zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect. (English) Zbl 1092.65083
A Korteweg-de Vries (KdV) equation with admissible boundary conditions is considered. Then an energy estimate for the KdV problem on the negative quarter-plane is obtained. A local discontinuous Galerkin method for solving KdV type equations with non-homogeneous boundary effect is proposed and its nonlinear $L^2$ stability is proved. Some wave patterns near the boundary are discussed and numerical results, consistent with these wave patterns, are presented.

MSC:
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
35Q53KdV-like (Korteweg-de Vries) equations
65M12Stability and convergence of numerical methods (IVP of PDE)
82D10Plasmas (statistical mechanics)
WorldCat.org
Full Text: DOI
References:
[1] Bona, J. L.; Pritchard, W. G.; Scott, L. R.: An evaluation of a model equation for water waves. Philos. trans. Roy. soc. London ser. A 302, No. 1471, 457-510 (1981) · Zbl 0497.76023
[2] Bona, J. L.; Sun, S. -M.; Zhang, B. -Y.: The initial boundary-value problem for the KdV equation on a quarter plane. Trans. amer. Math. soc. 354, 427-490 (2001) · Zbl 0988.35141
[3] Bona, J. L.; Sun, S. M.; Zhang, B. -Y.: Forced oscillations of a damped Korteweg-de Vries equation in a quarter plane. Commun. contemp. Math. 5, No. 3, 369-400 (2003) · Zbl 1054.35076
[4] Bona, J. L.; Sun, S. M.; Zhang, B. -Y.: A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain. Comm. partial differential equations 28, No. 7-8, 1391-1436 (2003) · Zbl 1057.35049
[5] Bona, J. L.; Winther, R.: The Korteweg-de Vries equation, posed in a quarter-plane. SIAM J. Math. anal. 14, No. 6, 1056-1106 (1983) · Zbl 0529.35069
[6] Bubnov, B. A.: Generalized boundary value problems for the KdV equation in bounded domain. Differential equations 15, 17-21 (1979) · Zbl 0422.35069
[7] Bubnov, B. A.: Solvability in the large of nonlinear boundary value problems for the KdV equations. Differential equations 16, 24-30 (1980) · Zbl 0451.35066
[8] Camassa, R.; Wu, T. Yao-Tsu: The Korteweg-de Vries equation with boundary forcing. Wave motion 11, 495-506 (1989) · Zbl 0699.35225
[9] Chu, X. -L.; Xiang, L. W.; Baransky, Y.: Solitary waves induced by boundary motion. Comm. pure appl. Math. 36, 495-504 (1983) · Zbl 0534.76034
[10] S.R. Clarke, J. Imberge, Nonlinear effects in the unsteady, critical withdrawal of a stratified fluid, in: Fourth International Symposium on Stratified Flows, LEGI, 1994, pp. 1-8.
[11] Cockburn, B.; Shu, C. -W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. anal. 35, No. 6, 2440-2463 (1998) · Zbl 0927.65118
[12] Cockburn, B.; Shu, C. -W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. sci. Comput. 16, No. 3, 173-261 (2001) · Zbl 1065.76135
[13] Colliander, J. E.; Kenig, C. E.: The generalized KdV equation on the half plane. Commun. partial differential equations 27, 2187-2266 (2002) · Zbl 1041.35064
[14] Kruskal, M. D.; Gardner, C. S.; Green, J. M.; Miura, R. M.: Method for solving Korteweg-de Vries equation. Phys. rev. Lett. 19, 1095-1098 (1967) · Zbl 1061.35520
[15] Fokas, A. S.: Ehrenpreis type representations and their Riemann-Hilbert nonlinearisation. J. nonlinear math. Phys. 10, No. Suppl. 1, 47-61 (2003)
[16] Fokas, A. S.; Ablowitz, M. J.: Forced nonlinear evolution equations and the inverse scattering transform. Stud. appl. Math. 80, No. 3, 253-272 (1989) · Zbl 0696.35135
[17] Fornberg, B.; Whitham, G. B.: A numerical and theoretical study of certain nonlinear wave phenomena. Philos. trans. Roy. soc. London ser. A 289, No. 1361, 373-404 (1978) · Zbl 0384.65049
[18] Jr., G. L. Lamb: Elements of soliton theory. (1980)
[19] Guo, B. -Y.; Shen, J.: On spectral approximations using modified Legendre rational functions: application to the Korteweg-de Vries equation on the half line. Indiana univ. Math. J. 50, No. Special Issue, 181-204 (2001) · Zbl 0992.65111
[20] Gurevich, A. V.; Pitayevskiı\breve{}, L. P.: Nonstationary structure of a collisionless shock wave. Sov. phys. JETP 33, 291-297 (1974)
[21] Huang, W. -Z; Sloan, D. M.: The pseudospectral method for third-order differential equations. SIAM J. Numer. anal. 29, No. 6, 1626-1647 (1992) · Zbl 0764.65058
[22] Kichenassamy, S.; Olver, P. J.: Existence and nonexistence of solitary wave solutions to high-order model evolution equations. SIAM J. Appl. math. 23, 1141-1166 (1992) · Zbl 0755.76023
[23] Kreiss, H. -O.; Lorenz, J.: Initial-boundary value problems and the Navier-Stokes equations. Classics in applied mathematics 47 (2004) · Zbl 1097.35113
[24] Levy, D.; Shu, C. -W.; Yan, J.: Local discontinuous Galerkin methods for nonlinear dispersive equations. J. comput. Phys. 196, No. 2, 751-772 (2004) · Zbl 1055.65109
[25] Liu, H.; Slemrod, M.: KdV dynamics in the plasma-sheath transition. Appl. math. Lett. 17, No. 4, 401-410 (2004) · Zbl 1060.76137
[26] Ma, H. -P.; Sun, W. -W.: A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations. SIAM J. Numer. anal. 38, No. 5, 1425-1438 (2000) · Zbl 0986.65095
[27] Ma, H. -P.; Sun, W. -W.: Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation. SIAM J. Numer. anal. 39, No. 4, 1380-1394 (2001) · Zbl 1008.65070
[28] Marchant, T. R.; Smyth, N. F.: The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane. Proc. roy. Soc. London ser. A math. Phys. eng. Sci. 458, No. 2020, 857-871 (2002) · Zbl 0997.35079
[29] Shen, J.: A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: application to the KdV equation. SIAM J. Numer. anal. 41, No. 5, 1595-1619 (2003) · Zbl 1053.65085
[30] Shu, C. -W.; Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. comput. Phys. 77, No. 2, 439-471 (1988) · Zbl 0653.65072
[31] G.B. Whitham, Linear and Nonlinear Waves, in: Pure and Applied Mathematics, Wiley-Interscience, John Wiley, New York, 1974.
[32] Xu, Y.; Shu, C. -W.: Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. comput. Phys. 205, 52-97 (2005) · Zbl 1072.65130
[33] Yan, J.; Shu, C. -W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. anal. 40, No. 2, 769-791 (2002) · Zbl 1021.65050
[34] Yan, J.; Shu, C. -W.: Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J. sci. Comput. 17, 27-47 (2002) · Zbl 1003.65115