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Absorbing boundary conditions for the Westervelt equation
1.  Imperial College London, Department of Mathematics, London, SW7 2AZ, United Kingdom 
2.  AlpenAdriaUniversität Klagenfurt, Institute of Mathematics, Klagenfurt, A9020, Austria 
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, Twodimensional 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 thermoacoustic lens effect during ultrasoundbased temperature estimation,, in Proceedings of 2002 IEEE Ultrasonics Symposium, (2002), 1447. Google Scholar 
[6] 
C. Le Floch, M. Tanter and M. Fink, Selfdefocusing 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 thermoacoustic lens effect during focused ultrasound surgery,, J. Acoust. Soc. Am., 109 (2001), 2245. Google Scholar 
[8] 
C. Connor and K. Hynynen, Bioacoustic 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, Highorder local nonreflecting 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, Highorder 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: Twodimensional 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 pseudodifferential operators,, Commun. Pure Appl. Math., 18 (1965), 269. Google Scholar 
[20] 
L. Hörmander, Pseudodifferential 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: PseudoDifferential Operators,, SpringerVerlag, (1985). Google Scholar 
[23] 
M. W. Wong, An introduction to pseudodifferential 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. HaDuong 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, WellPosedness 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, Selfadapting 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, Twodimensional 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 thermoacoustic lens effect during ultrasoundbased temperature estimation,, in Proceedings of 2002 IEEE Ultrasonics Symposium, (2002), 1447. Google Scholar 
[6] 
C. Le Floch, M. Tanter and M. Fink, Selfdefocusing 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 thermoacoustic lens effect during focused ultrasound surgery,, J. Acoust. Soc. Am., 109 (2001), 2245. Google Scholar 
[8] 
C. Connor and K. Hynynen, Bioacoustic 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, Highorder local nonreflecting 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, Highorder 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: Twodimensional 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 pseudodifferential operators,, Commun. Pure Appl. Math., 18 (1965), 269. Google Scholar 
[20] 
L. Hörmander, Pseudodifferential 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: PseudoDifferential Operators,, SpringerVerlag, (1985). Google Scholar 
[23] 
M. W. Wong, An introduction to pseudodifferential 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. HaDuong 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, WellPosedness 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, Selfadapting 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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