Error self-canceling of a difference scheme maintaining two conservation laws for linear advection equation
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- by Cui Yanfen and Mao De-kang PDF
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Abstract:
In recent years, Mao and his co-workers developed a new type of difference schemes for evolution partial differential equations. The core of the new schemes is to simulate, in addition to the original unknowns of the equations, some quantities that are nonlinear functions of the unknowns; therefore, they maintain additional nonlinear discrete structures of the equations. The schemes show a super-convergence property, and their numerical solutions are far better than that of traditional difference schemes at both accuracy and long-time behavior.
In this paper, to understand the super-convergence properties of the schemes, we carry out a truncation error investigation on the scheme maintaining two conservation laws for the linear advection equation. This scheme is the simplest one of this type. Our investigation reveals that the numerical errors of the scheme produced in different time steps are accumulated in a nonlinear fashion, in which they cancel each other. As to our knowledge, such an error self-canceling feature has not been seen in other numerical methods, and it is this feature that brings the super-convergence property of the scheme.
References
- Uri M. Ascher and Robert I. McLachlan, Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math. 48 (2004), no. 3-4, 255–269. Workshop on Innovative Time Integrators for PDEs. MR 2056917, DOI 10.1016/j.apnum.2003.09.002
- U. M. Ascher and R. I. McLachlan, On symplectic and multisymplectic schemes for the KdV equation, J. Sci. Comput. 25 (2005), no. 1-2, 83–104. MR 2231944, DOI 10.1007/s10915-004-4634-6
- R. Chen , Several numerical methods for hyperbolic conservation laws, Doctoral thesis, No. 10280-06810055, Shanghai University (in Chinese).
- Bernardo Cockburn, San Yih Lin, and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems, J. Comput. Phys. 84 (1989), no. 1, 90–113. MR 1015355, DOI 10.1016/0021-9991(89)90183-6
- Yanfen Cui and De-kang Mao, Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation, J. Comput. Phys. 227 (2007), no. 1, 376–399. MR 2361527, DOI 10.1016/j.jcp.2007.07.031
- Bruno Després and Frédéric Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput. 16 (2001), no. 4, 479–524 (2002). MR 1881855, DOI 10.1023/A:1013298408777
- Ami Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), no. 3, 357–393. MR 701178, DOI 10.1016/0021-9991(83)90136-5
- Ami Harten, Björn Engquist, Stanley Osher, and Sukumar R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes. III, J. Comput. Phys. 71 (1987), no. 2, 231–303. MR 897244, DOI 10.1016/0021-9991(87)90031-3
- Guo Benyu, The finite difference methods for Partial Differential Equations, Science Publication, 1988 (in Chinese).
- Guang-Shan Jiang and Chi-Wang Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996), no. 1, 202–228. MR 1391627, DOI 10.1006/jcph.1996.0130
- Randall J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. MR 1925043, DOI 10.1017/CBO9780511791253
- Randall J. LeVeque, Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1990. MR 1077828, DOI 10.1007/978-3-0348-5116-9
- H. Li, Entropy dissipating scheme for hyperbolic system of conservation laws in one space dimension, Doctoral thesis, No. 11903-02820022, Shanghai University (in Chinese).
- H. Li, Second-order entropy dissipation scheme for scalar conservation laws in one space dimension, Master’s thesis, No. 11903-99118086, Shanghai University (in Chinese).
- H. Li and D. Mao, The design of the entropy dissipator of the entropy dissipating scheme for scalar conservation law, Chinese J. Comput. Phys., 21, (2004), pp. 319-326 (in Chinese).
- Hongxia Li, Zhigang Wang, and De-kang Mao, Numerically neither dissipative nor compressive scheme for linear advection equation and its application to the Euler system, J. Sci. Comput. 36 (2008), no. 3, 285–331. MR 2434848, DOI 10.1007/s10915-008-9192-x
- K. W. Morton and D. F. Mayers, Numerical solution of partial differential equations, 2nd ed., Cambridge University Press, Cambridge, 2005. An introduction. MR 2153063, DOI 10.1017/CBO9780511812248
- Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1994. MR 1275838
- Philip L. Roe, Some contributions to the modelling of discontinuous flows, Large-scale computations in fluid mechanics, Part 2 (La Jolla, Calif., 1983) Lectures in Appl. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1985, pp. 163–193. MR 818787
- Z. Wang, Finitie difference Scsemes satisfying multiconservation laws for linear advection equations, Master’s thesis, No. 11903-99118086, Shanghai University (in Chinese).
- Zhi Gang Wang and De Kang Mao, A finite difference scheme satisfying three conservation laws for a linear advection equation, J. Shanghai Univ. Nat. Sci. 12 (2006), no. 6, 588–592 (Chinese, with English and Chinese summaries). MR 2281178
Additional Information
- Cui Yanfen
- Affiliation: Shanghai Institute of Applied Mathematics and Mechanics, Shanghai, 200072, People’s Republic of China
- Address at time of publication: Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China
- Mao De-kang
- Affiliation: Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China
- Received by editor(s): March 14, 2009
- Received by editor(s) in revised form: January 22, 2011
- Published electronically: July 26, 2011
- Additional Notes: This research was supported by China National Science Foundation Grant No.10971132 and Shanghai Pu Jiang Program [2006] 118, and also by Shanghai Leading Academic Discipline project (J50101)
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 81 (2012), 715-741
- MSC (2010): Primary 65M06, 65M15
- DOI: https://doi.org/10.1090/S0025-5718-2011-02523-8
- MathSciNet review: 2869034