Abstract
In this chapter we demonstrate how to apply the methods we presented in the previous chapters to stochastic nonlinear conservations laws. Specifically, since the problem is nonlinear we apply the stochastic collocation method (SCM) using Smolyak’s sparse grid for a one-dimensional piston problem and test its computational performance. This is a classical problem in every aerodynamics textbook with an analytical solution if the piston velocity is fixed. However, here we consider a piston with a velocity perturbed by Brownian motion moving into a straight tube filled with a perfect gas at rest. The shock generated ahead of the piston can be located by solving the one-dimensional Euler equations driven by white noise using the Stratonovich or Ito formulations. We apply the Lie-Trotter splitting method before we approximate the Brownian motion with its spectral truncation and subsequently apply stochastic collocation using either sparse grid or the quasi-Monte Carlo (QMC) method. Numerical results verify the Stratonovich-Euler and Ito-Euler models against stochastic perturbation results, and demonstrate the efficiency of sparse grid collocation and QMC for small and large random piston motions, respectively.
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Zhang, Z., Karniadakis, G.E. (2017). Application of collocation method to stochastic conservation laws. In: Numerical Methods for Stochastic Partial Differential Equations with White Noise. Applied Mathematical Sciences, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-319-57511-7_9
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