Numerical solution of the Stratonovich- and Ito–Euler equations: Application to the stochastic piston problem

Journal Article · Fri Mar 01 04:00:00 EST 2013 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2012.11.017· OSTI ID:22233565

Zhang, Zhongqiang; Yang, Xiu ^[1]; Lin, Guang ^[2]

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1. Division of Applied Mathematics, Brown University, Providence, RI 02912 (United States)
2. Pacific Northwest National Laboratory, Richland, WA 99352 (United States)

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.

• OSTI ID: 22233565
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 236; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
97 MATHEMATICS AND COMPUTING
APPROXIMATIONS
BROWNIAN MOVEMENT
EQUATIONS
MONTE CARLO METHOD
NOISE
NUMERICAL SOLUTION
ONE-DIMENSIONAL CALCULATIONS
PERTURBATION THEORY
PISTONS
RANDOMNESS
SHOCK TUBES
STOCHASTIC PROCESSES
TIME DEPENDENCE
VELOCITY