An efficient computational scheme for solving nonlinear, two-point boundary-value problems via the method of adjoint variables
Coker, Estelle M.
Master of Science
A method for solving nonlinear differential equations of the form x - <f>(x,t) =, £ t <_ 1, subject to boundary conditions of the form w(x()) = , ÿ(x(l)) = / is developed. It is assumed that t is a scalar, x and <J> are n-vectors, u is a p-vector, and ^ is a q-vector, with p + q ** n. The method is based on the consideration of the performance index P, the cumulative error in the differential equations and the boundary conditions. The differential equations and the boundary conditions are linearized about a nominal function x(t); the linearized system is embedded into a more general system by means of a scaling factor a, <_ a <_ 1, applied to each forcing term. The variations per unit stepsize A(t) = Ax(t)/a are governed by a system of n linear differential equations, subject to p separated initial conditions and q separated final conditions. Then, the system is solved employing the method of adjoint variables. The scaling factor a is determined by a bisection process, starting from a = 1, so as to ensure the decrease of the performance index P. Convergence to the desired solution is achieved when the inequality P £ e is met, where e is a small, preselected number. Two updating schemes are considered, called Scheme (a) and Scheme (b) for easy identification. In Scheme (a), the initial point x() is updated according tox(O) =x() +aA(), and the new nominal function x(t) is obtained by forward integration of the nonlinear differential equations. In Scheme (b), the function xCt) is updated according to x(t) = x(t) + aA(t). Four numerical examples are solved using the ITEL AS/6 computer of Rice University. The computational scheme developed here for the method of adjoint variables is particularly efficient, in that it minimizes the algorithmic work per iteration, namely, the number of integrations to be performed in order to solve the linear, two-point boundary-value problem. In the method developed by Roberts and Shipman (Ref. 4), the number of integrations is n, where n is the number of state variables. In this thesis, we show that the number of integrations can be reduced to q, where q < n is the number of final conditions.