Decoupling and stability of algorithms for boundary value problems

The ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes. Therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals. It is shown that most methods (like multiple shooting, collocation, invariant imbedding and difference methods) derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later (for instance when solving a linear system). This decoupling is related to the dichotomy of the ordinary differential equations. In fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed. How this stability is related to the use of the boundary conditions is also investigated

A Modification and Application of Parametric Continuation Method to Variety of Nonlinear Boundary Value Problems in Applied Mechanics

In the field of engineering, researches often come across strong nonlinear boundary value problems which cannot be solved easily. Numerical convergence for many problems, typically solved by the Newton-Raphson linearization algorithm, is sensitive to the initial approach, relaxation parameters and differential topology. Emphasis in the present work is placed on the alternative approach, the so called parametric imbedding of a particular problem into the family of problems. While this may appear to complicate rather than to simplify the problem, its justification lies in the fact that a relation between infinitesimally close neighboring processes results in a simple Cauchy problem with respect to the introduced parameter. Many problems in applied mechanics are reduced to the solutions of systems of nonlinear algebraic, transcendental, differential or integral-differential equations containing an explicit parameter. These are problems in the areas of thermo-fluids, gas dynamics, deformable solids, heat transfer, biomechanics, analytical dynamics, catastrophe theory, optimal control and others. A parameter found in these models is not unique, and may be easily identified as a load which could be geometric, structural, and physical or it could be introduced artificially. An important aspect of these problems is a question of the variation of the solution when parameter is incrementally changed. The growing interest in nonlinear problems in engineering has been intensified by the use of digital computers. This paved a way in development of the solution procedures which can be applied to a large class of nonlinear problems containing a parameter. An important aspect of these problems is the variation of the solution of with the parameter. Hence, method of continuing the solution with respect to the parameter is a natural and universal tool for the analysis. It was originally introduced by Ambarzumian and Chandrasekar, and intensively studied by Bellman, Kalaba and others. Different problems of applied mechanics and physics with dominant nonlinearities due to convective phenomena, constituent models, finite deformation, bifurcation and others are analyzed and solved in the present work. The choice of the optimal continuation parameter, which ensures the best conditioning of the corresponding system of nonlinear equations, is discussed. Some modifications for stiff systems of ordinary nonlinear differential equations are suggested and applied. Effectiveness of the continuation method is demonstrated by comparing the results with the stiff boundary value problem numerical solvers implemented using commercial softwares. The objective of the research is to investigate applicability of the method as a universal approach to the wide range of nonlinear boundary value problems in different areas of mechanics: nonlinear mechanics of solids, bifurcation problems, Newtonian and Non-Newtonian fluids, thermo-fluids, gas-dynamics, control, inverse problems

Automatic computation of quantum-mechanical bound states and wavefunctions

We discuss the automatic solution of the multichannel Schr\"odinger equation. The proposed approach is based on the use of a CP method for which the step size is not restricted by the oscillations in the solution. Moreover, this CP method turns out to form a natural scheme for the integration of the Riccati differential equation which arises when introducing the (inverse) logarithmic derivative. A new Pr\"ufer type mechanism which derives all the required information from the propagation of the inverse of the log-derivative, is introduced. It improves and refines the eigenvalue shooting process and implies that the user may specify the required eigenvalue by its index

Stable continuous orthonormalization techniques for linear boundary value problems

An investigation is made of a hybrid method inspired by Riccati transformations and marching algorithms employing (parts of) orthogonal matrices, both being decoupling algorithms. It is shown that this so-called continuous orthonormalisation is stable and practical as well. Nevertheless, if the problem is stiff and many output points are required the method does not give much gain over, say, multiple shootin

Locally one dimensional numerical methods for multidimensional free surface problems

Issued as Progress reports [1-5], and Final fiscal report, Project no. G-37-60

A dynamic programming approach to the formulation and solution of finite element equations

A method for formulating and algorithmically solving the equations of finite element problems is presented. The method starts with a parametric partition of the domain in juxtaposed strips that permits sweeping the whole region by a sequential addition (or removal) of adjacent strips. The solution of the difference equations constructed over that grid proceeds along with the addition removal of strips in a manner resembling the transfer matrix approach, except that different rules of composition that lead to numerically stable algorithms are used for the stiffness matrices of the strips. Dynamic programming and invariant imbedding ideas underlie the construction of such rules of composition. Among other features of interest, the present methodology provides to some extent the analyst's control over the type and quantity of data to be computed. In particular, the one-sweep method presented in Section 9, with no apparent counterpart in standard methods, appears to be very efficient insofar as time and storage is concerned. The paper ends with the presentation of a numerical exampl

Numerical simulation of wall-bounded turbulent shear flows

Developments in three dimensional, time dependent numerical simulation of turbulent flows bounded by a wall are reviewed. Both direct and large eddy simulation techniques are considered within the same computational framework. The computational spatial grid requirements as dictated by the known structure of turbulent boundary layers are presented. The numerical methods currently in use are reviewed and some of the features of these algorithms, including spatial differencing and accuracy, time advancement, and data management are discussed. A selection of the results of the recent calculations of turbulent channel flow, including the effects of system rotation and transpiration on the flow are included

Investigation of optimization of attitude control systems, volume i

Optimization of attitude control systems by development of mathematical model and computer program for space vehicle simulatio