# American Institute of Mathematical Sciences

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

 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

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.
Citation: Shi Jin, Dongsheng Yin. Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction. Kinetic & Related Models, 2011, 4 (1) : 295-316. doi: 10.3934/krm.2011.4.295
##### References:
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Cohen, "Higher-Order Numerical Methods for Transient Wave Equations,", Springer, (2002). Google Scholar [8] M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations,, Trans. Amer. Math. Soc., 277 (1983), 1. doi: 10.1090/S0002-9947-1983-0690039-8. Google Scholar [9] G. A. Deschamps, High frequency diffraction by wedges,, IEEE Transactions on Antennas and Propagation. AP-\textbf{33} (1985), 33 (1985), 357. doi: 10.1109/TAP.1985.1143598. Google Scholar [10] B. Engquist and O. Runborg, Computational high frequency wave propagation,, Acta Numerica, 12 (2003), 181. doi: 10.1017/S0962492902000119. Google Scholar [11] B. Engquist, O. Runborg, and A.-K. Tornberg, High frequency wave propagation by the segment projection method,, J. Comput. Phys., 178 (2002), 373. doi: 10.1006/jcph.2002.7033. Google Scholar [12] B. Engquist, A. -K. Tornberg and R. Tsai, Discretization of dirac delta functions in level set methods,, J. Comput. Phys., 207 (2005), 28. doi: 10.1016/j.jcp.2004.09.018. Google Scholar [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. doi: 10.1006/jcph.1995.1154. Google Scholar [14] S. Fomel and J. A. Sethian, Fast phase space computation of multiple arrivals,, Proc. Natl. Acad. Sci. USA, 99 (2002), 7329. doi: 10.1073/pnas.102476599. Google Scholar [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. doi: 10.1016/j.jcp.2005.05.020. Google Scholar [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. doi: 10.1016/S0167-2789(03)00124-6. Google Scholar [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. doi: 10.1142/S0219891606000999. Google Scholar [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. doi: 10.1016/j.jcp.2005.04.020. Google Scholar [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. Google Scholar [20] S. Jin and X. Wen, Hamiltonian-preserving scheme for the Liouville equation with discontinuous potentials,, Comm. Math. Sci., 3 (2005), 285. Google Scholar [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. doi: 10.1137/050631343. 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Soc. of America, 52 (1962), 116. doi: 10.1364/JOSA.52.000116. Google Scholar [27] J. B. Keller and R. Lewis, Asymptotic methods for partial differential equations: The reduced wave equation and maxwell's equations,, In, (1995). Google Scholar [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. doi: 10.1109/PROC.1974.9651. Google Scholar [29] R. LeVeque, "Numerical Methods for Conservation Laws,", Birkhauser, (1992). Google Scholar [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. doi: 10.1016/S0021-7824(00)00158-6. Google Scholar [31] M. Motamed and O. Runborg, A fast phase space method for computing creeping rays,, J. Comput. Phys., 219 (2006), 276. doi: 10.1016/j.jcp.2006.03.024. Google Scholar [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. Google Scholar [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. doi: 10.1006/jcph.2002.7080. Google Scholar [34] L. Ryzhik, G. Papanicolaou and J. Keller, Transport equations for elastic and other waves in random media,, Wave Motion, 24 (1996), 327. doi: 10.1016/S0165-2125(96)00021-2. Google Scholar [35] C. Sparber, N. Mauser and P. A. Markowich, Wigner functions vs. WKB techniques in multivalued geometric optics,, J. Asympt. Anal., 33 (2003), 153. Google Scholar [36] P. Smereka, The numerical approximation of a delta function with application to level set methods,, J. Comput. Phys., 211 (2006), 77. doi: 10.1016/j.jcp.2005.05.005. Google Scholar [37] N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields,, J. Comp. Phys., 228 (2009), 8856. doi: 10.1016/j.jcp.2009.08.028. Google Scholar [38] J. D. Towers, Two methods for discretizing a delta function supported on a level set,, J. Comput. Phys., 220 (2007), 915. doi: 10.1016/j.jcp.2006.05.037. Google Scholar [39] X. Wen, High order numerical methods to a type of delta function integrals,, J. Comput. Phys., 226 (2007), 1952. doi: 10.1016/j.jcp.2007.06.025. Google Scholar [40] X. Wen, High order numerical methods to two dimensional delta function integrals in level set methods,, J. Comput. Phys., 228 (2009), 4273. doi: 10.1016/j.jcp.2009.03.004. Google Scholar [41] X. Wen, High order numerical methods to three dimensional delta function integrals in level set methods,, SIAM J. Sci. Comput., 32 (2010), 1288. doi: 10.1137/090758295. Google Scholar [42] L. Ying and E. J. Candés, Fast geodesics computation with the phase flow method,, J. Comput. Phys., 220 (2006), 6. doi: 10.1016/j.jcp.2006.07.032. Google Scholar

show all references

##### References:
 [1] J.-D. Benamou, An introduction to Eulerian geometrical optics(1992-2002),, J. Sci. Comp., 19 (2001), 63. doi: 10.1023/A:1025339522111. Google Scholar [2] A. K. Bhattacharyya, "High-Frequency Electromagnetic Techniques: Recent Advances and Application,", John Wiley $&$ Sons, (1995). Google Scholar [3] Y. Brenier and E. Grenier, Strickly particles and scalar conservation laws,, SIAM, 38 (1998), 2317. doi: 10.1137/S0036142997317353. Google Scholar [4] R. N. Buchal and J. B. Keller, Boundary layer problems in diffraction theory,, Comm. Pure Appl. Math., 13 (1960), 85. doi: 10.1002/cpa.3160130109. Google Scholar [5] V. Cĕrvený, "Seismic Ray Theory,", Cambridge University Press, (2001). Google Scholar [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. Google Scholar [7] G. Cohen, "Higher-Order Numerical Methods for Transient Wave Equations,", Springer, (2002). Google Scholar [8] M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations,, Trans. Amer. Math. Soc., 277 (1983), 1. doi: 10.1090/S0002-9947-1983-0690039-8. Google Scholar [9] G. A. Deschamps, High frequency diffraction by wedges,, IEEE Transactions on Antennas and Propagation. AP-\textbf{33} (1985), 33 (1985), 357. doi: 10.1109/TAP.1985.1143598. Google Scholar [10] B. Engquist and O. Runborg, Computational high frequency wave propagation,, Acta Numerica, 12 (2003), 181. doi: 10.1017/S0962492902000119. Google Scholar [11] B. Engquist, O. Runborg, and A.-K. Tornberg, High frequency wave propagation by the segment projection method,, J. Comput. Phys., 178 (2002), 373. doi: 10.1006/jcph.2002.7033. Google Scholar [12] B. Engquist, A. -K. Tornberg and R. Tsai, Discretization of dirac delta functions in level set methods,, J. Comput. Phys., 207 (2005), 28. doi: 10.1016/j.jcp.2004.09.018. Google Scholar [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. doi: 10.1006/jcph.1995.1154. Google Scholar [14] S. Fomel and J. A. Sethian, Fast phase space computation of multiple arrivals,, Proc. Natl. Acad. Sci. USA, 99 (2002), 7329. doi: 10.1073/pnas.102476599. Google Scholar [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. doi: 10.1016/j.jcp.2005.05.020. Google Scholar [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. doi: 10.1016/S0167-2789(03)00124-6. Google Scholar [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. doi: 10.1142/S0219891606000999. Google Scholar [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. doi: 10.1016/j.jcp.2005.04.020. Google Scholar [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. Google Scholar [20] S. Jin and X. Wen, Hamiltonian-preserving scheme for the Liouville equation with discontinuous potentials,, Comm. Math. Sci., 3 (2005), 285. Google Scholar [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. doi: 10.1137/050631343. Google Scholar [22] S. Jin and X. Wen, Computation of transmissions and reflections in geometric optics via the reduced Liouville equation,, Wave Motion, 43 (2006), 667. doi: 10.1016/j.wavemoti.2006.06.001. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jcp.2008.02.029. Google Scholar [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. Google Scholar [26] J. B. Keller, Geometric theory of diffraction,, J. Opt. Soc. of America, 52 (1962), 116. doi: 10.1364/JOSA.52.000116. Google Scholar [27] J. B. Keller and R. Lewis, Asymptotic methods for partial differential equations: The reduced wave equation and maxwell's equations,, In, (1995). Google Scholar [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. doi: 10.1109/PROC.1974.9651. Google Scholar [29] R. LeVeque, "Numerical Methods for Conservation Laws,", Birkhauser, (1992). Google Scholar [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. doi: 10.1016/S0021-7824(00)00158-6. Google Scholar [31] M. Motamed and O. Runborg, A fast phase space method for computing creeping rays,, J. Comput. Phys., 219 (2006), 276. doi: 10.1016/j.jcp.2006.03.024. Google Scholar [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. Google Scholar [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. doi: 10.1006/jcph.2002.7080. Google Scholar [34] L. Ryzhik, G. Papanicolaou and J. Keller, Transport equations for elastic and other waves in random media,, Wave Motion, 24 (1996), 327. doi: 10.1016/S0165-2125(96)00021-2. Google Scholar [35] C. Sparber, N. Mauser and P. A. Markowich, Wigner functions vs. WKB techniques in multivalued geometric optics,, J. Asympt. Anal., 33 (2003), 153. Google Scholar [36] P. Smereka, The numerical approximation of a delta function with application to level set methods,, J. Comput. Phys., 211 (2006), 77. doi: 10.1016/j.jcp.2005.05.005. Google Scholar [37] N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields,, J. Comp. Phys., 228 (2009), 8856. doi: 10.1016/j.jcp.2009.08.028. Google Scholar [38] J. D. Towers, Two methods for discretizing a delta function supported on a level set,, J. Comput. Phys., 220 (2007), 915. doi: 10.1016/j.jcp.2006.05.037. Google Scholar [39] X. Wen, High order numerical methods to a type of delta function integrals,, J. Comput. Phys., 226 (2007), 1952. doi: 10.1016/j.jcp.2007.06.025. Google Scholar [40] X. Wen, High order numerical methods to two dimensional delta function integrals in level set methods,, J. Comput. Phys., 228 (2009), 4273. doi: 10.1016/j.jcp.2009.03.004. Google Scholar [41] X. Wen, High order numerical methods to three dimensional delta function integrals in level set methods,, SIAM J. Sci. Comput., 32 (2010), 1288. doi: 10.1137/090758295. Google Scholar [42] L. Ying and E. J. Candés, Fast geodesics computation with the phase flow method,, J. Comput. Phys., 220 (2006), 6. doi: 10.1016/j.jcp.2006.07.032. Google Scholar
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