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Convergence of the Finite-Difference Method for the 1—d Wave Equation with Homogeneous Dirichlet Boundary Conditions

  • Sylvain Ervedoza
  • Enrique Zuazua
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

This chapter of the book is devoted to the study of the convergence of the numerical scheme
$$\left \{\begin{array}{ll} \partial _{tt}\varphi _{j,h} - \frac{1} {{h}^{2}}\left (\varphi _{j+1,h} - 2\varphi _{j,h} + \varphi _{j-1,h}\right ) = 0,& \\ &(t,j) \in (0,T) \times \{ 1,\ldots ,N\}, \\ \varphi _{0,h}(t) = \varphi _{N+1,h}(t) = 0,\quad &t \in (0,T), \\ (\varphi _{h}(0),\partial _{t}\varphi _{h}(0)) = (\varphi _{0h},\varphi _{1h}), & \end{array} \right .$$
(3.1)
towards the continuous wave equation
$$\left \{\begin{array}{ll} \partial _{tt}\varphi - \partial _{xx}\varphi = 0,\quad &(t,x) \in (0,T) \times (0,1), \\ \varphi (t,0) = \varphi (t,1) = 0,\quad &t \in (0,T), \\ (\varphi (0),\partial _{t}\varphi (0)) = (\varphi _{0},\varphi _{1}).&\end{array} \right .$$
(3.2)
Of course, first of all, one needs to explain how discrete and continuous solutions can be compared. This will be done in Sect. 3.2. In Sect. 3.3, we will present our main convergence result. We shall then present some further convergence results in Sect. 3.4 and illustrate them in Sect. 3.5.

References

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    G.A. Baker, J.H. Bramble, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations. RAIRO Anal. Numér. 13(2), 75–100 (1979)MathSciNetMATHGoogle Scholar
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    I. Lasiecka, J.-L. Lions, R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. (9) 65(2), 149–192 (1986)Google Scholar
  3. 36.
    J.-L. Lions, in Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte, RMA, vol. 8 (Masson, Paris, 1988)Google Scholar
  4. 45.
    J. Rauch, On convergence of the finite element method for the wave equation. SIAM J. Numer. Anal. 22(2), 245–249 (1985)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Sylvain Ervedoza and Enrique Zuazua 2013

Authors and Affiliations

  • Sylvain Ervedoza
    • 1
  • Enrique Zuazua
    • 2
  1. 1.Institut de Mathématiques de Toulouse & CNRSUniversité Paul SabatierToulouseFrance
  2. 2.BCAM-Basque Center for Applied Mathematics IkerbasqueBilbaoSpain

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