### 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 language | English |
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Pages (from-to) | 657-684 |

Number of pages | 28 |

Journal | Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis |

Volume | 14 |

Issue number | 5 |

Publication status | Published - 2007 Oct 1 |

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### 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

**Two-step MaCcormack method for statistical moments of a stochastic Burger's equation.** / Kim, Hongjoong.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Two-step MaCcormack method for statistical moments of a stochastic Burger's equation

AU - Kim, Hongjoong

PY - 2007/10/1

Y1 - 2007/10/1

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.

KW - Burger's equation

KW - Lax-Wendroff scheme

KW - MacCormack scheme

KW - Monte Carlo method

KW - Stochastic differential equation

UR - http://www.scopus.com/inward/record.url?scp=35248816580&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35248816580&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:35248816580

VL - 14

SP - 657

EP - 684

JO - Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis

JF - Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis

SN - 1201-3390

IS - 5

ER -