The Boundary Forced MKdV Equation

An unconditionally stable numerical algorithm for the modified Korteweg-de Vries equation based on the B-spline finite element method is described. The algorithm is validated through a single soliton simulation. In further numerical experiments forced boundary conditions u = U0 are applied at the en...

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Bibliographic Details
Published in:Journal of computational physics 1994-07, Vol.113 (1), p.5-12
Main Authors: Gardner, L.R.T., Gardner, G.A., Geyikli, T.
Format: Article
Language:English
Subjects:
990200 > Mathematics & Computers
ALGORITHMS
CALCULATION METHODS
Classical and quantum physics: mechanics and fields
Classical mechanics of continuous media: general mathematical aspects
Computational techniques
COMPUTERIZED SIMULATION
DIFFERENTIAL EQUATIONS
ENGINEERING
EQUATIONS
Exact sciences and technology
FINITE ELEMENT METHOD
Finite-element and galerkin methods
GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
KORTEWEG-DE VRIES EQUATION
MATHEMATICAL LOGIC
Mathematical methods in physics
Numerical approximation and analysis
NUMERICAL SOLUTION
Ordinary and partial differential equations, boundary value problems
PARTIAL DIFFERENTIAL EQUATIONS
Physics
SIMULATION 420400 > Engineering-- Heat Transfer & Fluid Flow
Vibrations and mechanical waves
WAVE PROPAGATION
Waves and wave propagation: general mathematical aspects
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Description
Summary:An unconditionally stable numerical algorithm for the modified Korteweg-de Vries equation based on the B-spline finite element method is described. The algorithm is validated through a single soliton simulation. In further numerical experiments forced boundary conditions u = U0 are applied at the end x = 0 and the generated states of solitary waves are studied. By long impulse experiments these are shown to be generated periodically with period ΔTB proportional to U-30 and to have a limiting amplitude proportional to U0. This limit is achieved by all waves, after the first, provided the experiment proceeds long enough. The temporal development of the derivatives U'(0, t), U"(0, t) and U'"(0, t ) is also periodic, with period ΔTB. The effect of negative forcing is to generate a train of negative waves. The solitary wave states generated by applying a positive impulse followed immediately by an negative impulse, of equal amplitude and duration, is dependent on the period of forcing. The solitary waves generated by these various forcing functions possess many of the attributes of free solitons.
ISSN:0021-9991
1090-2716
DOI:10.1006/jcph.1994.1113