Numerical Methods for SPDEs with Tempered Stable Processes

Abstract

We develop new probabilistic and deterministic approaches for moment statistics of stochastic partial differential equations with pure jump tempered $\alpha$-stable (T$\alpha$S) Lévy processes. With the compound Poisson (CP) approximation or the series representation of the T$\alpha$S process, we simulate the moment statistics of stochastic reaction-diffusion equations with additive T$\alpha$S white noises by the probability collocation method (PCM) and the Monte Carlo (MC) method. PCM is shown to be more efficient and accurate than MC in relatively low dimensions. Then as an alternative approach, we solve the generalized Fokker--Planck equation that describes the evolution of the density for stochastic overdamped Langevin equations to obtain the density and the moment statistics for the solution following two different approaches. First, we solve an integral equation for the density by approximating the T$\alpha$S processes as CP processes; second, we directly solve the tempered fractional PDE (TFPDE). We show that the numerical solution of TFPDE achieves higher accuracy than PCM at a lower cost and we also demonstrate agreement between the histogram from MC and the density from the TFPDE.