Construction and analysis of fourth order, energy consistent, family of explicit time discretizations for dissipative linear wave equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 845-878.

This paper deals with the construction of a family of fourth order, energy consistent, explicit time discretizations for dissipative linear wave equations. The schemes are obtained by replacing the inversion of a matrix, that comes naturally after using the technique of the Modified Equation on the second order Leap Frog scheme applied to dissipative linear wave equations, by explicit approximations of its inverse. The stability of the schemes are studied using an energy analysis and a convergence analysis is carried out. Numerical results in 1D illustrate the space/time convergence properties of the schemes and their efficiency is compared to more classical time discretizations.

DOI : 10.1051/m2an/2019079
Classification : 00A71, 35L05, 85A20, 33C55, 65M60
Mots-clés : Dissipative linear wave equation, high order time discretisation, energy analysis 
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     author = {Chabassier, Juliette and Diaz, Julien and Imperiale, S\'ebastien},
     title = {Construction and analysis of fourth order, energy consistent, family of explicit time discretizations for dissipative linear wave equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {845--878},
     publisher = {EDP-Sciences},
     volume = {54},
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     doi = {10.1051/m2an/2019079},
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     url = {http://www.numdam.org/articles/10.1051/m2an/2019079/}
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Chabassier, Juliette; Diaz, Julien; Imperiale, Sébastien. Construction and analysis of fourth order, energy consistent, family of explicit time discretizations for dissipative linear wave equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 845-878. doi : 10.1051/m2an/2019079. http://www.numdam.org/articles/10.1051/m2an/2019079/

S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. 3rd ed. Springer, New York (2008). | DOI | MR | Zbl

M. Cassier, P. Joly and M. Kachanovska, Mathematical models for dispersive electromagnetic waves: an overview. Comput. Math. Appl. 74 (2017) 2792–2830. | DOI | MR | Zbl

J. Chabassier and S. Imperiale, Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string. Wave Motion 50 (2012) 456–480. | DOI | MR | Zbl

J. Chabassier and S. Imperiale, Introduction and study of fourth order theta schemes for linear wave equations. J. Comput. Appl. Math. 245 (2013) 194–212. | DOI | MR | Zbl

J. Chabassier and S. Imperiale, Space/time convergence analysis of a class of conservative schemes for linear wave equations. C.R. Math. 355 (2017) 282–289. | DOI | MR | Zbl

G. Cohen, High-order Numerical Methods for Transient Wave Equations. Springer-Verlag (2001). | MR | Zbl

G. Cohen, P. Joly and N. Tordjman, Higher-order finite elements with mass-lumping for the 1D wave equation. Finite Element Anal. Des. 16 (1994) 329–336. | DOI | MR | Zbl

G. Cohen, P. Joly, J.E. Roberts and N. Tordjman, Higher order triangular finite elements with mass lumping for the wave equation. SIAM J. Numer. Anal. 38 (2001) 2047–2078. | DOI | MR | Zbl

R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology, In: Vols 5 and 6 of Evolution Problems I and II. Springer-Verlag, Berlin (2000). | MR | Zbl

S.S. Dragomir, Some Gronwall type inequalities and applications. Nova Science Pub Incorporated (2003). | MR | Zbl

S. Ervedoza, A. Marica and E. Zuazua, Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis. IMA J. Numer. Anal. 36 (2016) 503–542. | DOI | MR | Zbl

K. Feng, Difference schemes for Hamiltonian formalism and symplectic geometry. J. Comput. Math. 4 (1986) 279–289. | MR | Zbl

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping. J. Differ. Equ. 132 (1996) 338–352. | DOI | MR | Zbl

P. Freitas, M. Grinfeld and P.A. Knight, Stability of finite-dimensional systems with indefinite damping. Adv. Math. Sci. App. 17 (1997) 435–446. | MR | Zbl

J.C. Gilbert and P. Joly, Higher order time stepping for second order hyperbolic problems and optimal CFL conditions. Partial Differ. Equ. 16 (2008) 67–93. | MR | Zbl

P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR | Zbl

M.J. Grote and T. Mitkova, High-order explicit local time-stepping methods for damped wave equations. J. Comput. Appl. Math. 239 (2013) 270–289. | DOI | MR | Zbl

M.J. Grote, A. Schneebeli and D. Schötzau, Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44 (2006) 2408–2431. | DOI | MR | Zbl

E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics (2006). | MR | Zbl

P. Joly, Topics in computational wave propagation. “Variational methods for time-dependent wave propagation problems”. In: Vol. 31 of  Lecture Notes in Computational Science and Engineering. Springer, Berlin (2003) 201–264. | DOI | MR | Zbl

P. Joly, The mathematical model for elastic wave propagation. Effective computational methods for wave propagation. Numer. Insights Chapman Hall/CRC 5 (2008) 247–266. | DOI | MR | Zbl

T. Kato, Perturbation theory for linear operators. In: Vol. 132 of Classics in Mathematics. Springer-Verlag, Berlin Heidelberg (1995). | MR | Zbl

G.R. Shubin and J.B. Bell, A modified equation approach to constructing fourth-order methods for acoustic wave propagation. Soc. Ind. Appl. Math. J. Sci. Stat. Comput. 8 (1987) 135–151. | DOI | MR | Zbl

M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A 146 (1990) 319–323. | DOI | MR

J.G. Verwer, Runge-Kutta methods and viscous wave equations. Numer. Math. 112 (2009) 485–507. | DOI | MR | Zbl

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