A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives
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- by Yingda Cheng and Chi-Wang Shu;
- Math. Comp. 77 (2008), 699-730
- DOI: https://doi.org/10.1090/S0025-5718-07-02045-5
- Published electronically: September 6, 2007
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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.References
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Bibliographic Information
- Yingda Cheng
- Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
- MR Author ID: 811395
- Email: ycheng@dam.brown.edu
- Chi-Wang Shu
- Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
- MR Author ID: 242268
- Email: shu@dam.brown.edu
- 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.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 699-730
- MSC (2000): Primary 65M60
- DOI: https://doi.org/10.1090/S0025-5718-07-02045-5
- MathSciNet review: 2373176