Abstract
A general framework for constructing energy dissipative or conservative Galerkin schemes for time-dependent partial differential equations (PDEs) is presented. The framework embraces a variety of dissipative and conservative PDEs with variational structure. An advantageous feature is that implementation of the resulting scheme requires only P1 elements. The schemes and their underlying \(H^1\)-weak forms are derived from the concept of formal weak form and an \(L^2\)-projection technique.
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References
Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer, Heidelberg (2009)
Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
Camassa, R., Holm, D.D., Hyman, J.M.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)
Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D.I., O’Neale, D., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method. J. Comput. Phys. 231, 6770–6789 (2012)
Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25, 303–328 (1998)
Constantin, A., Strauss, W.A.: Stability of peakons. Comm. Pure Appl. Math. 53, 603–610 (2000)
Dahlby, M., Owren, B.: A general framework for deriving integral preserving numerical methods for PDEs. SIAM J. Sci. Comput. 33, 2318–2340 (2011)
Degasperis, A., Procesi, M.: Asymptotic Integrability in Symmetry and Perturbation theory, pp. 23–37. World Scientific, Singapore (1999)
Du, Q., Nicolaides, R.A.: Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28, 1310–1322 (1991)
Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D 4, 47–66 (1981)
Furihata, D.: Finite difference schemes for \(\frac{\partial u}{\partial t} = \left( \frac{\partial }{\partial x} \right) ^{\alpha } \frac{\delta {G}}{\delta u}\) that inherit energy conservation or dissipation property. J. Comput. Phys. 156, 181–205 (1999)
Furihata, D., Matsuo, T.: Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations. Chapman & Hall/CRC, Boca Raton (2011)
Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6, 449–467 (1996)
Itoh, T., Abe, K.: Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J. Comput. Phys. 76, 85–102 (1988)
Kawahara, T.: Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn 33, 260–264 (1972)
Kuramae, H.: An alternating discrete variational derivative method and its applications (in Japanese). Master’s thesis, University of Tokyo (2012)
Matsuo, T.: Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations. J. Comput. Appl. Math. 218, 506–521 (2008)
Matsuo, T.: A Hamiltonian-conserving Galerkin scheme for the Camassa–Holm equation. J. Comput. Appl. Math. 234, 1258–1266 (2010)
Matsuo, T., Furihata, D.: Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171, 425–447 (2001)
Matsuo, T., Yamaguchi, H.: An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations. J. Comput. Phys. 228, 4346–4358 (2009)
Miyatake, Y., Matsuo, T.: Energy-preserving \({H}^1\)-Galerkin schemes for shallow water wave equations with peakon solutions. Phys. Lett. A 376, 2633–2639 (2012)
Miyatake, Y., Yaguchi, T., Matsuo, T.: Numerical integration of the Ostrovsky equation based on its geometric structures. J. Comput. Phys. 231, 4542–4559 (2012)
Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A 41, 045206 (2008)
Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319–328 (1977)
Wazwaz, A.M.: New solitary wave solutions to the Kuramoto–Sivashinsky and the Kawahara equations. Appl. Math. Comput. 182, 1642–1650 (2006)
Yaguchi, T., Matsuo, T., Sugihara, M.: An extension of the discrete variational method to nonuniform grids. J. Comput. Phys. 229, 4382–4423 (2010)
Yaguchi, T., Matsuo, T., Sugihara, M.: The discrete variational derivative method based on discrete differential forms. J. Comput. Phys. 231, 3963–3986 (2012)
Acknowledgments
We are thankful for various comments by the reviewers. We are also grateful to Takaharu Yaguchi and Norikazu Saito for many useful discussions. This work was partly supported by JSPS KAKENHI Grant Numbers 25287030, 23560063. It was also partly supported by the Aihara Project, the FIRST program from JSPS, initiated by CSTP. The first author is supported by the Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists.
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Communicated by Anne Kværnø.
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Miyatake, Y., Matsuo, T. A general framework for finding energy dissipative/conservative \(H^1\)-Galerkin schemes and their underlying \(H^1\)-weak forms for nonlinear evolution equations. Bit Numer Math 54, 1119–1154 (2014). https://doi.org/10.1007/s10543-014-0483-3
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DOI: https://doi.org/10.1007/s10543-014-0483-3
Keywords
- Discrete partial derivative method
- Dissipation
- Conservation
- Galerkin method
- Structure-preserving integration
- \(L^2\)-projection
- Discrete gradient method