Two-step MaCcormack method for statistical moments of a stochastic Burger's equation

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2 Citations (Scopus)

Abstract

The two-step MacCormack scheme has been modified to solve a stochastic Burger's equation driven by a random force with a random initial condition. Statistical moments of a solution are expressed by Hermite-Fourier coefficients so that the stochastic equation is transformed into a deterministic propagator system. The resultant system needs to be solved only once and computational loads are reduced accordingly. The numerical stability, accuracy and efficiency of the scheme have been analyzed and compared with the Monte Carlo method and the Lax-Wendroff scheme. The modified MacCormack scheme shows less diffusion near discontinuities than the Lax-Wendroff scheme. While maintaining the same accuracy, the MacCormack scheme improves numerical efficiency over the Lax-Wendroff scheme in the ratio of (N+11/6) when the length is JV. Compared to the Monte Carlo method, the scheme saves more than 98% of CPU time and removes dependence upon a random number generator.

Original languageEnglish
Pages (from-to)657-684
Number of pages28
JournalDynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis
Volume14
Issue number5
Publication statusPublished - 2007 Oct 1

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Two-step Method
Burgers Equation
Stochastic Equations
Monte Carlo methods
Moment
Convergence of numerical methods
Program processors
Monte Carlo method
Random number Generator
Numerical Stability
Fourier coefficients
CPU Time
Propagator
Hermite
Numerical Scheme
Discontinuity
Initial conditions

Keywords

  • Burger's equation
  • Lax-Wendroff scheme
  • MacCormack scheme
  • Monte Carlo method
  • Stochastic differential equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

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title = "Two-step MaCcormack method for statistical moments of a stochastic Burger's equation",
abstract = "The two-step MacCormack scheme has been modified to solve a stochastic Burger's equation driven by a random force with a random initial condition. Statistical moments of a solution are expressed by Hermite-Fourier coefficients so that the stochastic equation is transformed into a deterministic propagator system. The resultant system needs to be solved only once and computational loads are reduced accordingly. The numerical stability, accuracy and efficiency of the scheme have been analyzed and compared with the Monte Carlo method and the Lax-Wendroff scheme. The modified MacCormack scheme shows less diffusion near discontinuities than the Lax-Wendroff scheme. While maintaining the same accuracy, the MacCormack scheme improves numerical efficiency over the Lax-Wendroff scheme in the ratio of (N+11/6) when the length is JV. Compared to the Monte Carlo method, the scheme saves more than 98{\%} of CPU time and removes dependence upon a random number generator.",
keywords = "Burger's equation, Lax-Wendroff scheme, MacCormack scheme, Monte Carlo method, Stochastic differential equation",
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AU - Kim, Hongjoong

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N2 - The two-step MacCormack scheme has been modified to solve a stochastic Burger's equation driven by a random force with a random initial condition. Statistical moments of a solution are expressed by Hermite-Fourier coefficients so that the stochastic equation is transformed into a deterministic propagator system. The resultant system needs to be solved only once and computational loads are reduced accordingly. The numerical stability, accuracy and efficiency of the scheme have been analyzed and compared with the Monte Carlo method and the Lax-Wendroff scheme. The modified MacCormack scheme shows less diffusion near discontinuities than the Lax-Wendroff scheme. While maintaining the same accuracy, the MacCormack scheme improves numerical efficiency over the Lax-Wendroff scheme in the ratio of (N+11/6) when the length is JV. Compared to the Monte Carlo method, the scheme saves more than 98% of CPU time and removes dependence upon a random number generator.

AB - The two-step MacCormack scheme has been modified to solve a stochastic Burger's equation driven by a random force with a random initial condition. Statistical moments of a solution are expressed by Hermite-Fourier coefficients so that the stochastic equation is transformed into a deterministic propagator system. The resultant system needs to be solved only once and computational loads are reduced accordingly. The numerical stability, accuracy and efficiency of the scheme have been analyzed and compared with the Monte Carlo method and the Lax-Wendroff scheme. The modified MacCormack scheme shows less diffusion near discontinuities than the Lax-Wendroff scheme. While maintaining the same accuracy, the MacCormack scheme improves numerical efficiency over the Lax-Wendroff scheme in the ratio of (N+11/6) when the length is JV. Compared to the Monte Carlo method, the scheme saves more than 98% of CPU time and removes dependence upon a random number generator.

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