# 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. Google Scholar [2] M. Averkiou and R. Cleveland, Modeling of an electrohydraulic lithotripter with the KZK equation,, J. Acoust. Soc. Am., 106 (1999), 102. Google Scholar [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. Google Scholar [4] C. Simon, P. VanBaren and E. Ebbini, Two-dimensional temperature estimation using diagnostic ultrasound,, IEEE Trans. Ultrason. Ferr., 45 (1998), 1088. Google Scholar [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. Google Scholar [6] C. Le Floch, M. Tanter and M. Fink, Self-defocusing in ultrasonic hyperthermia: Experiment and simulation,, Appl. Phys. Lett., 74 (1999), 3062. Google Scholar [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. Google Scholar [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. Google Scholar [9] S. Tsynkov, Numerical solution of problems on unbounded domains. A review,, Appl. Numer. Math., 27 (1998), 465. Google Scholar [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, (2003), 1. Google Scholar [11] D. Givoli, High-order local non-reflecting boundary conditions: a review,, Wave motion, 39 (2004), 319. Google Scholar [12] D. Givoli, Computational absorbing boundaries,, in Computational Acoustics of Noise Propagation in Fluids (S. Marburg and B. Nolte), (2008), 145. Google Scholar [13] B. Engquist and A. Majda, Radiation Boundary Conditions for Acoustic and Elastic Wave Calculations,, Comm. Pure Appl. Math., 32 (1979), 313. Google Scholar [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. Google Scholar [15] G. W. Hedstrom, Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems,, J. Comput. Phys., 30 (1979), 222. Google Scholar [16] J. Szeftel, Absorbing boundary conditions for nonlinear scalar partial differential equations,, Comput. Method. Appl. M., 195 (2006), 3760. Google Scholar [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. Google Scholar [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. Google Scholar [19] J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators,, Commun. Pure Appl. Math., 18 (1965), 269. Google Scholar [20] L. Hörmander, Pseudo-differential operators,, Commun. Pure Appl. Math., 18 (1965), 501. Google Scholar [21] L. Nirenberg, Lectures on linear partial differential equations,, CBMS.: Regional Conference Series in Mathematics, 17 (1973), 1. Google Scholar [22] L. Hörmander, The analysis of linear partial differential operators III: Pseudo-Differential Operators,, Springer-Verlag, (1985). Google Scholar [23] M. W. Wong, An introduction to pseudo-differential operators,, World Scientific Publishing, (1999). Google Scholar [24] A. Majda and S. Osher, Reflection of singularities at the boundary,, Comm. Pure Appl. Math., 28 (1975), 479. Google Scholar [25] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves,, Math. Comp., 31 (1977), 629. Google Scholar [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. Google Scholar [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. Google Scholar [28] B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation.,, Discret. Contin. Dyn. S., 2 (2009), 503. Google Scholar [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. Google Scholar [30] I. Shevchenko and B. Wohlmuth, Self-adapting absorbing boundary conditions for the wave equation,, Wave Motion, 49 (2012), 461. Google Scholar [31] T. Hughes, The finite element method: linear static and dynamic finite element analysis,, Dover Publications, (2000). Google Scholar [32] B. Kaltenbacher and I. Shevchenko, Absorbing boundary conditions for the Westervelt equation,, arXiv:1408.5031, (). Google Scholar

show all references

##### References:
 [1] P. J. Westervelt, Parametric acoustic array,, J. Acoust. Soc. Am., 35 (1963), 535. Google Scholar [2] M. Averkiou and R. Cleveland, Modeling of an electrohydraulic lithotripter with the KZK equation,, J. Acoust. Soc. Am., 106 (1999), 102. Google Scholar [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. Google Scholar [4] C. Simon, P. VanBaren and E. Ebbini, Two-dimensional temperature estimation using diagnostic ultrasound,, IEEE Trans. Ultrason. Ferr., 45 (1998), 1088. Google Scholar [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. Google Scholar [6] C. Le Floch, M. Tanter and M. Fink, Self-defocusing in ultrasonic hyperthermia: Experiment and simulation,, Appl. Phys. Lett., 74 (1999), 3062. Google Scholar [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. Google Scholar [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. Google Scholar [9] S. Tsynkov, Numerical solution of problems on unbounded domains. A review,, Appl. Numer. Math., 27 (1998), 465. Google Scholar [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, (2003), 1. Google Scholar [11] D. Givoli, High-order local non-reflecting boundary conditions: a review,, Wave motion, 39 (2004), 319. Google Scholar [12] D. Givoli, Computational absorbing boundaries,, in Computational Acoustics of Noise Propagation in Fluids (S. Marburg and B. Nolte), (2008), 145. Google Scholar [13] B. Engquist and A. Majda, Radiation Boundary Conditions for Acoustic and Elastic Wave Calculations,, Comm. Pure Appl. Math., 32 (1979), 313. Google Scholar [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. Google Scholar [15] G. W. Hedstrom, Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems,, J. Comput. Phys., 30 (1979), 222. Google Scholar [16] J. Szeftel, Absorbing boundary conditions for nonlinear scalar partial differential equations,, Comput. Method. Appl. M., 195 (2006), 3760. Google Scholar [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. Google Scholar [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. Google Scholar [19] J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators,, Commun. Pure Appl. Math., 18 (1965), 269. Google Scholar [20] L. Hörmander, Pseudo-differential operators,, Commun. Pure Appl. Math., 18 (1965), 501. Google Scholar [21] L. Nirenberg, Lectures on linear partial differential equations,, CBMS.: Regional Conference Series in Mathematics, 17 (1973), 1. Google Scholar [22] L. Hörmander, The analysis of linear partial differential operators III: Pseudo-Differential Operators,, Springer-Verlag, (1985). Google Scholar [23] M. W. Wong, An introduction to pseudo-differential operators,, World Scientific Publishing, (1999). Google Scholar [24] A. Majda and S. Osher, Reflection of singularities at the boundary,, Comm. Pure Appl. Math., 28 (1975), 479. Google Scholar [25] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves,, Math. Comp., 31 (1977), 629. Google Scholar [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. Google Scholar [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. Google Scholar [28] B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation.,, Discret. Contin. Dyn. S., 2 (2009), 503. Google Scholar [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. Google Scholar [30] I. Shevchenko and B. Wohlmuth, Self-adapting absorbing boundary conditions for the wave equation,, Wave Motion, 49 (2012), 461. Google Scholar [31] T. Hughes, The finite element method: linear static and dynamic finite element analysis,, Dover Publications, (2000). Google Scholar [32] B. Kaltenbacher and I. Shevchenko, Absorbing boundary conditions for the Westervelt equation,, arXiv:1408.5031, (). Google Scholar
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