Numerical Methods for Stiff Nonlinear and Quadratic Differential Equations.
Abstract
Generalizations of multistep and one-leg methods were defined for implicit differential systems with algebraic constraints. Such systems occur frequently in circuit analysis. An error analysis was carried out for three particular methods of this type, associated with an A() - contractive second-order Adams formula resulting from earlier research under this contract. Successful numerical tests of these methods were carried out. A complete set of affine invariants was derived for all two-by-two systems of first-order quadratic differential equations. These invariants provide a criterion for deciding to which affine equivalence class any given system belongs. It was shown that each contains a system which is equivalent to a single second-order equation in one unknown. Efficient numerical methods were developed for finding running solutions of the Josephson interferometer equations. Starting from perturbation solutions for small nonlinearities, the problem was formulated as a periodic boundary value problem and solved by finite differences and by continuation with respect to a parameter governing the 'degree of nonlinearity' of the problem. The existence of discontinuum of running solutions of the above mentioned equations was proved for arbitrarily strong nonlinearities with dissipation. For vanishing dissipation, the existence of a continuum of solutions was proved. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1979
- Accession Number
- ADA079303
Entities
People
- Werner Liniger
Organizations
- IBM Thomas J. Watson Research Center