Numerical Methods for Stiff Differential Equations, and Josephson Interferometer Equations

摘要

Results for stiff differential equation include: (1) For all k > or = 1, the (k-1)-parameter family of all A-contractive k-step formulas of order of accuracy p = 2 was derived for variable steps. A-contractive one-leg methods are guaranteed to be stable for the variable coefficient test problem x(dot)=lambda(t)x with any lambda(t), Re lambda(t) < or = 0, and arbitrary step sequences; (2) A stability theorem was proved for A sub 0-contractive one-leg methods applied to diagonally dominant (but not necessarily monotone) nonlinear systems; (3) A procedure was devised for selecting an A-contractive two-step method at every step of the integration in such a way as to minimize a global error bound; and (4) It was shown that the sets of all A-stable and A-contractive formulas with p = k = 2 can be identified by a local analysis in the transfer or time domain pertaining to slowly varying oscillatory solutions. Results on Josephson equations include: (1) The response of an extended junction to a time-dependent quasi-periodic voltage source was studied analytically. The existence and uniqueness of a quasi-periodic solution was established in the presence of dissipation and for small non-linearities; (2) The numerical procedure for studying the response of the two-junction interferometer was modified to increase its accuracy and efficiency. Standard birfucation of the solution branch, as well as period-doubling bifurcation were found, causing the I-V curves to be multi-valued and hysteretic.