \frac{1}{2}]\), and \(O(\tau^{\max(\gamma, 2\gamma - \frac{1}{2}-)})\) in three dimensions for \(\gamma \in (0, \frac{3}{4}]\), where \(\tau\) 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.

翻译：随机非线性波动方程解的正则性对数值算法的精度与效率具有关键影响。粗糙或不连续的初始条件带来了显著挑战，常导致现有方法精度损失与计算效率降低。本研究通过开发一种新颖高效、专门用于计算随机非线性波动方程粗糙解的数值算法来应对这些挑战，同时大幅放宽对初始数据的正则性要求。通过利用随机非线性波动方程的内在结构并运用调和分析中的先进工具，我们构建了一种时间离散化方法，该方法对所有\(\gamma > 0\)的初始值\((u^{0}, v^{0}) \in H^{\gamma} \times H^{\gamma-1}\)均能实现鲁棒收敛。值得注意的是，我们的方法在\(\gamma \in (0, \frac{1}{2}]\)时，于一维和二维空间达到了\(O(\tau^{2\gamma-})\)的改进误差率；在\(\gamma \in (0, \frac{3}{4}]\)时，于三维空间达到了\(O(\tau^{\max(\gamma, 2\gamma - \frac{1}{2}-)})\)的误差率，其中\(\tau\)表示时间步长。这些收敛率超越了相同正则性条件下现有数值方法的性能，凸显了我们方法的优势。为验证方法的性能，我们进行了大量的数值实验，结果表明相较于最先进的方法，我们的方法具有更高的精度和计算效率。这些结果凸显了我们的方法即使在具有挑战性的初始条件下，也能实现随机波动现象精确高效模拟的潜力。