Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Computational investigations of low-discrepancy sequences
ACM Transactions on Mathematical Software (TOMS)
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
On the L2-discrepancy for anchored boxes
Journal of Complexity
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications
Journal of Computational Physics
A Multistage Wiener Chaos Expansion Method for Stochastic Advection-Diffusion-Reaction Equations
SIAM Journal on Scientific Computing
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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 approximate the Brownian motion with its spectral truncation and subsequently apply stochastic collocation using either sparse grid or the quasi-Monte Carlo (QMC) method. In particular, we first transform the Euler equations with an unsteady stochastic boundary into stochastic Euler equations over a fixed domain with a time-dependent stochastic source term. We then solve the transformed equations by splitting them up into two parts, i.e., a 'deterministic part' and a 'stochastic part'. Numerical results verify the Stratonovich-Euler and Ito-Euler models against stochastic perturbation results, and demonstrate the efficiency of sparse grid and QMC for small and large random piston motions, respectively. The variance of shock location of the piston grows cubically in the case of white noise in contrast to colored noise reported in [1], where the variance of shock location grows quadratically with time for short times and linearly for longer times.