Numerical methods for nonlinear partial differential equations of Gross-Pitaevskii type and their discrete analogues

Date of Completion

January 2002






The subject of this thesis work is numerical solution of the time-independent Gross-Pitaevskii equation -Du+Vr,s,t u+ku3=mu, 0.1 lim r,s,t→∞ u=0,- ∞

ur,s,t2 drdsdt=1, 0.2 which we write here in the unit-less form. ^ This equation plays a central role in modeling of the Bose-Einstein condensates. The constant of non-linearity, k, is proportional to the number of particles in the condensate, and therefore can be very large. The macroscopic behavior of Bose-Einstein condensate is highly sensitive to the shape of the trapping potential V(r, s, t). Typically in our numerical experiments it is taken to be in the form of the a-harmonic well: V (r, s, t) = ar2 + bs2 + ct2, where constants are such that a > 0, b > 0, and c > 0. ^ The phenomena of the Bose-Einstein Condensation and corresponding mathematical models are subject of active theoretical and numerical research. ^ We analyze application of various discretization schemes and properties of the resulting discrete equations. We also study the convergence behavior of the obtained iterative processes with the purpose of developing of fast and efficient numerical procedures for computing the positive unitary solution of (0.1), (0.2). ^ A new efficient method for the solution of the time-independent Gross-Pitaevskii partial differential equation in three spatial variables is the main result of this work. High accuracy of the solution, combined with the speed of the algorithm with which the accuracy is achieved, are the main reasons for presenting the algorithm. ^ We also study the application of the Newton-type iteration for solving the discretized Gross-Pitaevskii equation. We show that the resulting Newton iteration is monotone and converges globally. ^