Stochastic Transport with Lévy Noise -- Fully Discrete Numerical Approximation

Semilinear hyperbolic stochastic partial differential equations have various applications in the natural and engineering sciences. From a modeling point of view the Gaussian setting may be too restrictive, since applications in mathematical finance and phenomena such as porous media or pollution models indicate an influence of noise of a different nature. In order to capture temporal discontinuities and allow for heavy-tailed distributions, Hilbert space-valued L\'evy processes (or L\'evy fields) as driving noise terms are considered. The numerical discretization of the corresponding SPDE involves several difficulties: Low spatial and temporal regularity of the solution to the problem entails slow convergence rates and instabilities for space/time-discretization schemes. Furthermore, the L\'evy process admits values in a possibly infinite-dimensional Hilbert space, hence projections onto a finite-dimensional subspace for each discrete point in time are necessary. Finally, unbiased sampling from the resulting L\'evy field may not be possible. We introduce a novel fully discrete approximation scheme that addresses all of these aspects. Our central contribution is a novel discontinuous Petrov-Galerkin scheme for the spatial approximation that naturally arises from the weak formulation of the SPDE. We prove optimal convergence of this approach and couple it with a suitable time stepping scheme to avoid numerical oscillations. Moreover, we approximate the driving noise process by truncated Karhunen-Lo\'eve expansions. The latter essentially yields a sum of scaled and uncorrelated one-dimensional L\'evy processes, which may be simulated with controlled bias by Fourier inversion techniques.