Open Access

ARTICLE

A Fictitious Time Integration Method for the Burgers Equation

by ,

1 Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liuchei.nshan@msa.hinet.net

Computers, Materials & Continua 2009, 9(3), 229-252. https://doi.org/10.3970/cmc.2009.009.229

Abstract

When the given input data are corrupted by an intensive noise, most numerical methods may fail to produce acceptable numerical solutions. Here, we propose a new numerical scheme for solving the Burgers equation forward in time and backward in time. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ )u(x,t) =: v(x,t,τ), such that the original Burgers equation is written as a new parabolic type partial differential equation in the space of (x,t,τ). A fictitious damping coefficient can be used to strengthen the stability in the numerical integration of a semi-discretized ordinary differential equations set on the spatial-temporal grid points. Even for a very large final time and under a large noise, the present Fictitious Time Integration Method (FTIM) can be used to retrieve the initial data very well. When the FTIM is used to solve the direct problems of Burgers equation, with a large Reynolds number and the input data being noised seriously, we can still reconstruct the solution rather accurately. This result however cannot be achieved by other conventional numerical methods. It is interesting that both the forward and backward problems of Burgers equation can be unifiedly treated by the FTIM.

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Keywords

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