Sufficient Stability Criteria and Uniform Stability of Difference Schemes

We prove two new criteria for the sufficiency of the von Neumann condition for stability of difference schemes. The first criterion is that the von Neumann criterion is sufficient for stability if a finite power of the amplification matrix is a uniformly diagonalizable matrix. The second criterion relaxes the uniform diagonalizability requirement for the amplification matrix: The uniform diagonalizability is needed only in some subregion of the parameter values, and for the remaining parameter values, all the eigenvalues of the amplification matrix should be strictly less than unity in modulus. The numerical investigation of the behavior of the norms of powers of amplification matrix has pointed to the advisability of introducing a new definition, the uniform stability. We prove constructive criteria for uniform stability. We investigate the satisfaction of the obtained uniform stability criteria for a number of well-known difference schemes for the numerical solution of fluid dynamics problems.