Superposition of multi-valued solutions in high frequency wave dynamics

Journal of Scientific Computing

Volumn 35, Issue 2-3, 2008, Pages 192-218

  Liu, Hailiang ^a   Wang, Zhongming ^a  

^a IOWA STATE UNIVERSITY   (United States)

Author keywords

Level set method; Multi valued solution; Superposition; WKB system

Indexed keywords

APPLICATIONS.; COUPLED WKB; HAMILTON-JACOBI EQUATIONS; HIGH FREQUENCIES; LEVEL SET METHOD; LEVEL SETS; MULTI-VALUED SOLUTION; NUMERICAL TESTING; POSITION DENSITIES; SUPERPOSITION; WAVE DYNAMICS; WKB SYSTEM;

DYNAMICS; WAVEFRONTS; WAVES;

LEVEL MEASUREMENT;

EID: 49749100791     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-008-9198-4     Document Type: Article

References (32)

1. Beyer, R.P., LeVeque, R.J.: Analysis of a one-dimensional model for the immersed boundary method. SIAM J. Numer. Anal. 29(2), 332-364 (1992)
2. Brenier, Y., Corrias, L.: A kinetic formulation for multi-branch entropy solutions of scalar conservation laws. Ann. Inst. Henri Poincaré Anal. Non Linéaire 15(2), 169-190 (1998)
3. Cheng, L.-T.: An Efficient level set method for constructing wave fronts in three space dimensions. UCLA CAM Report (15) (2006)
4. Cheng, L.-T., Liu, H., Osher, S.: Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations. Commun. Math. Sci. 1(3), 593-621 (2003)
5. Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277(1), 1-42 (1983)
6. Engquist, B., Runborg, O.: Multi-phase computations in geometrical optics. J. Comput. Appl. Math. 74(1-2), 175-192 (1996). TICAM Symposium (Austin, TX, 1995)
7. Engquist, B., Runborg, O.: Computational high frequency wave propagation. In: Acta Numerica, vol. 12, pp. 181-266. Cambridge University Press, Cambridge (2003)
8. Engquist, B., Tornberg, A.-K., Tsai, R.: Discretization of Dirac delta functions in level set methods. J. Comput. Phys. 207(1), 28-51 (2005)
9. Gosse, L.: Using K-branch entropy solutions for multivalued geometric optics computations. J. Comput. Phys. 180(1), 155-182 (2002)
10. Gosse, L., Jin, S., Li, X.: Two moment systems for computing multiphase semiclassical limits of the Schrödinger equation. Math. Models Methods Appl. Sci. 13(12), 1689-1723 (2003)
11. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89-112 (2001) (electronic)
12. Harten, A.: Preliminary results on the extension of ENO schemes to two-dimensional problems. In: Nonlinear Hyperbolic Problems, (St. Etienne, 1986). Lect. Notes in Math., vol. 1270, pp. 23-40. Springer, Berlin (1987)
13. Harten, A.: ENO schemes with subcell resolution. J. Comput. Phys. 83(1), 148-184 (1989)
14. Jin, S., Li, X.: Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs. Wigner. Physica D 182(1-2), 46-85 (2003)
15. Jin, S., Liu, H., Osher, S., Tsai, R.: Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems. J. Comput. Phys. 210(2), 497-518 (2005)
16. Jin, S., Liu, H., Osher, S., Tsai, Y.-H.R.: Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation. J. Comput. Phys. 205(1), 222-241 (2005)
17. Jin, S., Osher, S.: A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations. Commun. Math. Sci. 1(3), 575-591 (2003)
18. Kružkov, S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123), 228-255 (1970)
19. Liu, H., Cheng, L.-T., Osher, S.: A level set framework for capturing multi-valued solutions to nonlinear first-order equations. J. Sci. Comput. 29(3), 353-373 (2006)
20. Liu, H., Osher, S., Tsai, R.: Multi-valued solution and level set methods in computational high frequencywave propagation. Commun. Comput. Phys. 1(5), 765-804 (2006)
21. Liu, H., Wang, Z.: Computing multi-valued velocity and electric fields for 1d Euler-Poisson equations. Appl. Numer. Math. 57(5-7), 821-836 (2007)
22. Liu, H., Wang, Z.: A filed-space based level set method for computing mult-valued solutions to 1d Euler-Poisson equations. J. Comput. Phys. 225(1), 591-614 (2007)
23. Min, C.: Local level set method in high dimension and codimension. J. Comput. Phys. 200(1), 368-382 (2004)
24. Osher, S., Cheng, L.-T., Kang, M., Shim, H., Tsai, Y.-H.: Geometric optics in a phase-space-based level set and Eulerian framework. J. Comput. Phys. 179(2), 622-648 (2002)
25. Osher, S., Shu, C.-W.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907-922 (1991)
26. Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDE-based fast local level set method. J. Comput. Phys. 155(2), 410-438 (1999)
27. Raviart, P.-A.: On the numerical analysis of particle simulations in plasma physics. In: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. IV, Paris, 1981/1982. Res. Notes in Math., vol. 84, pp. 173-193. Pitman, Boston (1983)
28. Shu, C.-W.: High order ENO and WENO schemes for computational fluid dynamics. In: High-Order Methods for Computational Physics. Lect. Notes Comput. Sci. Eng., vol. 9, pp. 439-582. Springer, Berlin (1999)
29. Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439-471 (1988)
30. Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. J. Comput. Phys. 83(1), 32-78 (1989)
31. Towers, J.: Two methods for discretizing a delta function supported on a level set. J. Comput. Phys. 220(2), 915-931 (2007)
32. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974). Pure Appl. Math.