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A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives


Authors: Yingda Cheng and Chi-Wang Shu
Journal: Math. Comp. 77 (2008), 699-730
MSC (2000): Primary 65M60
DOI: https://doi.org/10.1090/S0025-5718-07-02045-5
Published electronically: September 6, 2007
MathSciNet review: 2373176
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Abstract: In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal $ (k+1)$-th order of accuracy when using piecewise $ k$-th degree polynomials, under the condition that $ k+1$ is greater than or equal to the order of the equation.


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Additional Information

Yingda Cheng
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: ycheng@dam.brown.edu

Chi-Wang Shu
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: shu@dam.brown.edu

DOI: https://doi.org/10.1090/S0025-5718-07-02045-5
Keywords: Discontinuous Galerkin method, partial differential equations with higher order derivatives, stability, error estimate, high order accuracy
Received by editor(s): August 25, 2006
Received by editor(s) in revised form: February 17, 2007
Published electronically: September 6, 2007
Additional Notes: This research was supported in part by ARO grant W911NF-04-1-0291, NSF grant DMS-0510345 and AFOSR grant FA9550-05-1-0123.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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