Semi-discrete approximations to nonlinear systems of conservation laws; consistency and L(infinity)-stability imply convergence. Final report

Technical Report · Fri Jul 01 04:00:00 EDT 1988

OSTI ID:6718936

Tadmor, E

A convergence theory for semi-discrete approximations to nonlinear systems of conservation laws is developed. It is shown, by a series of scalar counter-examples, that consistency with the conservation law alone does not guarantee convergence. Instead, a notion of consistency which takes into account both the conservation law and its augmenting entropy condition is introduced. In this context it is concluded that consistency and L(infinity)-stability guarantee for a relevant class of admissible entropy functions, that their entropy production rate belongs to a compact subset of H(loc)sup -1 (x,t). One can now use compensated compactness arguments in order to turn this conclusion into a convergence proof. The current state of the art for these arguments includes the scalar and a wide class of 2 x 2 systems of conservation laws. The general framework of the vanishing viscosity method is studied as an effective way to meet the consistency and L(infinity)-stability requirements. How this method is utilized to enforce consistency and stability for scalar conservation laws is shown. In this context we prove, under the appropriate assumptions, the convergence of finite difference approximations (e.g., the high resolution TVD and UNO methods), finite element approximations (e.g., the Streamline-Diffusion methods) and spectral and pseudospectral approximations (e.g., the Spectral Viscosity methods).

• Research Organization: National Aeronautics and Space Administration, Hampton, VA (USA). Langley Research Center
• OSTI ID: 6718936
• Report Number(s): N-88-26157; NASA-CR-181683; ICASE-88-41; NAS-1.26:181683
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CONSERVATION LAWS
CONVERGENCE
DATA
DIFFERENTIAL EQUATIONS
DOCUMENT TYPES
EQUATIONS
FINITE ELEMENT METHOD
INFORMATION
NUMERICAL DATA
NUMERICAL SOLUTION
PROGRESS REPORT
SCALARS
THEORETICAL DATA