Numerical solutions of Burgers' equation with random initial conditions using the Wiener chaos expansion and the Lax-Wendroff scheme

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3 Citations (Scopus)

Abstract

The work is concerned with efficient computation of statistical moments of solutions to Burgers' equation with random initial conditions. When the Lax-Wendroff scheme is expanded using the Wiener chaos expansion (WCE), it introduces an infinite system of deterministic equations with respect to non-random Hermite-Fourier coefficients. One of important properties of the system is that all the statistical moments of the solution can be computed using simple formulae that involve only the solution of the system. The stability, accuracy, and efficiency of the WCE approach to computing statistical moments have been numerically tested and compared to those for the Monte Carlo (MC) method. Strong evidence has been given that the WCE approach is as accurate as but substantially faster than the MC method, at least for certain classes of initial conditions.

Original languageEnglish
Pages (from-to)545-550
Number of pages6
JournalApplied Mathematics Letters
Volume20
Issue number5
DOIs
Publication statusPublished - 2007 May 1

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Chaos Expansion
Burgers Equation
Chaos theory
Initial conditions
Numerical Solution
Moment
Monte Carlo method
Monte Carlo methods
Statistical Computing
Infinite Systems
Fourier coefficients
Hermite

Keywords

  • Burgers' equation
  • Lax-Wendroff scheme
  • Monte Carlo method
  • Wiener chaos expansion

ASJC Scopus subject areas

  • Computational Mechanics
  • Control and Systems Engineering
  • Applied Mathematics
  • Numerical Analysis

Cite this

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abstract = "The work is concerned with efficient computation of statistical moments of solutions to Burgers' equation with random initial conditions. When the Lax-Wendroff scheme is expanded using the Wiener chaos expansion (WCE), it introduces an infinite system of deterministic equations with respect to non-random Hermite-Fourier coefficients. One of important properties of the system is that all the statistical moments of the solution can be computed using simple formulae that involve only the solution of the system. The stability, accuracy, and efficiency of the WCE approach to computing statistical moments have been numerically tested and compared to those for the Monte Carlo (MC) method. Strong evidence has been given that the WCE approach is as accurate as but substantially faster than the MC method, at least for certain classes of initial conditions.",
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AB - The work is concerned with efficient computation of statistical moments of solutions to Burgers' equation with random initial conditions. When the Lax-Wendroff scheme is expanded using the Wiener chaos expansion (WCE), it introduces an infinite system of deterministic equations with respect to non-random Hermite-Fourier coefficients. One of important properties of the system is that all the statistical moments of the solution can be computed using simple formulae that involve only the solution of the system. The stability, accuracy, and efficiency of the WCE approach to computing statistical moments have been numerically tested and compared to those for the Monte Carlo (MC) method. Strong evidence has been given that the WCE approach is as accurate as but substantially faster than the MC method, at least for certain classes of initial conditions.

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