Computing multi-valued velocity and electric fields for 1D Euler-Poisson equations

Applied Numerical Mathematics

Volumn 57, Issue 5-7 SPEC. ISS., 2007, Pages 821-836

  Liu, Hailiang ^a   Wang, Zhongming ^a  

^a IOWA STATE UNIVERSITY   (United States)

Author keywords

Euler Poisson equations; Field space; Level set method; Multi valued solutions

Indexed keywords

APPROXIMATION THEORY; ELECTRIC FIELDS; MATHEMATICAL MODELS; SCHRODINGER EQUATION; VELOCITY MEASUREMENT;

EULER-POISSON EQUATIONS; FIELD SPACE; LEVEL SET METHOD; MULTI-VALUED SOLUTIONS;

POISSON EQUATION;

EID: 34147167264     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2006.07.021     Document Type: Article

References (37)

1. Brenier Y., and Corrias L. A kinetic formulation for multi-branch entropy solutions of scalar conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 2 (1998) 169-190
2. Benamou J.-D., Castella F., Katsaounis T., and Perthame B. High frequency limit of the Helmholtz equations. Rev. Mat. Iberoamericana 18 1 (2002) 187-209
3. 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 2 (2001) 720-741 76M25 (76D17)
4. Cheng L.-T., Kang M., Osher S., 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
5. Cheng L.-T., Liu H., and 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 (2003) 593-621
6. Engelberg S., Liu H., 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 (2001) 109-157 (special issue)
7. Engquist B., and Runborg O. Multi-phase computations in geometrical optics. TICAM Symposium (Austin, TX, 1995). J. Comput. Appl. Math. 74 1-2 (1996) 175-192
8. Engquist B., Runborg O., and Tornberg A.-K. High-frequency wave propagation by the segment projection method. J. Comput. Phys. 178 2 (2002) 373-390
9. Fatemi E., Engquist B., and Osher S. Numerical solution of the high frequency asymptotic expansion for the scalar wave equation. J. Comput. Phys. 120 1 (1995) 145-155
10. Fomel S., and Sethian J.A. Fast-phase space computation of multiple arrivals. Proc. Natl. Acad. Sci. USA 99 11 (2002) 7329-7334 (electronic)
11. Gosse L. Using K-branch entropy solutions for multivalued geometric optics computations. J. Comput. Phys. 180 2 (2002) 155-182
12. Gosse L. Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. II. Impurities, confinement and Bloch oscillations. J. Comput. Phys. 201 1 (2004) 344-375
13. Gosse L., Jin S., and Li X. Two moment systems for computing multiphase semiclassical limits of the Schrödinger equation. Math. Models Methods Appl. Sci. 13 12 (2003) 1689-1723
14. 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
15. Harten A. Preliminary results on the extension of ENO schemes to two-dimensional problems. Nonlinear Hyperbolic Problems. St. Etienne, 1986. Lecture Notes in Math. vol. 1270 (1987), Springer, Berlin 23-40
16. Harten A. ENO schemes with subcell resolution. J. Comput. Phys. 83 1 (1989) 148-184
17. Jin S., and Li X. Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs. Wigner. Phys. D 182 1-2 (2003) 46-85
18. Jin S., Liu H., Osher S., and Tsai Y.-H.R. Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation. J. Comput. Phys. 205 1 (2005) 222-241
19. S. Jin, H. Liu, S. Osher, Y.-H.R. Tsai, Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems. J. Comput. Phys., in press
20. Jin S., and 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 (2003) 575-591
21. Jiang G.-S., and Wu C.-C. A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 150 2 (1999) 561-594
22. H. Liu, L.-T. Cheng, S. Osher, A level set framework for capturing multi-valued solutions to nonlinear first-order equations, J. Sci. Comput., in press
23. Leung S., Qian J., 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
24. Liu H., and Tadmor E. Semiclassical limit of the nonlinear Schrödinger-Poisson equation with subcritical initial data. Methods Appl. Anal. 9 4 (2002) 517-531
25. H. Liu, Z. Wang, A field space based level set method for comnputing multi-valued solutions to 1D Euler-Poisson equations, in preparation
26. Li X., Wöhlbier J.G., Jin S., and Booske J.H. An Eulerian method for computing multi-valued solutions of the Euler-Poisson equations and applucaions to wave breaking in klystrons. Phys. Rev. E 70 (2004) 016502
27. Markowich P.A., Ringhofer C.A., and Schmeiser C. Semiconductor equations (1990), Springer, Vienna
28. Montarnal P., and Shu C.-W. Real gas computation using an energy relaxation method and high-order WENO schemes. J. Comput. Phys. 148 1 (1999) 59-80
29. 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 2 (2002) 622-648
30. Osher S., and Fedkiw R.P. Level Set Methods and Dynamic Implicit Surface (2002), Springer, Berlin
31. Osher S., and Sethian J.A. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 1 (1988) 12-49
32. Qian J., Cheng L.-T., and Osher S. A level set-based Eulerian approach for anisotropic wave propagation. Wave Motion 37 4 (2003) 365-379
33. Qian J., and Leung S. A level set based Eulerian method for paraxial multivalued traveltimes. J. Comput. Phys. 197 2 (2004) 711-736
34. Runborg O. Some new results in multiphase geometrical optics. M2AN Math. Model. Numer. Anal. 34 6 (2000) 1203-1231
35. Sethian J.A. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (1999), Cambridge University Press, Cambridge
36. 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 439-582
37. Zhang P., Zheng Y., and Mauser N.J. The limit from the Schrödinger-Poisson to the Vlasov-Poisson equations with general data in one dimension. Comm. Pure Appl. Math. 55 5 (2002) 582-632