zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On discreteness of the Hopf equation. (English) Zbl 1149.65071
Summary: The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.

65M06Finite difference methods (IVP of PDE)
35L65Conservation laws
[1]Ahmed, H., Liu, H. Formulation and analysis of alternating evolution (AE) schemes for hyperbolic conservation laws. (preprint)
[2]Bianco, F., Puppo, G., Russo, G. High-order central schemes for hyperbolic systems of conservation laws. SIAM J. Sci. Comput., 21(1): 294–322, (1999) (electronic) · Zbl 0940.65093 · doi:10.1137/S1064827597324998
[3]Bouchut, F. James, F. Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Comm. Partial Differential Equations, 24(11–12): 2173–2189 (1999) · Zbl 0937.35098 · doi:10.1080/03605309908821498
[4]Camassa, R., Holm, D.D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71: 1661–1664 (1993) · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661
[5]Chen, G.Q. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics. III. Acta Math. Sci. (English Ed.), 6(1): 75–120 (1986)
[6]Chen, G.Q., Liu, H.L. Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J. Math. Anal., 34(4): 925–938 (2003)(electronic) · Zbl 1038.35035 · doi:10.1137/S0036141001399350
[7]Chen, G.Q., Liu, H.L. Concentration and cavitation in solutions of the euler equations for nonisentropic fluids as the pressure vanishes. Phys. D., 189(1–2): 141–165 (2004) · Zbl 1098.76603 · doi:10.1016/j.physd.2003.09.039
[8]Cheng, L.T., Liu, H.L., Osher, S. Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations. Comm. Math. Sci., 1(3): 593–621 2003
[9]Cheng, L.T., Osher, S., Kang, M., Shim, H., Tsai, Y.H. Reflection in a level set framework for geometric optics. Comput. Model Eng. Sci., 5(4): 347–360 (2004)
[10]Cockburn, B., Shu, C.W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16(3): 173–261 (2001) · Zbl 1065.76135 · doi:10.1023/A:1012873910884
[11]Coclite, G.M., Karlsen, K.H. On the well-posedness of the degasperisprocesi equation. J. Funct. Anal., 233: 60–91 (2006) · Zbl 1090.35142 · doi:10.1016/j.jfa.2005.07.008
[12]Colella, P., Woodward, P.R. The piecewise parabolic method (ppm) for gas-dynamical simulations. J. Comput. Phys., 54(1): 174–201 (1984) · Zbl 0531.76082 · doi:10.1016/0021-9991(84)90143-8
[13]Courant, R., Hilbert, D. Methods of mathematical physics. Vol. II. John Wiley & Sons Inc., New York, 1989. Partial differential equations, Reprint of the 1962 original, A Wiley-Interscience Publication.
[14]Courant, R., Isaacson, E., Rees, M. On the solution of nonlinear hyperbolic differential equations by finite differences. Comm. Pure. Appl. Math., 5:243–255 (1952) · Zbl 0047.11704 · doi:10.1002/cpa.3160050303
[15]Crandall, M.G., Lions, P.L. Two approximations of solutions of Hamilton-Jacobi equations. Math. Comp., 43(167): 1–19 (1984) · Zbl 0557.65066 · doi:10.1090/S0025-5718-1984-0744921-8
[16]Degasperis, A., Procesi, M. Asymptotic integrability. In: Symmetry and perturbation theory. In Rome, pages 23–37, World Scientific, River Edge, NJ, 1999
[17]Ding, X.Q., Chen, G.Q., Luo, P.Z. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics. I, II. Acta Math. Sci., (English Ed.), 5(4): 415–472 (1985)
[18]Ding, X.Q., Chen, G.Q., Luo, P.Z. Convergence of the Lax-Friedrichs scheme for the system of equations of isentropic gas dynamics. I. Acta Math. Sci., 7(4): 467–480 (1987) (in Chinese)
[19]E, W., Rykov, Y.G., Sinai, Y.G. Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm. Math. Phys., 177(2): 349–380 (1996) · Zbl 0852.35097 · doi:10.1007/BF02101897
[20]Engquist, B., Osher, S. Stable and entropy satisfying approximations for transonic flow calculations. Math. Comp., 34(149): 45–75 (1980) · doi:10.1090/S0025-5718-1980-0551290-1
[21]Engquist, B., Runborg, O. Computational high frequency wave propagation. Acta Numer., 12: 181–266 (2003) · Zbl 1049.65098 · doi:10.1017/S0962492902000119
[22]Evans, L.C. A geometric interpretation of the heat equation with multivalued initial data. SIAM J. Math. Anal., 27(4): 932–958 (1996) · Zbl 0862.35043 · doi:10.1137/S0036141094275439
[23]Francesco, M.D., Fellner, K., Liu, H. A non-local conservation law with nonlinear ’radiation’ inhomogeneity. To appear in J. Hyperbolic Differ. Equ., (2008)
[24]Fuchssteiner, B., Fokas, A. Symplectic structures, their backlund transformations and hereditary symmetries. Physica D, 4(1): 47–66 (1881/82) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[25]Giga, Y., Sato, M.H. A level set approach to semicontinuous viscosity solutions for Cauchy problems. Comm. Partial Differential Equations, 26(5–6): 813–839 (2001) · Zbl 1005.49025 · doi:10.1081/PDE-100002379
[26]Godunov, S.K. A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.), 47(89): 271–306 (1959)
[27]Goodman, J., Lax, P.D. On dispersive difference schemes. I. Comm. Pure Appl. Math., 41(5): 591–613 (1988) · Zbl 0647.65062 · doi:10.1002/cpa.3160410506
[28]Gosse, L., James, F. Convergence results for an inhomogeneous system arising in various high frequency approximations. Numer. Math., 90(4): 721–753 (2002) · Zbl 0994.65095 · doi:10.1007/s002110100309
[29]Gottlieb, S., Shu, C.W., Tadmor, E. Strong stability-preserving high-order time discretization methods. SIAM Rev., 43(1): 89–112 (2001) (electronic) · Zbl 0967.65098 · doi:10.1137/S003614450036757X
[30]Harten, A. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49(3): 357–393 (1983) · Zbl 0565.65050 · doi:10.1016/0021-9991(83)90136-5
[31]Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R. Uniformly high-order accurate essentially nonoscillatory schemes. III. J. Comput. Phys., 71(2): 231–303 (1987) · Zbl 0652.65067 · doi:10.1016/0021-9991(87)90031-3
[32]Harten, A., Lax, P.D., van Leer, B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev., 25(1): 35–61 (1983) · Zbl 0565.65051 · doi:10.1137/1025002
[33]Hou, T.Y., Lax, P.D. Dispersive approximations in fluid dynamics. Comm. Pure Appl. Math., 44(1): 1–40 (1991) · Zbl 0729.76065 · doi:10.1002/cpa.3160440102
[34]Jiang, G.S., Tadmor, E. Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput., 19(6): 1892–1917 (1998) (electronic) · Zbl 0914.65095 · doi:10.1137/S106482759631041X
[35]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) · Zbl 1098.65094 · doi:10.1016/j.jcp.2005.04.020
[36]Jin, S., Liu, H., Osher, S., Tsai, R. Computing multi-valued physical observables for the semiclassical limit of the Schrödinger equation. J. Comput. Phys., 205(1): 222–241 (2005) · Zbl 1072.65132 · doi:10.1016/j.jcp.2004.11.008
[37]Jin, S., 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): 575–591 (2003)
[38]Kac, M., yon Moerbeke, P. On an explicitly soluble system of nonlinear differential equations related to certain toda lattices. Adv. in Math., 16: 160–169 (1975) · Zbl 0306.34001 · doi:10.1016/0001-8708(75)90148-6
[39]A. Kurganov, S. Noelle, and G. Petrova. Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput., 23(3): 707–740 (electronic), 2001. · Zbl 0998.65091 · doi:10.1137/S1064827500373413
[40]Kurganov, A., Petrova, G. A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems. Numer. Math., 88(4): 683–729 (2001) · doi:10.1007/PL00005455
[41]Kurganov, A., Tadmor, E. New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations. J. Comput. Phys., 160(2): 720–742 (2000) · Zbl 0961.65077 · doi:10.1006/jcph.2000.6485
[42]Lax, P., Levermore, C. The zero dispersion limit of the korteweg-de vries equation. Comm. Pure Appl. Math., 36(I, II, III): 253–290, 571-593, 809-829 (1983) · Zbl 0532.35067 · doi:10.1002/cpa.3160360302
[43]Lax, P.D. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm. Pure Appl. Math., 7: 159–193 (1954) · Zbl 0055.19404 · doi:10.1002/cpa.3160070112
[44]Lax, P.D. On dispersive difference schemes. Physica D, 18: 250–254 (1986) · Zbl 0613.65127 · doi:10.1016/0167-2789(86)90185-5
[45]Lax, P.D. Oscillatory solutions of partial differential and difference equations. In Mathematics Applied to Science (New Orleans, La., 1986), pages 155–170. Academic Press, Boston, MA, 1988
[46]Lax, P.D., Wendroff, B. Difference schemes for hyperbolic equations with high order of accuracy. Comm. Pure Appl. Math., 17: 381–398 (1964) · Zbl 0233.65050 · doi:10.1002/cpa.3160170311
[47]Levermore, C.D., Liu, J.G. Large oscillations arising in a dispersive numerical scheme. Phys. D, 99(2–3): 191–216 (1996) · Zbl 0887.65098 · doi:10.1016/S0167-2789(96)00157-1
[48]Levy, D., Puppo, G., Russo, G. Central WENO schemes for hyperbolic systems of conservation laws. M2AN Math. Model. Numer. Anal., 33(3): 547–571 (1999) · Zbl 0938.65110 · doi:10.1051/m2an:1999152
[49]Liu, H. Wave breaking in a class of nonlocal dispersive wave equations. J. Nonlinear Math. Phys., 13(3): 441–466 (2006) · Zbl 1110.35069 · doi:10.2991/jnmp.2006.13.3.8
[50]Liu, H. An alternating evolution approximation to systems of hyperbolic conservation laws. To appear in J. Hyperbolic Differ. Equ., (2008)
[51]Liu, H., Cheng, L.T., Osher, S. A level set framework for tracking multi-valued solutions to nonlinear first-order equations, December 07, 2005 (electronically). J. Sci. Comp., 29(3): 353–373 (2006) · Zbl 1109.65080 · doi:10.1007/s10915-005-9016-1
[52]Liu, H., Osher, S., Tsai, R. Multi-valued solution and level set methods in computing high frequency wave propagation. Comm. in Comput. Phys., 1(5): 765–804 (2006)
[53]Liu, H., Tadmor, E. Critical thresholds in a convolution model for nonlinear conservation laws. SIAM J. Math. Anal., 33(4): 930–945 (2001) · Zbl 1002.35085 · doi:10.1137/S0036141001386908
[54]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) · Zbl 1146.78012 · doi:10.1016/j.apnum.2006.07.021
[55]Liu, H., Wang, Z. A field-space-based level set method for computing multi-valued solutions to 1D Euler-Poisson equations. J. Comput. Phys., 225(1): 591–614 (2007) · Zbl 1140.81399 · doi:10.1016/j.jcp.2006.12.018
[56]Liu, H., Wang, Z. Superposition of multi-valued solutions in high frequency wave dynamics. J. Sci. Comput., publication online 3/11/2008
[57]Liu, X.D., Osher, S., Chan, T. Weighted essentially non-oscillatory schemes. J. Comput. Phys., 115(1): 200–212 (1994) · Zbl 0811.65076 · doi:10.1006/jcph.1994.1187
[58]Liu, X.D., Tadmor, E. Third order nonoscillatory central scheme for hyperbolic conservation laws. Numer. Math., 79(3): 397–425 (1998) · Zbl 0906.65093 · doi:10.1007/s002110050345
[59]Liu, Y. Central schemes on overlapping cells. J. Comput. Phys., 209(1): 82–104 (2005) · Zbl 1076.65076 · doi:10.1016/j.jcp.2005.03.014
[60]Miller, P.D., Ercolani, N.M., Levermore, C.D. Modulation of multiphase waves in the presence of resonance. Phys. D, 92(1–2): 1–27 (1996) · Zbl 0887.34060 · doi:10.1016/0167-2789(95)00281-2
[61]Min, C. Local level set method in high dimension and codimension. J. Comput. Phys., 200(1): 368–382 (2004) · Zbl 1086.65088 · doi:10.1016/j.jcp.2004.04.019
[62]Moser, J. Three integrable Hamiltonian systems connected with isospectral deformations. In Surveys in applied mathematics (Proc. First Los Alamos Sympos. Math. in Natural Sci., Los Alamos, N.M., 1974), pages 235–258. Academic Press, New York, 1976
[63]Nessyahu, H., Tadmor, E. Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys., 87(2): 408–463 (1990) · Zbl 0697.65068 · doi:10.1016/0021-9991(90)90260-8
[64]Osher, S. Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal., 21(2): 217–235 (1984) · Zbl 0592.65069 · doi:10.1137/0721016
[65]Osher, S. A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations. SIAM J. Math. Anal., 24(5): 1145–1152 (1993) · Zbl 0804.35021 · doi:10.1137/0524066
[66]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) · Zbl 0999.78002 · doi:10.1006/jcph.2002.7080
[67]Osher, S., Fedkiw, R. Level set methods and dynamic implicit surfaces. Springer-Verlag, New York, 2002
[68]Osher, S., Sethian, J.A. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys., 79(1): 12–49 (1988) · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[69]Qian, J., Cheng, L.T., Osher. S. A level set-based Eulerian approach for anisotropic wave propagation. Wave Motion, 37(4): 365–379 2003 · Zbl 1163.74426 · doi:10.1016/S0165-2125(02)00101-4
[70]Sheng, W., Zhang, T. The Riemann problem for the transportation equations in gas dynamics. Mem. Amer. Math. Soc., 137(654): viii+77 (1999)
[71]Shu, C.W., Osher, S. Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys., 77(2): 439–471 (1988) · Zbl 0653.65072 · doi:10.1016/0021-9991(88)90177-5
[72]Shu, C.W., Osher. S. Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. J. Comput. Phys., 83(1): 32–78 1989 · Zbl 0674.65061 · doi:10.1016/0021-9991(89)90222-2
[73]Tadmor, E. Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comp., 43(168): 369–381 (1984) · doi:10.1090/S0025-5718-1984-0758189-X
[74]Toda, M. Theory of nonlinear lattices. Vol.20 of Solid-State Sciences. Springer-Verlag, New York, Second edition, 1988
[75]Tsai, Y.H.R., Giga, Y., Osher, S. A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations. Math. Comp., 72(241): 159–181 (2003) (electronic) · Zbl 1013.65088 · doi:10.1090/S0025-5718-02-01438-2
[76]Turner, C.V., Rosales, R.R. The small dispersion limit for a nonlinear semidiscrete system of equations. Stud. Appl. Math., 99(3): 205–254 (1997) · Zbl 0886.65100 · doi:10.1111/1467-9590.00060
[77]van Leer, B. Upwind differencing for hyperbolic systems of conservation laws. In Numerical methods for engineering, 1 (Paris, 1980), pages 137–149, Dunod, Paris, 1980
[78]van Leer, B. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32(1): 101–136 (1979) J. Comput. Phys., 135(2): 227-248 (1997) With an introduction by Ch. Hirsch, Commemoration of the 30th anniversary of J. Comput. Phys.. · doi:10.1016/0021-9991(79)90145-1
[79]Venakides, S. The zero dispersion limit of the Korteweg-de Vries equation for initial potentials with non-trivial reflection coefficient. Comm. Pure Appl. Math., 38(2): 125–155 (1985) · Zbl 0571.35095 · doi:10.1002/cpa.3160380202
[80]Venakides, S., Deift, P., Oba, R. The Toda shock problem. Comm. Pure Appl. Math., 44(8–9): 1171–1242 (1991) · Zbl 0749.35054 · doi:10.1002/cpa.3160440823
[81]Von Neumann, J. Proposal and analysis of a new numerical method in the treatment of hydrodynamical shock problems. In Collected Works VI. Pergamon, New York, 1961
[82]Wang, Z. Spectral volume method for conservation laws on unstructured grids: basic formulation. J. Comp. Phys., 178: 210–251 (2002) · Zbl 0997.65115 · doi:10.1006/jcph.2002.7041
[83]Yin, Z. Global weak solutions for a new periodic integrable equation with peakon solutions. J. Funct. Anal., 212(1): 182–194 (2004) · Zbl 1059.35149 · doi:10.1016/j.jfa.2003.07.010
[84]N. Zabusky, N., Kruskal, M. Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 15: 240–243 (1965) · Zbl 1201.35174 · doi:10.1103/PhysRevLett.15.240