A stochastic transport problem with Lévy noise: Fully discrete numerical approximation
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Semilinear hyperbolic stochastic partial differential equations have various applications in the natural and engineering sciences. From a modelling point of view the Gaussian setting can be too restrictive, since phenomena as porous media, pollution models or applications in mathamtical finance indicate an influence of noise of a different nature. In order to capture temporal discontinuities and allow for heavy-tailed distributions, Hilbert space valued-Lévy processes (or Lévy 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 Lévy process admits values in a possibly infinite-dimensional Hilbert space, hence projections into a finite-dimensional subspace for each discrete point in time are necessary. Finally, unbiased sampling from the resulting Lévy field may not be possible. We introduce a fully discrete approximation scheme that addresses these issues. A discontinuous Galerkin approach for the spatial approximation is coupled with a suitable time stepping scheme to avoid numerical oscillations. Moreover, we approximate the driving noise process by truncated Karhunen-Loéve expansions. The latter essentially yields a sum of scaled and uncorrelated one-dimensional Lévy processes, which may be simulated with controlled bias by Fourier inversion techniques.
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