Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques

In this thesis, an efficient algorithm and a code BVPDI is developed for solving Boundary Value Problems (BVPs) for Ordinary Differential Equations (ODEs). A generalised variable order variable stepsize Direct Integration (01) method, a generalised Backward Differentiation method (BDF) and shooting techniques are used to solve the given BVP. When using simple shooting technique, sometimes stability difficulties arise when the differential operator of the given ODE contains rapidly growing and decaying fundamental solution modes. Then the initial value solution is very sensitive to small changes in the initial condition. In order to decrease the bound of this error, the size of domains over which the Initial Value Problems (IVPs) are integrated has to be restricted. This leads to the multiple shooting technique, which is generalisation of the simple shooting technique. Multiple shooting technique for higher order ODEs with automatic partitioning is designed and successfully implemented in the code BVPDI, to solve the underlying IVP. The well conditioning of a higher order BVP is shown to be related to bounding quantities, one involving the boundary conditions and the other involving the Green's function. It is also shown that the conditioning of the multiple shooting matrix is related to the given BVP. The numerical results are then compared with the only existing direct method code COLNEW. The advantages in computational time and the accuracy of the computed solution, especially, when the range of interval is large, are pointed out. Also the advantages of BVPDI are clearer when the results are compared with the NAG subroutine D02SAF (reduction method). Stiffness tests for the system of first order ODEs and the techniques of identifying the equations causing stiffness in a system a rediscussed. The analysis is extended for the higher order ODEs. Numerical results are discussed indicating the advantages of BVPDI code over COLNEW. The success of the BVP DI code applied to the general class of BVPs is the motivation to con sider the same code for a special class of second order BVPs called Sturm-Liouville (SL) problems. By the application of Floquet theory and shooting algorithm, eigenvalues of SL problems with periodic boundary conditions are determined without reducing to the first order system of equations. Some numerical examples are given to illustrate the success of the method. The results are then compared, when the same problem is reduced to the first order system of equations and the advantages are indicated. The code BVPDI developed in this thesis clearly demonstrates the efficiency of using DI Method and shooting techniques for solving higher order BVP for ODEs

Unitary Integrators and Applications to Continuous Orthonormalization Techniques

This is the published version, also available here: http://dx.doi.org/10.1137/0731014.In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coefficient matrices in the linear case and a related structure in the nonlinear case. These skew systems arise in a number of applications, and interest originates from application to continuous orthogonal decoupling techniques. In this case, the matrix system has a cubic nonlinearity. Numerical integration schemes that compute a unitary approximate solution for all stepsizes are studied. These schemes can be characterized as being of two classes: automatic and projected unitary schemes. In the former class, there belong those standard finite difference schemes which give a unitary solution; the only ones are in fact the Gauss–Legendre point Runge–Kutta (Gauss RK) schemes. The second class of schemes is created by projecting approximations computed by an arbitrary scheme into the set of unitary matrices. In the analysis of these unitary schemes, the stability considerations are guided by the skew-Hermitian character of the problem. Various error and implementation issues are considered, and the methods are tested on a number of examples

An efficient shooting algorithm for Evans function calculations in large systems

In Evans function computations of the spectra of asymptotically constant-coefficient linear operators, a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ODE. For small systems, a fast, shooting algorithm may be obtained by representing subspaces as single exterior products \cite{AS,Br.1,Br.2,BrZ,BDG}. For large systems, however, the dimension of the exterior-product space quickly becomes prohibitive, growing as $\binom{n}{k}$, where $n$ is the dimension of the system written as a first-order ODE and $k$ (typically $\sim n/2$) is the dimension of the subspace. We resolve this difficulty by the introduction of a simple polar coordinate algorithm representing ``pure'' (monomial) products as scalar multiples of orthonormal bases, for which the angular equation is a numerically optimized version of the continuous orthogonalization method of Drury--Davey \cite{Da,Dr} and the radial equation is evaluable by quadrature. Notably, the polar-coordinate method preserves the important property of analyticity with respect to parameters.Comment: 21 pp., two figure

Unsteady residual distribution schemes for transition prediction

In this work, the unsteady simulation of the Navier–Stokes equations is carried out by using a Residual Distribution Schemes (RDS) methodology. This algorithm has a compact stencil (cell-based computations) and uses a finite element like method to compute the residual over the cell. The RDS method has been successfully proven in steady Navier–Stokes computation but its application to fully unsteady configurations is still not closed, because some of the properties of the steady counterpart can be lost. Here, we proposed a numerical solution for unsteady problems that is fully compatible with the original approach. In order to check the method, we chose a very demanding test case, namely the numerical simulation of a Tollmien–Schlichting (TS) wave in a 2D boundary layer. The evolution of this numerical perturbation is accurately computed and checked against theoretical results

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

Stability of highly cooled hypervelocity boundary layers

The influence of high levels of wall cooling on the stability of hypervelocity boundary layers is investigated. Such conditions are relevant to experiments in high-enthalpy impulse facilities, where the wall temperature is much smaller than the free-stream temperature, as well as to some real flight scenarios. Some effects of wall cooling are well known, for instance, the stabilization of the first mode and destabilization of the second mode. In this paper, several new instability phenomena are investigated that arise only for high Mach numbers and high levels of wall cooling. In particular, certain unstable modes can travel supersonically with respect to the free stream, which changes the nature of the dispersion curve and leads to instability over a much wider band of frequencies. The cause of this phenomenon, the range of parameters for which it occurs and its implications for boundary layer stability are examined. Additionally, growth rates are systematically reported for a wide range of conditions relevant to high-enthalpy impulse facilities, and the stability trends in terms of Mach number and wall temperature are mapped out. Thermal non-equilibrium is included in the analysis and its influence on the stability characteristics of flows in impulse facilities is assessed

Stable Continuous Orthonormalization Techniques for Linear Boundary Value Problems

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim