An entropy-based approach to nonlinear stability

Technical Report · Tue Feb 28 23:00:00 EST 1989

OSTI ID:6973642

Merriam, M L

Many numerical methods used in computational fluid dynamics (CFD) incorporate an artificial dissipation term to suppress spurious oscillations and control nonlinear instabilities. The same effect can be accomplished by using upwind techniques, sometimes augmented with limiters to form Total Variation Diminishing (TVD) schemes. An analysis based on numerical satisfaction of the second law of thermodynamics allows many such methods to be compared and improved upon. A nonlinear stability proof is given for discrete scalar equations arising from a conservation law. Solutions to such equations are bounded in the L sub 2 norm if the second law of thermodynamics is satisfied in a global sense over a periodic domain. It is conjectured that an analogous statement is true for discrete equations arising from systems of conservation laws. Analysis and numerical experiments suggest that a more restrictive condition, a positive entropy production rate in each cell, is sufficient to exclude unphysical phenomena such as oscillations and expansion shocks. Construction of schemes which satisfy this condition is demonstrated for linear and nonlinear wave equations and for the one-dimensional Euler equations.

• Research Organization: National Aeronautics and Space Administration, Moffett Field, CA (USA). Ames Research Center
• OSTI ID: 6973642
• Report Number(s): N-90-17376; NASA-TM--101086; A--89078; NAS--1.15:101086
• Country of Publication: United States
• Language: English

Related Subjects

42 ENGINEERING
420400* -- Engineering-- Heat Transfer & Fluid Flow
CALCULATION METHODS
CONSERVATION LAWS
DIFFERENTIAL EQUATIONS
ENTROPY
EQUATIONS
EQUATIONS OF MOTION
FLOW MODELS
FLUID FLOW
MATHEMATICAL MODELS
MATHEMATICS
NAVIER-STOKES EQUATIONS
NUMERICAL ANALYSIS
OSCILLATIONS
PARTIAL DIFFERENTIAL EQUATIONS
PHYSICAL PROPERTIES
SCALARS
STABILITY
THERMODYNAMIC PROPERTIES
THERMODYNAMICS
WAVE EQUATIONS