Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction

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March 2011, 4(1): 295-316. doi: 10.3934/krm.2011.4.295

Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction

Shi Jin ^1, and Dongsheng Yin ^2,

1.	Department of Mathematics, University of Wisconsin, Madison, WI 53706
2.	Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

Received August 2010 Revised November 2010 Published January 2011

Abstract
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We construct a numerical scheme based on the Liouville equation of geometric optics coupled with the Geometric Theory of Diffraction (GTD) to simulate the high frequency linear waves diffracted by a corner. While the reflection boundary conditions are used at the boundary, a diffraction condition, based on the GTD theory, is introduced at the vertex. These conditions are built into the numerical flux for the discretization of the geometrical optics Liouville equation. Numerical experiments are used to verify the validity and accuracy of this new Eulerian numerical method which is able to capture the physical observable of high frequency and diffracted waves without fully resolving the high frequency numerically.

Keywords: Geometrical Theory of Diffraction, Liouville equation, High frequency wave..

Mathematics Subject Classification: Primary: 65M06, 78A45, 35L05; Secondary: 65Z05, 34E0.

Citation: Shi Jin, Dongsheng Yin. Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction. Kinetic and Related Models, 2011, 4 (1) : 295-316. doi: 10.3934/krm.2011.4.295

References:

[1]	J.-D. Benamou, An introduction to Eulerian geometrical optics(1992-2002), J. Sci. Comp., 19 (2001), 63-93. doi: 10.1023/A:1025339522111.
[2]	A. K. Bhattacharyya, "High-Frequency Electromagnetic Techniques: Recent Advances and Application," John Wiley $&$ Sons, Inc., 1995.
[3]	Y. Brenier and E. Grenier, Strickly particles and scalar conservation laws, SIAM, J. Num. Anal., 38 (1998), 2317-2328. doi: 10.1137/S0036142997317353.
[4]	R. N. Buchal and J. B. Keller, Boundary layer problems in diffraction theory, Comm. Pure Appl. Math., 13 (1960), 85-114. doi: 10.1002/cpa.3160130109.
[5]	V. Cĕrvený, "Seismic Ray Theory," Cambridge University Press, 2001.
[6]	L.-T. Cheng, H.-L. Liu and S. Osher, Computational high-frequency wave propagation using the Level Set method, with applications to the semi-classical limit of Schrödinger equations, Comm. Math. Sci., 1 (2003), 593-621.
[7]	G. Cohen, "Higher-Order Numerical Methods for Transient Wave Equations," Springer, Berlin; New York, 2002.
[8]	M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.
[9]	G. A. Deschamps, High frequency diffraction by wedges, IEEE Transactions on Antennas and Propagation. AP-33 (1985), 357-368. doi: 10.1109/TAP.1985.1143598.
[10]	B. Engquist and O. Runborg, Computational high frequency wave propagation, Acta Numerica, 12 (2003), 181-266. doi: 10.1017/S0962492902000119.
[11]	B. Engquist, O. Runborg, and A.-K. Tornberg, High frequency wave propagation by the segment projection method, J. Comput. Phys., 178 (2002), 373-390. doi: 10.1006/jcph.2002.7033.
[12]	B. Engquist, A. -K. Tornberg and R. Tsai, Discretization of dirac delta functions in level set methods, J. Comput. Phys., 207 (2005), 28-51. doi: 10.1016/j.jcp.2004.09.018.
[13]	E. Fatemi, B. Engquist and S. Osher, Numerical solution of the high frequency asymptotic expansion for the scalar wave equation, J. Comput. Phys., 120 (1995), 145-155. doi: 10.1006/jcph.1995.1154.
[14]	S. Fomel and J. A. Sethian, Fast phase space computation of multiple arrivals, Proc. Natl. Acad. Sci. USA, 99 (2002), 7329-7334. doi: 10.1073/pnas.102476599.
[15]	L. Gosse and N. J. Mauser, Multiphase semicalssical approximation of an electron in a one-dimensional crystalline lattice - III. From ab initio models to WKB for Schrödinger-Poisson, J. Comput. Phys., 211 (2006), 326-346. doi: 10.1016/j.jcp.2005.05.020.
[16]	S. Jin and X. Li, Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner, Physics D, 182 (2003), 46-85. doi: 10.1016/S0167-2789(03)00124-6.
[17]	S. Jin and X. Liao, A Hamiltonian-preserving scheme for high frequency elastic waves in heterogeneous media, J. Hyperbolic Diff Eqn., 3 (2006), 741-777. doi: 10.1142/S0219891606000999.
[18]	S. Jin, H. L. Liu, S. Osher and R. Tsai, Computing multi-valued physical observables for high frequency limit of symmetric hyperbolic systems, J. Comp. Phys., 210 (2005), 497-518. doi: 10.1016/j.jcp.2005.04.020.
[19]	S. Jin and S. Osher, A level set method for the computation of multi-valued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations, Comm. Math. Sci., 1 (2003), 575-591.
[20]	S. Jin and X. Wen, Hamiltonian-preserving scheme for the Liouville equation with discontinuous potentials, Comm. Math. Sci., 3 (2005), 285-315.
[21]	S. Jin and X. Wen, A Hamiltonian-preserving scheme for the Liouville equation of geometric optics with partial transmissions and reflections, SIAM J. Num. Anal., 44 (2006), 1801-1828. doi: 10.1137/050631343.
[22]	S. Jin and X. Wen, Computation of transmissions and reflections in geometric optics via the reduced Liouville equation, Wave Motion, 43 (2006), 667-688. doi: 10.1016/j.wavemoti.2006.06.001.
[23]	S. Jin, H. Wu and X. Yang, Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Comm. Math. Sci., 6 (2008), 995-1020.
[24]	S. Jin and D. S. Yin, Computational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction, J. Comput. Phys., 227 (2008), 6106-6139. doi: 10.1016/j.jcp.2008.02.029.
[25]	S. Jin and D. S. Yin, Computation of high frequency wave diffraction by a half plane via the Liouville equation and geometric theory of diffraction, Communications in Computational Physics, 4 (2008), 1106-1128.
[26]	J. B. Keller, Geometric theory of diffraction, J. Opt. Soc. of America, 52 (1962), 116-130. doi: 10.1364/JOSA.52.000116.
[27]	J. B. Keller and R. Lewis, Asymptotic methods for partial differential equations: The reduced wave equation and maxwell's equations, In "Surveys in Applied Mathematics"(eds. D. McLaughlin J. B. Keller and G. Papanicolaou), Plenum Press, New York, 1995.
[28]	R. G. Kouyoumjian and P. H. Parthak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proc. Of the IEEE, 62 (1974), 1448-1461. doi: 10.1109/PROC.1974.9651.
[29]	R. LeVeque, "Numerical Methods for Conservation Laws," Birkhauser, 1992.
[30]	L. Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary, J. Math. Pures Appl., 79 (2000), 227-269. doi: 10.1016/S0021-7824(00)00158-6.
[31]	M. Motamed and O. Runborg, A fast phase space method for computing creeping rays, J. Comput. Phys., 219 (2006), 276-295. doi: 10.1016/j.jcp.2006.03.024.
[32]	M. Motamed and O. Runborg, A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems, Commun. Math. Sci., 5 (2007), 617-648.
[33]	S. Osher, L. T. Cheng, M. Kang, H. Shim and Y. -H. Tsai, Geometric optics in a phase-space-based level set and Eulerian framework, J. Comput. Phys., 179 (2002), 622-648. doi: 10.1006/jcph.2002.7080.
[34]	L. Ryzhik, G. Papanicolaou and J. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370. doi: 10.1016/S0165-2125(96)00021-2.
[35]	C. Sparber, N. Mauser and P. A. Markowich, Wigner functions vs. WKB techniques in multivalued geometric optics, J. Asympt. Anal., 33 (2003), 153-187.
[36]	P. Smereka, The numerical approximation of a delta function with application to level set methods, J. Comput. Phys., 211 (2006), 77-90. doi: 10.1016/j.jcp.2005.05.005.
[37]	N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields, J. Comp. Phys., 228 (2009), 8856-8871. doi: 10.1016/j.jcp.2009.08.028.
[38]	J. D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys., 220 (2007), 915-931. doi: 10.1016/j.jcp.2006.05.037.
[39]	X. Wen, High order numerical methods to a type of delta function integrals, J. Comput. Phys., 226 (2007), 1952-1967. doi: 10.1016/j.jcp.2007.06.025.
[40]	X. Wen, High order numerical methods to two dimensional delta function integrals in level set methods, J. Comput. Phys., 228 (2009), 4273-4290. doi: 10.1016/j.jcp.2009.03.004.
[41]	X. Wen, High order numerical methods to three dimensional delta function integrals in level set methods, SIAM J. Sci. Comput., 32 (2010), 1288-1309. doi: 10.1137/090758295.
[42]	L. Ying and E. J. Candés, Fast geodesics computation with the phase flow method, J. Comput. Phys., 220 (2006), 6-18. doi: 10.1016/j.jcp.2006.07.032.

show all references

References:

[1]	J.-D. Benamou, An introduction to Eulerian geometrical optics(1992-2002), J. Sci. Comp., 19 (2001), 63-93. doi: 10.1023/A:1025339522111.
[2]	A. K. Bhattacharyya, "High-Frequency Electromagnetic Techniques: Recent Advances and Application," John Wiley $&$ Sons, Inc., 1995.
[3]	Y. Brenier and E. Grenier, Strickly particles and scalar conservation laws, SIAM, J. Num. Anal., 38 (1998), 2317-2328. doi: 10.1137/S0036142997317353.
[4]	R. N. Buchal and J. B. Keller, Boundary layer problems in diffraction theory, Comm. Pure Appl. Math., 13 (1960), 85-114. doi: 10.1002/cpa.3160130109.
[5]	V. Cĕrvený, "Seismic Ray Theory," Cambridge University Press, 2001.
[6]	L.-T. Cheng, H.-L. Liu and S. Osher, Computational high-frequency wave propagation using the Level Set method, with applications to the semi-classical limit of Schrödinger equations, Comm. Math. Sci., 1 (2003), 593-621.
[7]	G. Cohen, "Higher-Order Numerical Methods for Transient Wave Equations," Springer, Berlin; New York, 2002.
[8]	M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.
[9]	G. A. Deschamps, High frequency diffraction by wedges, IEEE Transactions on Antennas and Propagation. AP-33 (1985), 357-368. doi: 10.1109/TAP.1985.1143598.
[10]	B. Engquist and O. Runborg, Computational high frequency wave propagation, Acta Numerica, 12 (2003), 181-266. doi: 10.1017/S0962492902000119.
[11]	B. Engquist, O. Runborg, and A.-K. Tornberg, High frequency wave propagation by the segment projection method, J. Comput. Phys., 178 (2002), 373-390. doi: 10.1006/jcph.2002.7033.
[12]	B. Engquist, A. -K. Tornberg and R. Tsai, Discretization of dirac delta functions in level set methods, J. Comput. Phys., 207 (2005), 28-51. doi: 10.1016/j.jcp.2004.09.018.
[13]	E. Fatemi, B. Engquist and S. Osher, Numerical solution of the high frequency asymptotic expansion for the scalar wave equation, J. Comput. Phys., 120 (1995), 145-155. doi: 10.1006/jcph.1995.1154.
[14]	S. Fomel and J. A. Sethian, Fast phase space computation of multiple arrivals, Proc. Natl. Acad. Sci. USA, 99 (2002), 7329-7334. doi: 10.1073/pnas.102476599.
[15]	L. Gosse and N. J. Mauser, Multiphase semicalssical approximation of an electron in a one-dimensional crystalline lattice - III. From ab initio models to WKB for Schrödinger-Poisson, J. Comput. Phys., 211 (2006), 326-346. doi: 10.1016/j.jcp.2005.05.020.
[16]	S. Jin and X. Li, Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner, Physics D, 182 (2003), 46-85. doi: 10.1016/S0167-2789(03)00124-6.
[17]	S. Jin and X. Liao, A Hamiltonian-preserving scheme for high frequency elastic waves in heterogeneous media, J. Hyperbolic Diff Eqn., 3 (2006), 741-777. doi: 10.1142/S0219891606000999.
[18]	S. Jin, H. L. Liu, S. Osher and R. Tsai, Computing multi-valued physical observables for high frequency limit of symmetric hyperbolic systems, J. Comp. Phys., 210 (2005), 497-518. doi: 10.1016/j.jcp.2005.04.020.
[19]	S. Jin and S. Osher, A level set method for the computation of multi-valued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations, Comm. Math. Sci., 1 (2003), 575-591.
[20]	S. Jin and X. Wen, Hamiltonian-preserving scheme for the Liouville equation with discontinuous potentials, Comm. Math. Sci., 3 (2005), 285-315.
[21]	S. Jin and X. Wen, A Hamiltonian-preserving scheme for the Liouville equation of geometric optics with partial transmissions and reflections, SIAM J. Num. Anal., 44 (2006), 1801-1828. doi: 10.1137/050631343.
[22]	S. Jin and X. Wen, Computation of transmissions and reflections in geometric optics via the reduced Liouville equation, Wave Motion, 43 (2006), 667-688. doi: 10.1016/j.wavemoti.2006.06.001.
[23]	S. Jin, H. Wu and X. Yang, Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Comm. Math. Sci., 6 (2008), 995-1020.
[24]	S. Jin and D. S. Yin, Computational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction, J. Comput. Phys., 227 (2008), 6106-6139. doi: 10.1016/j.jcp.2008.02.029.
[25]	S. Jin and D. S. Yin, Computation of high frequency wave diffraction by a half plane via the Liouville equation and geometric theory of diffraction, Communications in Computational Physics, 4 (2008), 1106-1128.
[26]	J. B. Keller, Geometric theory of diffraction, J. Opt. Soc. of America, 52 (1962), 116-130. doi: 10.1364/JOSA.52.000116.
[27]	J. B. Keller and R. Lewis, Asymptotic methods for partial differential equations: The reduced wave equation and maxwell's equations, In "Surveys in Applied Mathematics"(eds. D. McLaughlin J. B. Keller and G. Papanicolaou), Plenum Press, New York, 1995.
[28]	R. G. Kouyoumjian and P. H. Parthak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proc. Of the IEEE, 62 (1974), 1448-1461. doi: 10.1109/PROC.1974.9651.
[29]	R. LeVeque, "Numerical Methods for Conservation Laws," Birkhauser, 1992.
[30]	L. Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary, J. Math. Pures Appl., 79 (2000), 227-269. doi: 10.1016/S0021-7824(00)00158-6.
[31]	M. Motamed and O. Runborg, A fast phase space method for computing creeping rays, J. Comput. Phys., 219 (2006), 276-295. doi: 10.1016/j.jcp.2006.03.024.
[32]	M. Motamed and O. Runborg, A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems, Commun. Math. Sci., 5 (2007), 617-648.
[33]	S. Osher, L. T. Cheng, M. Kang, H. Shim and Y. -H. Tsai, Geometric optics in a phase-space-based level set and Eulerian framework, J. Comput. Phys., 179 (2002), 622-648. doi: 10.1006/jcph.2002.7080.
[34]	L. Ryzhik, G. Papanicolaou and J. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370. doi: 10.1016/S0165-2125(96)00021-2.
[35]	C. Sparber, N. Mauser and P. A. Markowich, Wigner functions vs. WKB techniques in multivalued geometric optics, J. Asympt. Anal., 33 (2003), 153-187.
[36]	P. Smereka, The numerical approximation of a delta function with application to level set methods, J. Comput. Phys., 211 (2006), 77-90. doi: 10.1016/j.jcp.2005.05.005.
[37]	N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields, J. Comp. Phys., 228 (2009), 8856-8871. doi: 10.1016/j.jcp.2009.08.028.
[38]	J. D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys., 220 (2007), 915-931. doi: 10.1016/j.jcp.2006.05.037.
[39]	X. Wen, High order numerical methods to a type of delta function integrals, J. Comput. Phys., 226 (2007), 1952-1967. doi: 10.1016/j.jcp.2007.06.025.
[40]	X. Wen, High order numerical methods to two dimensional delta function integrals in level set methods, J. Comput. Phys., 228 (2009), 4273-4290. doi: 10.1016/j.jcp.2009.03.004.
[41]	X. Wen, High order numerical methods to three dimensional delta function integrals in level set methods, SIAM J. Sci. Comput., 32 (2010), 1288-1309. doi: 10.1137/090758295.
[42]	L. Ying and E. J. Candés, Fast geodesics computation with the phase flow method, J. Comput. Phys., 220 (2006), 6-18. doi: 10.1016/j.jcp.2006.07.032.

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