### 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 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics