An entropy based approach to nonlinear stability
Abstract
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. For example, certain TVD schemes tend to square a smooth pulse. These can be detected a priori by their negative entropy production rates. A nonlinear stability proof is given for discrete scalar equations arising from a conservation law. Solutions to such equations are bounded in the L2 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. Stability in the L2 norm is not sufficient to exclude expansion shocks, oscillations, and other unphysical phenomena. Numerical experiments suggest that a more restrictive condition, a positive entropy production rate in each cell, is both necessary and sufficient to exclude such phenomena.
 Publication:

Ph.D. Thesis
 Pub Date:
 1989
 Bibcode:
 1989PhDT........33M
 Keywords:

 Computational Fluid Dynamics;
 Entropy;
 Flow Stability;
 Conservation Laws;
 Nonlinearity;
 Scalars;
 Thermodynamics;
 Fluid Mechanics and Heat Transfer