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On Discreteness of the Hopf Equation

On Discreteness of the Hopf Equation
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摘要 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. 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.
作者 Hai-liang Liu
机构地区 Iowa State University
出处 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2008年第3期423-440,共18页 应用数学学报（英文版）
基金 the National Science Foundation under Grant DMS05-05975.
关键词 Hopf equation dispersive scheme shock capturing schemes multi-valued solutions level set equation Hopf equation, dispersive scheme, shock capturing schemes, multi-valued solutions, level set equation
分类号 O18 [理学—基础数学]
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参考文献84

  • 1Ahmed, H., Liu, H. Formulation and analysis of alternating evolution (AE) schemes for hyperbolic conservation laws. (preprint)
  • 2Bianco, 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)
  • 3Bouchut, F. James, F. Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Comm. Partial Differential Equations, 24(11-12): 2173-2189 (1999)
  • 4Camassa, R., Holm, D.D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71: 1661-1664 (1993)
  • 5Chen, G.Q. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics. Ⅲ. Acta Math. Sci. (English Ed.), 6(1): 75-120 (1986)
  • 6Chen, 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)
  • 7Chen, 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)
  • 8Cheng, 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 SchrSdinger equations. Comm. Math. Sci., 1(3): 593-621 2003
  • 9Cheng, 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)
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Acta Mathematicae Applicatae Sinica
2008年 第3期

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