Semilinear hyperbolic stochastic partial differential equations (SPDEs) find widespread applications in the natural and engineering sciences. However, the traditional Gaussian setting may prove too restrictive, as phenomena in mathematical finance, porous media, and pollution models often exhibit noise of a different nature. To capture temporal discontinuities and accommodate heavy-tailed distributions, Hilbert space-valued L\'evy processes or L\'evy fields are employed as driving noise terms. The numerical discretization of such SPDEs presents several challenges. The low regularity of the solution in space and time leads to slow convergence rates and instability in space/time discretization schemes. Furthermore, the L\'evy process can take values in an infinite-dimensional Hilbert space, necessitating projections onto finite-dimensional subspaces at each discrete time point. Additionally, unbiased sampling from the resulting L\'evy field may not be feasible. In this study, we introduce a novel fully discrete approximation scheme that tackles these difficulties. Our main contribution is a discontinuous Galerkin scheme for spatial approximation, derived naturally from the weak formulation of the SPDE. We establish optimal convergence properties for this approach and combine it with a suitable time stepping scheme to prevent numerical oscillations. Furthermore, we approximate the driving noise process using truncated Karhunen-Lo\`eve expansions. This approximation yields a sum of scaled and uncorrelated one-dimensional L\'evy processes, which can be simulated with controlled bias using Fourier inversion techniques.

翻译：半线性双曲型随机偏微分方程在自然科学和工程科学中有着广泛应用。然而，传统的高斯设定可能过于局限，因为数学金融、多孔介质和污染模型中的现象往往表现出不同性质的噪声。为捕捉时间不连续性并适应重尾分布，采用Hilbert空间值莱维过程或莱维场作为驱动噪声项。这类随机偏微分方程的数值离散化面临若干挑战：解在空间和时间上的低正则性导致时空离散化方案收敛速度慢且不稳定；莱维过程可在无限维Hilbert空间中取值，需在每个离散时间点投影到有限维子空间；此外，对生成的莱维场进行无偏抽样可能不可行。本研究提出一种新型全离散逼近方案以解决这些难题。主要贡献在于基于随机偏微分方程弱形式自然推导出的间断伽辽金空间逼近格式，并建立了该方法的优化收敛性质。通过结合合适的时域步进方案，有效避免了数值振荡。进一步采用截断卡洛-洛埃夫展开逼近驱动噪声过程，该逼近生成一组尺度化且不相关的一维莱维过程求和，并利用傅里叶反演技术实现可控偏差模拟。