A field-space-based level set method for computing multi-valued solutions to 1D Euler-Poisson equations

Journal of Computational Physics

Volumn 225, Issue 1, 2007, Pages 591-614

  Liu, Hailiang ^a   Wang, Zhongming ^a  

^a IOWA STATE UNIVERSITY   (United States)

Author keywords

Euler Poisson equations; Level set method; Multi valued solution

Indexed keywords

DELTA FUNCTIONS; DROP BREAKUP; LEVEL MEASUREMENT; NUMERICAL METHODS; POISSON EQUATION; SET THEORY;

EULER-POISSON EQUATION; FIELD TRANSPORT; LEVEL SET METHOD; LEVEL-SET FUNCTION; MULTI-VALUED; MULTI-VALUED SOLUTION; ONE-DIMENSIONAL; SEMI-CLASSICAL APPROXIMATION; SPACE-BASED; TRANSPORT EQUATION;

ELECTRIC FIELDS;

EID: 34447264074     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2006.12.018     Document Type: Article

References (56)

1. Brenier Y., and Corrias L. A kinetic formulation for multi-branch entropy solutions of scalar conservation laws. Ann. Inst. Henri Poincare 15 (1998) 169-190
2. Burchard P., Cheng L.-T., Merriman B., and Osher S. Motion of curves in three spatial dimensions using a level set approach. J. Comput. Phys. 170 (2001) 720-741
3. Beyer R.P., and LeVeque R.J. Analysis of a one-dimensional model for the immersed boundary method. SIAM J. Numer. Anal. 29 2 (1992) 332-364
4. Courant R., and Hilbert D. Methods of Mathematical Physics. Vol. II: Partial Differential Equations (Vol. II by R. Courant.) (1962), Interscience Publishers (a division of John Wiley & Sons), New York-London xxii+830, pp. 35.00
5. L.-T. Cheng, Efficient level set methods for constructing wave fronts in three dimensions, UCLA CAM report, 2006.
6. Cheng L.-T., Liu H.-L., and Osher S. Computational high-frequency wave propagation in Schrödinger equations using the Level Set Method, with applications to the semi-classical limit of Schrödinger equations. Commun. Math. Sci. 1 3 (2003) 593-621
7. Crandall M., and Lions P.L. Viscosity solutions of Hamilton-Jacobi equations. Trans. AMS. Math. Soc. 277 1 (1983) 1-42
8. Cheng L.-T., Osher S., Kang M., Shim H., and Tsai Y.-H. Geometry optics in a phase-space-based level set and Eulerian framework. J. Comput. Phys. 179 2 (2002) 622-648
9. Cheng L.-T., Osher S., Kang M., Shim H., and Tsai Y.-H. Reflection in a level set framework for geometric optics. CMES Comput. Model. Eng. Sci. 5 4 (2004) 347-360
10. Cockburn B., Qian J.L., Reitich F., and Wang J. An accurate spectral/discontinuous finite-element formulation of a phase-space-based level set approach to geometrical optics. (English. English summary). J. Comput. Phys. 208 1 (2005) 175-195
11. Engelberg S., Liu H.-L., and Tadmor E. Critical thresholds in Euler-Poisson equations. (Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000)). Indiana Univ. Math. J. 50 Special Issue (2001) 109-157
12. Engquist B., and Runborg O. Multi-phase computations in geometrical optics. J. Comp. Appl. Math. 74 (1996) 175-199
13. Engquist B., and Runborg O. Computational high frequency wave propagation. Acta Numer. 12 (2003) 181-266 (Cambridge Univ. Press, Cambridge)
14. Engquist B., Runborg O., and Tornberg T.-K. High frequency wave propagation by the segment projection method. J. Comput. Phys. 178 (2002) 373-390
15. Engquist B., Tornberg A.-K., and Tsai R. Discretization of Dirac delta functions in level set methods. J. Comput. Phys. 207 1 (2005) 28-51
16. Forest M.-G., Flaschka H., and Mclaughlin D.-W. Multi-phase averaging and the inverse spectral solution of the Korteweg-de Vries equation. Comm. Pure Appl. Math. 33 (1980) 739-784
17. Fomel S., and Sethian J.A. Fast-phase space computation of multiple arrivals. Proc. Natl. Acad. Sci. 99 11 (2002) 7329-7334
18. Gasser I., Levermore D., Markowich P., and Schmeiser C. The initial time layer problem and the quasineutral limit in the semiconductor drift-diffusion model. European J. Appl. Math. 12 4 (2001) 497-512
19. Gerard P., Markowich P.A., Mauser N.J., and Poupaud F. Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 323-380
20. Gosse L. Using K-branch entropy solutions for multivalued geometric optics computations. J. Comput. Phys. 180 (2002) 155-182
21. Gosse L., Jin S., and Li X. On two moment systems for computing multiphase semiclassical limits of the Schrödinger equation. Math. Models Methods Appl. Sci. 13 12 (2003) 1689-1723
22. Gosse L., and Markowich P.A. Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. I. Homogeneous problems. J. Comput. Phys. 197 2 (2004) 387-417
23. Gosse L., and Mauser N.J. Multiphase semiclassical 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
24. 3 + 1. Comm. Math. Phys. 195 2 (1998) 249-265 Reviewer: Woodford W. Zachary
25. Hutter R. Beam and Wave Electronics in Microwave Tubes (1960), D. Van Nostrand Company, Inc., Princeton
26. Jin S., and Li X. Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner. Physica D 182 (2003) 46-85
27. Jin S., Liu H.-L., Osher S., and Tsai Y.h. Computing multivalued physical observables for the semiclassical limit of the Schrö dinger equation. J. Comput. Phys. 205 1 (2005) 222-241
28. Jin S., Liu H.-L., Osher S., and Tsai Y.H. Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems. J. Comput. Phys. 210 (2005) 497-518
29. Jin S., and Osher S. A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDE's and Hamilton-Jacobi equations. Comm. Math. Sci. 1 3 (2003) 575-591
30. Liu H.-L., Cheng L.-T., and Osher S. A level set framework for capturing multi-valued solutions to nonlinear first-order equations. J. Sci. Comp. (2005) (December 07 (electronically))
31. Lax P., and Levermore D. The small dispersion limit of the Korteweg-de Vries equation. I. Comm. Pure Appl. Math. 36 3 (1983) 253-290
32. Lax P., and Levermore D. The small dispersion limit of the Korteweg-de Vries equation. II. Comm. Pure Appl. Math. 36 5 (1983) 571-593
33. Lax P., and Levermore D. The small dispersion limit of the Korteweg-de Vries equation. III. Comm. Pure Appl. Math. 36 6 (1983) 809-829
34. Liu H.-L., Osher S., and Tsai Y.H. Multi-valued solution and level set methods in computational high frequency wave propagation. Commun. Comput. Phys. 1 5 (2006) 765-804
35. Lions P.-L., and Paul T. Sur les measures de Wigner. Revista. Mat. Iberoamericana 9 (1993) 553-618
36. Leung S.Y., Qian J.L., and Osher S. A level set method for three-dimensional paraxial geometrical optics with multiple point sources. Commun. Math. Sci. 2 4 (2004) 643-672
37. Liu H.-L., and Tadmor E. Semi-classical limit of the nonlinear Schrödinger-Poisson equation with subcritical initial data. Methods Appl. Anal. 9 4 (2002) 517-532
38. H.-L. Liu, Z.-M. Wang, Computing multi-valued velocity and electrical fields for 1d Euler-Poisson equations, Appl. Numer. Math., in press.
39. Li X., Wöhlbier J., Jin S., and Booske J. An Eulerian method for computing multi-valued solutions of the Euler-Poisson equations and applications to wave breaking in klystrons. Phys. Rev. E 70 (2004) 016502
40. Markowich P., and Mauser N. The classical limit of a self-consistent quantum-Vlasov equation in 3-D. Math. Models Methods Appl. Sci. 3 1 (1993) 109.12
41. Markowich P., Ringhofer C., and Schmeiser C. Semiconductor Equations (1990), Springer, Vienna
42. Osher S., Cheng L.-T., Kang M., Shim H., and Tsai Y.-H. Geometric optics in a phase space based level set and Eulerian framework. J. Comput. Phys. 179 (2002) 622-648
43. Qian J., Cheng L.-T., and Osher S. A level set based Eulerian approach for anisotropic wave propagation. Wave Motion 37 (2003) 365-379
44. Raviart P.-A. An analysis of particle methods. Numerical Methods in Fluid Dynamics (Como, 1983). Lecture Notes in Math. vol. 1127 (1983), Springer, Berlin 243-324
45. Ryzhik L., Papanicolaou G., and Keller J.B. Transport equations for elastic and other waves in random media. Wave Motion 24 4 (1996) 327-370
46. Runborg O. Some new results in multi-phase geometric optics. Math. Model. Numer. Anal. 34 (2000) 1203-1231
47. Shu C.-W. High order ENO and WENO schemes for computational fluid dynamics. High-order Methods for Computational Physics. Lect. Notes Comput. Sci. Eng. vol. 9 (1999), Springer, Berlin
48. Sparber C., Markowich P.A., and Mauser N.J. Wigner functions versus WKB-methods in multivalued geometrical optics. Asymptot. Anal. 33 2 (2003) 153-187
49. Symes W., and Qian J. A slowness matching Eulerian method for multivalued solutions of Eikonal equations. (Special issue in honor of the sixtieth birthday of Stanley Osher). J. Sci. Comput. 19 1-3 (2003) 501-526
50. Tornberg A.-K., and Engquist B. Regularization techniques for numerical approximation of PDEs with singularities. Special issue in honor of the sixtieth birthday of Stanley Osher. J. Sci. Comput. 19 1-3 (2003) 527-552
51. Tornberg A.-K., and Engquist B. Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200 2 (2004) 462-488
52. Tsai Y.-H.R., Giga Y., and Osher S. A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations. Math. Comp. 72 (2003) 159-181
53. van Trier J., and Symes W.W. Upwind finite-difference calculation of traveltimes. Geophysics 56 (1991) 812-821
54. Wigner E. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40 (1932) 749-759
55. Whitham G. Linear and Nonlinear Waves (1974), Wiley, New York
56. Zhang P., Zheng Y.X., and Mauser N. The limit from the Schröinger-Poisson to the Vlasov-Poisson equations with general data in one dimension. Comm. Pure Appl. Math. 55 5 (2002) 582-632