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A characteristics-mixed finite element method for Burgers’ equation

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Abstract

In this paper, we propose a new mixed finite element method, called the characteristics-mixed method, for approximating the solution to Burgers’ equation. This method is based upon a space-time variational form of Burgers’ equation. The hyperbolic part of the equation is approximated along the characteristics in time and the diffusion part is approximated by a mixed finite element method of lowest order. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Our analysis show the new method approximate the scalar unknown and the vector flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. Numerical example is presented to show that the new scheme is easily implemented, shocks and boundary layers are handled with almost no oscillations.

One of the contributions of the paper is to show how the optimal error estimates inL 2(Ω) are obtained which are much more difficult than in the standard finite element methods. These results seem to be new in the literature of finite element methods.

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Authors and Affiliations

  1. Department of Mathematics, Shandong Normal University, 250014, Jinan, Shandong, P. R. China

    Huanzhen Chen & Ziwen Jiang

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  1. Huanzhen Chen
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  2. Ziwen Jiang
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Huanzhen Chen received his Ph. D in mathematics of computation at Shandong University under the direction of Yirang Yuan in September, 1998, and at the same time he was named a proffessor in Shandong Normal University. His research interests focus on numerical analysis for partial differencial equations, numerical methods for fluid mechanics. Also he does mathematical model and numerical simulation. He has issued more than 30 papers in these fields.

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Chen, H., Jiang, Z. A characteristics-mixed finite element method for Burgers’ equation. JAMC 15, 29–51 (2004). https://doi.org/10.1007/BF02935745

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  • DOI: https://doi.org/10.1007/BF02935745

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