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Semilinear hyperbolic stochastic partial differential equations (SPDEs) findwidespread applications in the natural and engineering sciences. However, thetraditional Gaussian setting may prove too restrictive, as phenomena inmathematical finance, porous media, and pollution models often exhibit noise ofa different nature. To capture temporal discontinuities and accommodateheavy-tailed distributions, Hilbert space-valued L\'evy processes or L\'evyfields are employed as driving noise terms. The numerical discretization ofsuch SPDEs presents several challenges. The low regularity of the solution inspace and time leads to slow convergence rates and instability in space/timediscretization schemes. Furthermore, the L\'evy process can take values in aninfinite-dimensional Hilbert space, necessitating projections ontofinite-dimensional subspaces at each discrete time point. Additionally,unbiased sampling from the resulting L\'evy field may not be feasible. In thisstudy, we introduce a novel fully discrete approximation scheme that tacklesthese difficulties. Our main contribution is a discontinuous Galerkin schemefor spatial approximation, derived naturally from the weak formulation of theSPDE. We establish optimal convergence properties for this approach and combineit with a suitable time stepping scheme to prevent numerical oscillations.Furthermore, we approximate the driving noise process using truncatedKarhunen-Lo\`eve expansions. This approximation yields a sum of scaled anduncorrelated one-dimensional L\'evy processes, which can be simulated withcontrolled bias using Fourier inversion techniques.

Stochastic Transport with L\'evy Noise -- Fully Discrete Numerical Approximation