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Perfectly matched layers in 1-d : energy decay for continuous and semi-discrete waves

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Abstract

In this paper we investigate the efficiency of the method of perfectly matched layers (PML) for the 1-d wave equation. The PML method furnishes a way to compute solutions of the wave equation for exterior problems in a finite computational domain by adding a damping term on the matched layer. In view of the properties of solutions in the whole free space, one expects the energy of solutions obtained by the PML method to tend to zero as t → ∞, and the rate of decay can be understood as a measure of the efficiency of the method. We prove, indeed, that the exponential decay holds and characterize the exponential decay rate in terms of the parameters and damping potentials entering in the implementation of the PML method. We also consider a space semi-discrete numerical approximation scheme and we prove that, due to the high frequency spurious numerical solutions, the decay rate fails to be uniform as the mesh size parameter h tends to zero. We show however that adding a numerical viscosity term allows us to recover the property of exponential decay of the energy uniformly on h. Although our analysis is restricted to finite differences in 1-d, most of the methods and results apply to finite elements on regular meshes and to multi-dimensional problems.

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References

  1. Abarbanel S., Gottlieb D.: A mathematical analysis of the PML method. J. Comput. Phys. 134(2), 357–363 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abarbanel S., Gottlieb D.: On the construction and analysis of absorbing layers in CEM. Appl. Numer. Math. 27(4), 331–340. Absorbing boundary conditions (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abarbanel S., Gottlieb D., Hesthaven J.S.: Well-posed perfectly matched layers for advective acoustics. J. Comput. Phys. 154(2), 266–283 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Appelö D., Hagstrom T., Kreiss G.: Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability. SIAM J. Appl. Math. 67(1), 1–23 (electronic) (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Banks, H.T., Ito, K., Wang, C.: Exponentially stable approximations of weakly damped wave equations. In: Estimation and control of distributed parameter systems (Vorau, 1990), volume 100 of Internat. Ser. Numer. Math., pp 1–33. Birkhäuser, Basel (1991)

  6. Bayliss A., Gunzburger M., Turkel E.: Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math. 42(2), 430–451 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bécache E., Fauqueux S., Joly P.: Stability of perfectly matched layers, group velocities and anisotropic waves. J. Comput. Phys. 188(2), 399–433 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bécache E., Joly P.: On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations. M2AN Math. Model. Numer. Anal. 36(1), 87–119 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bécache E., Petropoulos P.G., Gedney S.D.: On the long-time behavior of unsplit perfectly matched layers. IEEE Trans. Antennas Propagat. 52(5), 1335–1342 (2004)

    Article  Google Scholar 

  10. Bérenger J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bermúdez A., Hervella-Nieto L., Prieto A., Rodríguez R.: An exact bounded PML for the Helmholtz equation. C. R. Math. Acad. Sci. Paris 339(11), 803–808 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Bermúdez A., Hervella-Nieto L., Prieto A., Rodríguez R.: Numerical simulation of time-harmonic scattering problems with an optimal PML. Sci. Ser. A Math. Sci. (N.S.) 13, 58–71 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Bermúdez A., Hervella-Nieto L., Prieto A., Rodríguez R.: An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems. J. Comput. Phys. 223(2), 469–488 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bermúdez, A., Hervella-Nieto, L.M., Prieto, A., Rodríguez, R.: An exact bounded perfectly matched layer for time-harmonic scattering problems. SIAM J. Sci. Comput., to be published (2007)

  15. Bramble J.H., Pasciak J.E.: Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem. Math. Comp. 77(261), 1–10 (electronic) (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Castro C., Micu S.: Boundary controllability of a linear semi-discrete 1-d wave equation derived from a mixed finite element method. Numer. Math. 102(3), 413–462 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Collino F., Monk P.: The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput. 19(6), 2061–2090 (electronic) (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cox S., Castro C.: Achieving arbitrarily large decay in the damped wave equation. SIAM J. Control Optim. 39(6), 1748–1755 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cox S., Zuazua E.: The rate at which energy decays in a damped string. Commun. Partial Differ. Equat. 19(1-2), 213–243 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cox S., Zuazua E.: The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44(2), 545–573 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Engquist B., Majda A.: Absorbing boundary conditions for the numerical simulation of waves. Math. Comp. 31(139), 629–651 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  22. Glowinski R.: Ensuring well-posedness by analogy: stokes problem and boundary control for the wave equation. J. Comput. Phys. 103(2), 189–221 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hébrard P., Henrot A.: A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim. 44(1), 349–366 (electronic) (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ignat L.I., Zuazua E.: A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence. C. R. Math. Acad. Sci. Paris 341(6), 381–386 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Infante J.A., Zuazua E.: Boundary observability for the space semi discretizations of the 1-d wave equation. Math. Model. Num. Ann. 33, 407–438 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lassas M., Somersalo E.: On the existence and convergence of the solution of PML equations. Computing 60(3), 229–241 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lebeau, G.: Équations des ondes amorties. Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, École Polytech. (1994)

  28. Lions, J.-L.: Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte volume RMA 8. Masson (1988)

  29. Macià F.: Propagación y control de vibraciones en medios discretos y continuos. PhD thesis, Universidad Complutense de Madrid (2001)

  30. Macià, F.: The effect of group velocity in the numerical analysis of control problems for the wave equation. In Mathematical and numerical aspects of wave propagation—WAVES 2003, pp 195–200. Springer, Berlin (2003)

  31. Petropoulos P.G.: Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell equations in rectangular, cylindrical, and spherical coordinates. SIAM J. Appl. Math. 60(3), 1037–1058 (electronic) (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Singer I., Turkel E.: A perfectly matched layer for the Helmholtz equation in a semi-infinite strip. J. Comput. Phys. 201(2), 439–465 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tcheugoué Tébou L.R., Zuazua E.: Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math. 95(3), 563–598 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Trefethen L.N.: Group velocity in finite difference schemes. SIAM Rev. 24(2), 113–136 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tsynkov S.V.: Numerical solution of problems on unbounded domains—a review. Appl. Numer. Math. 27(4), 465–532 Absorbing boundary conditions (1998)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tsynkov, S.V., Turkel, E.: A Cartesian perfectly matched layer for the Helmholtz equation. In: Absorbing boundaries and layers, domain decomposition methods, pp 279–309. Nova Sci. Publ., Huntington (2001)

  37. Turkel E., Yefet A.: Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math. 27(4), 533–557 Absorbing boundary conditions (1998)

    Article  MathSciNet  MATH  Google Scholar 

  38. Young R.M.: An introduction to nonharmonic Fourier series first edition. Academic Press Inc., San Diego (2001)

    Google Scholar 

  39. Zuazua E.: Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. 78(5), 523–563 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université de Versailles Saint Quentin en Yvelines, 45 avenue des États Unis, 78000, Versailles, France

    S. Ervedoza

  2. IMDEA Matemáticas & Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma, 28049, Madrid, Spain

    E. Zuazua

Authors
  1. S. Ervedoza
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  2. E. Zuazua
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Correspondence to S. Ervedoza.

Additional information

This work started while the first author was visiting the Department of Mathematics of the Universidad Autónoma de Madrid, in the frame of the European program “New materials, adaptive systems and their nonlinearities: modeling, control and numerical simulation” HPRN-CT-2002-00284. The work was finished while both authors visited the Isaac Newton Institute of Cambridge within the Program “Highly Oscillatory Problems”.

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Ervedoza, S., Zuazua, E. Perfectly matched layers in 1-d : energy decay for continuous and semi-discrete waves. Numer. Math. 109, 597–634 (2008). https://doi.org/10.1007/s00211-008-0153-y

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  • DOI: https://doi.org/10.1007/s00211-008-0153-y

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