Dispersive approximations for hyperbolic conservation laws

Technical Report · Tue Dec 01 04:00:00 EST 1981

OSTI ID:5230665

Lucier, B J

Necessary and sufficient conditions are given so that the Sobolev-type partial differential equations: u/sub t/ + f(u)/sub x/ - ..nu..g(u)/sub xx/ - ..beta..u/sub xxt/ = 0, x epsilon R, t > 0, ..beta.. > 0, u(x,0) = u/sub 0/(x), x epsilon R, generate a contraction semigroup on L/sup 1/(R). It is shown that any nonlinear contraction from L/sup 1/(R) to itself that preserves the integral and commutes with translations satisfies maximum and minimum principles. This lemma is applied to the solution operator S/sub t/ to give necessary and sufficient conditions that S/t/ satisfy a maximum principle, despite the dispersive nature. Sufficient conditions are given so that the solutions converge, as ..nu.. and ..beta.. tend to zero, to the entropy solution of the conservation law: u/sub t/ + f(u)/sub x/ = 0, x epsilon R, t > 0, u(x,0) = u/sup 0/(x), x epsilon R. A larger class of monotone finite-difference schemes for the numerical solution of the conservation law motivated by finite-difference discretizations of the Sobolev equations, is introduced, and convergence results are proved for methods in this class. The methods analyzed include some that were previously used to approximate the solution of a linear waterflood problem in petroleum engineering.

• Research Organization: Argonne National Lab., IL (USA)
• DOE Contract Number: W-31109-ENG-38
• OSTI ID: 5230665
• Report Number(s): ANL-81-74; ON: DE82007821
• Country of Publication: United States
• Language: English

Related Subjects

645301* -- High Energy Physics-- Particle Invariance Principles & Symmetries-- General-- (-1987)
657006 -- Theoretical Physics-- Statistical Physics & Thermodynamics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
ANALYTICAL SOLUTION
CONSERVATION LAWS
DIFFERENTIAL EQUATIONS
DISPERSION RELATIONS
ENTROPY
EQUATIONS
FINITE DIFFERENCE METHOD
ITERATIVE METHODS
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
PHYSICAL PROPERTIES
THERMODYNAMIC PROPERTIES