# American Institute of Mathematical Sciences

2015, 2015(special): 1000-1008. doi: 10.3934/proc.2015.1000

## Absorbing boundary conditions for the Westervelt equation

 1 Imperial College London, Department of Mathematics, London, SW7 2AZ, United Kingdom 2 Alpen-Adria-Universität Klagenfurt, Institute of Mathematics, Klagenfurt, A-9020, Austria

Received  September 2014 Revised  December 2014 Published  November 2015

The focus of this work is on the construction of a family of nonlinear absorbing boundary conditions for the Westervelt equation in one and two space dimensions. The principal ingredient used in the design of such conditions is pseudo-differential calculus. This approach enables to develop high order boundary conditions in a consistent way which are typically more accurate than their low order analogs. Under the hypothesis of small initial data, we establish local well-posedness for the Westervelt equation with the absorbing boundary conditions. The performed numerical experiments illustrate the efficiency of the proposed boundary conditions for different regimes of wave propagation.
Citation: Igor Shevchenko, Barbara Kaltenbacher. Absorbing boundary conditions for the Westervelt equation. Conference Publications, 2015, 2015 (special) : 1000-1008. doi: 10.3934/proc.2015.1000
##### References:
 [1] P. J. Westervelt, Parametric acoustic array, J. Acoust. Soc. Am., 35 (1963), 535-537. [2] M. Averkiou and R. Cleveland, Modeling of an electrohydraulic lithotripter with the KZK equation, J. Acoust. Soc. Am., 106 (1999), 102-112. [3] C. Le Floch and M. Fink, Ultrasonic mapping of temperature in hyperthermia: the thermal lens effect, in Proceedings of 1997 IEEE Ultrasonics Symposium, (1997), 1301-1304. [4] C. Simon, P. VanBaren and E. Ebbini, Two-dimensional temperature estimation using diagnostic ultrasound, IEEE Trans. Ultrason. Ferr., 45 (1998), 1088-1099. [5] M. Pernot, K. Waters, J. Bercoff, M. Tanter and M. Fink, Reduction of the thermo-acoustic lens effect during ultrasound-based temperature estimation, in Proceedings of 2002 IEEE Ultrasonics Symposium, (2002), 1447-1450. [6] C. Le Floch, M. Tanter and M. Fink, Self-defocusing in ultrasonic hyperthermia: Experiment and simulation, Appl. Phys. Lett., 74 (1999), 3062-3064. [7] I. Hallaj, R. Cleveland and K. Hynynen, Simulations of the thermo-acoustic lens effect during focused ultrasound surgery, J. Acoust. Soc. Am., 109 (2001), 2245-2253. [8] C. Connor and K. Hynynen, Bio-acoustic thermal lensing and nonlinear propagation in focused ultrasound surgery using large focal spots: a parametric study, Phys. Med. Biol., 47 (2002), 1911-1928. [9] S. Tsynkov, Numerical solution of problems on unbounded domains. A review, Appl. Numer. Math., 27 (1998), 465-532. [10] T. Hagstrom, New results on absorbing layers and radiation boundary conditions, in Topics in computational wave propagation. Direct and inverse problems. Lect. Notes Comput. Sci. Eng. (M. Ainsworth, P. Davies, D. Duncan, P. Martin and B. Rynne), Springer-Verlag, (2003), 1-42. [11] D. Givoli, High-order local non-reflecting boundary conditions: a review, Wave motion, 39 (2004), 319-326. [12] D. Givoli, Computational absorbing boundaries, in Computational Acoustics of Noise Propagation in Fluids (S. Marburg and B. Nolte), Springer-Verlag, (2008), 145-166. [13] B. Engquist and A. Majda, Radiation Boundary Conditions for Acoustic and Elastic Wave Calculations, Comm. Pure Appl. Math., 32 (1979), 313-357. [14] E. Bécache, D. Givoli and T. Hagstrom, High-order absorbing boundary conditions for anisotropic and convective wave equations, J. Comput. Phys., 229 (2010), 1099-1129. [15] G. W. Hedstrom, Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems, J. Comput. Phys., 30 (1979), 222-237. [16] J. Szeftel, Absorbing boundary conditions for nonlinear scalar partial differential equations, Comput. Method. Appl. M., 195 (2006), 3760-3775. [17] J. Zhang, Z. Xu and X. Wu, Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations: Two-dimensional case, Phys. Rev. E, 79 (2009), 046711-1-046711-8. [18] R. R. Paz, M. A. Storti and L. Garelli, Absorbing Boundary Condition for Nonlinear Hyperbolic Partial Differential Equations with Unknown Riemann Invariants, Fluid Mechanics (C), XXVIII (2009), 1593-1620. [19] J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Commun. Pure Appl. Math., 18 (1965), 269-305. [20] L. Hörmander, Pseudo-differential operators, Commun. Pure Appl. Math., 18 (1965), 501-517. [21] L. Nirenberg, Lectures on linear partial differential equations, CBMS.: Regional Conference Series in Mathematics, 17 (1973), 1-58. [22] L. Hörmander, The analysis of linear partial differential operators III: Pseudo-Differential Operators, Springer-Verlag, Berlin Heidelberg, 1985. [23] M. W. Wong, An introduction to pseudo-differential operators, World Scientific Publishing, Singapore, 1999. [24] A. Majda and S. Osher, Reflection of singularities at the boundary, Comm. Pure Appl. Math., 28 (1975), 479-499. [25] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651. [26] C. Clason, B. Kaltenbacher and S. Veljovic, Boundary optimal control of the Westervelt and the Kuznetsov equations, J. Math. Anal. Appl., 356 (2009), 738-751. [27] T. Ha-Duong and P. Joly, On the Stability Analysis of Boundary Conditions for the Wave Equation by Energy Methods. Part I: The Homogeneous Case, Math. Comp., 62 (1994), 539-563. [28] B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation., Discret. Contin. Dyn. S., 2 (2009), 503-525. [29] B. Kaltenbacher and I. Lasiecka, Well-Posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, in The Proceedings of the 8th AIMS Conference, (2011), 763-773. [30] I. Shevchenko and B. Wohlmuth, Self-adapting absorbing boundary conditions for the wave equation, Wave Motion, 49 (2012), 461-473. [31] T. Hughes, The finite element method: linear static and dynamic finite element analysis, Dover Publications, 2000. [32] B. Kaltenbacher and I. Shevchenko, Absorbing boundary conditions for the Westervelt equation,, arXiv:1408.5031, ().

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##### References:
 [1] P. J. Westervelt, Parametric acoustic array, J. Acoust. Soc. Am., 35 (1963), 535-537. [2] M. Averkiou and R. Cleveland, Modeling of an electrohydraulic lithotripter with the KZK equation, J. Acoust. Soc. Am., 106 (1999), 102-112. [3] C. Le Floch and M. Fink, Ultrasonic mapping of temperature in hyperthermia: the thermal lens effect, in Proceedings of 1997 IEEE Ultrasonics Symposium, (1997), 1301-1304. [4] C. Simon, P. VanBaren and E. Ebbini, Two-dimensional temperature estimation using diagnostic ultrasound, IEEE Trans. Ultrason. Ferr., 45 (1998), 1088-1099. [5] M. Pernot, K. Waters, J. Bercoff, M. Tanter and M. Fink, Reduction of the thermo-acoustic lens effect during ultrasound-based temperature estimation, in Proceedings of 2002 IEEE Ultrasonics Symposium, (2002), 1447-1450. [6] C. Le Floch, M. Tanter and M. Fink, Self-defocusing in ultrasonic hyperthermia: Experiment and simulation, Appl. Phys. Lett., 74 (1999), 3062-3064. [7] I. Hallaj, R. Cleveland and K. Hynynen, Simulations of the thermo-acoustic lens effect during focused ultrasound surgery, J. Acoust. Soc. Am., 109 (2001), 2245-2253. [8] C. Connor and K. Hynynen, Bio-acoustic thermal lensing and nonlinear propagation in focused ultrasound surgery using large focal spots: a parametric study, Phys. Med. Biol., 47 (2002), 1911-1928. [9] S. Tsynkov, Numerical solution of problems on unbounded domains. A review, Appl. Numer. Math., 27 (1998), 465-532. [10] T. Hagstrom, New results on absorbing layers and radiation boundary conditions, in Topics in computational wave propagation. Direct and inverse problems. Lect. Notes Comput. Sci. Eng. (M. Ainsworth, P. Davies, D. Duncan, P. Martin and B. Rynne), Springer-Verlag, (2003), 1-42. [11] D. Givoli, High-order local non-reflecting boundary conditions: a review, Wave motion, 39 (2004), 319-326. [12] D. Givoli, Computational absorbing boundaries, in Computational Acoustics of Noise Propagation in Fluids (S. Marburg and B. Nolte), Springer-Verlag, (2008), 145-166. [13] B. Engquist and A. Majda, Radiation Boundary Conditions for Acoustic and Elastic Wave Calculations, Comm. Pure Appl. Math., 32 (1979), 313-357. [14] E. Bécache, D. Givoli and T. Hagstrom, High-order absorbing boundary conditions for anisotropic and convective wave equations, J. Comput. Phys., 229 (2010), 1099-1129. [15] G. W. Hedstrom, Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems, J. Comput. Phys., 30 (1979), 222-237. [16] J. Szeftel, Absorbing boundary conditions for nonlinear scalar partial differential equations, Comput. Method. Appl. M., 195 (2006), 3760-3775. [17] J. Zhang, Z. Xu and X. Wu, Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations: Two-dimensional case, Phys. Rev. E, 79 (2009), 046711-1-046711-8. [18] R. R. Paz, M. A. Storti and L. Garelli, Absorbing Boundary Condition for Nonlinear Hyperbolic Partial Differential Equations with Unknown Riemann Invariants, Fluid Mechanics (C), XXVIII (2009), 1593-1620. [19] J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Commun. Pure Appl. Math., 18 (1965), 269-305. [20] L. Hörmander, Pseudo-differential operators, Commun. Pure Appl. Math., 18 (1965), 501-517. [21] L. Nirenberg, Lectures on linear partial differential equations, CBMS.: Regional Conference Series in Mathematics, 17 (1973), 1-58. [22] L. Hörmander, The analysis of linear partial differential operators III: Pseudo-Differential Operators, Springer-Verlag, Berlin Heidelberg, 1985. [23] M. W. Wong, An introduction to pseudo-differential operators, World Scientific Publishing, Singapore, 1999. [24] A. Majda and S. Osher, Reflection of singularities at the boundary, Comm. Pure Appl. Math., 28 (1975), 479-499. [25] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651. [26] C. Clason, B. Kaltenbacher and S. Veljovic, Boundary optimal control of the Westervelt and the Kuznetsov equations, J. Math. Anal. Appl., 356 (2009), 738-751. [27] T. Ha-Duong and P. Joly, On the Stability Analysis of Boundary Conditions for the Wave Equation by Energy Methods. Part I: The Homogeneous Case, Math. Comp., 62 (1994), 539-563. [28] B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation., Discret. Contin. Dyn. S., 2 (2009), 503-525. [29] B. Kaltenbacher and I. Lasiecka, Well-Posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, in The Proceedings of the 8th AIMS Conference, (2011), 763-773. [30] I. Shevchenko and B. Wohlmuth, Self-adapting absorbing boundary conditions for the wave equation, Wave Motion, 49 (2012), 461-473. [31] T. Hughes, The finite element method: linear static and dynamic finite element analysis, Dover Publications, 2000. [32] B. Kaltenbacher and I. Shevchenko, Absorbing boundary conditions for the Westervelt equation,, arXiv:1408.5031, ().
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