HIGH ACCURACY NUMERICAL METHODS FOR THE SOLUTION OF NON-LINEAR BOUNDARY VALUE PROBLEMS

Transcription

1 ABSTRACT Of The Thesis Entitled HIGH ACCURACY NUMERICAL METHODS FOR THE SOLUTION OF NON-LINEAR BOUNDARY VALUE PROBLEMS Submitted To The University of Delhi In Partial Fulfillment For The Award of The Degree of DOCTOR OF PHILOSOPHY (MATHEMATICS) By DEEPIKA DHALL DEPARTMENT OF MATHEMATICS UNIVERSITY OF DELHI DELHI INDIA MARCH, 202

2 ABSTRACT Title of the thesis: High Accuracy Numerical Methods for the Solution of Non-linear Boundary Value Problems In current times, nonlinear differential equations have a lot of attention, because many physical problems in science and engineering are described mathematically by nonlinear differential equations in one or more than one dependent/ independent variables. The closed form solution of these differential equations often arising in applications cannot be obtained although the existence and uniqueness of the solution is easier to establish. Consequently, one is obliged to devise appropriate stable numerical techniques for determination of approximate solutions using modern day computers. Two-point nonlinear singular boundary value problems occur in a number of areas of applied mathematics and engineering, such as in beam theory, electric circuit analysis etc. Elliptic equations typically model steady-state or equilibrium phenomena, and so there is no temporal dependence. Elliptic equations may also arise in solving time-dependent problems if we are modelling some phenomena that are always in local equilibrium and equilibrate on time scales that are much faster than the time scale being modelled. Elliptic equations give boundary value problems where the solutions at all points must be simultaneously determined based on the boundary conditions all around the domain. High-order accurate finite difference schemes are important in scientific computation because they offer a means to obtain accurate solutions with less work than may be required for methods of lower accuracy. Elliptic problems, common in engineering applications can be solved to second order using simple (or central) finite difference schemes that are mostly compact. Fourth order finite difference schemes have become quite popular as against the other lower order accurate schemes which require high mesh refinement and hence are computationally inefficient. On the other hand, the higher order accuracy of the fourth order compact methods combined with the compactness of the difference stencil yields highly accurate numerical solutions on relatively coarser grids with greater computational efficiency. The high order compact method which we consider here is different in that the governing differential equation is used to approximate the lower order derivative terms with the imbedding technique. The scheme is difficult to develop due to the need of extensive algebraic manipulation, especially for non-linear problems. However, once high order method developed, it can be incorporated easily in application. A number of approaches to derive suitable digital computer algorithms have been developed. Finite Difference Methods are one means of obtaining approximate solutions to ordinary or partial differential equations. Other methods include finite elements, finite volumes, spectral methods, various spline approximations etc. Finite difference methods are attractive because of the relative ease of implementation and flexibility. The advances in computer technology and the introduction of numerical computing applications like MATLAB, Mathematica has led to improvements in the numerical methods that are used. Consequently, many recalcitrant scientific and engineering problems that involve linear and nonlinear singular ordinary and partial differential equations that were previously unsolved can now be resolved by using Page 2

3 appropriate numerical methods. In this thesis, an attempt has been made to develop third order discretizations for the solution of nonlinear two point boundary value problems using variable mesh, when the forcing function is in integral form. We also devote our attention on the application of iterative algorithms, Two-Parameter Alternating Group Explicit (TAGE) method and Newton TAGE iterative method to a class of singular differential equations. These methods being explicit in nature and coupled compactly are suited for use on parallel architectures. We have also developed some new efficient high order accurate methods based on cubic spline approximations for the solution of two spatial dimensions nonlinear second order elliptic partial differential equations. The thesis is set up into seven chapters followed by the list of references useful for the development and application of the methods discussed in the thesis. Each chapter contains a set of example problems solved to validate the developed methods. A brief description of the contents of each chapter is as follows. The first chapter is an introductory chapter which presents some formal mathematical concepts needed to develop highly accurate numerical schemes for solutions of boundary value problems. We review important concepts of linear matrix algebra which plays a crucial role in setting up and in analyzing the convergence properties of the numerical methods. Various iterative methods along with its convergence for the solution of linear and nonlinear system of equations are studied. It is known that the main problem which arises in the solutions of elliptic problems is the solution of large sparse sets of algebraic equations. The principal weapon in the solution of these equations is the iterative solution. A brief introduction of the Alternating Group Explicit (AGE) iterative method and the Newton-AGE iterative method for the solution of two point boundary value problems are given in this chapter. This chapter also presents a summary of the work done in the thesis. In the second chapter, using three grid points, on a variable mesh we derive a third order accurate numerical method based on Numerov type discretization for the solution of second order nonlinear two-point boundary value problem with integral homogeneous functions 0 y f ( x, y, y) K x, s ds, 0 x, 0 s subject to natural boundary conditions y 0 A, y B where A and B are finite constants. We assume that the conditions required for the above boundary value problem to have a unique solution are satisfied. These equations arise naturally in different fields of physics, fluid dynamics, biological models, chemical kinetics such as electric circuit analysis, scattering theory, colloidal dispersion and many body problems. For the numerical solution of the given two-point BVP, we discretize the solution region with the non-uniform mesh such that 0 x0 x... xn. Our method consists of three grid points xl, xl and xl, where xl xl hl and x l xl hl. The mesh ratio is l hl hl. For l, it reduces to the constant mesh case. The standard 5- point discretization of integro-differential equation under consideration is obtained using Page 3

4 third order variable mesh approximations for y and y. This requires the use of fictitious points outside the solution region. However, the third order variable mesh method, which we present in this chapter, requires no fictitious points for incorporating the boundary conditions. Throughout our discussion, we have considered N as odd, that is the proposed method is applicable only when the internal grid points of the solution space are odd in number. It is observed that the proposed technique is not directly applicable in case of equations with a singularity present in the solution domain. In that case, we have modified our method in such a way that the solutions retain their order and accuracy everywhere in the solution region including the vicinity of the singularity. We also discuss the application of Two Parameter Alternating Group Explicit (TAGE) and Newton-TAGE iteration methods along with its convergence to solve both linear and nonlinear variable mesh difference equations. The application of TAGE and Newton-TAGE methods confirm the superiority over the corresponding SOR and Newton-SOR methods in terms of number of iterations. Furthermore, it is evident from the numerical results that the order of the method for constant mesh case is nearly equal to four. During last four decades, there has been growing interest in developing and using highly accurate numerical methods based on cubic spline approximations for the solution of nonlinear differential equations. In third chapter, we report an efficient high order numerical method on a non-uniform mesh based on cubic spline approximation and application of TAGE and Newton-TAGE iterative methods for the solution of two-point nonlinear boundary value problems, whose forcing functions are in integral form 0 u ( x, u, u) K x, s ds, 0 x, s The two point Dirichlet type boundary conditions are given by u 0, u 0 where 0, are finite constants. The method is applicable when the internal grid points of solution interval are odd in number. This cubic spline based method is applicable to integrodifferential equations, both linear and nonlinear, having singularities. We also discuss the application of TAGE and Newton-TAGE iterative methods and error analysis of the method in details. The proposed TAGE and Newton-TAGE iteration methods show superiority over the corresponding successive over relaxation (SOR) iteration methods in terms of number of iterations. However, for k = the proposed method reduces to a constant mesh method of accuracy of order four. In Chapter 4, using three variable mesh points, we discuss a new numerical method of 3 accuracy of Oh ( k ) based on arithmetic average discretizations for the solution of the second order nonlinear boundary value problem with source function in integral form 0 u F( x, u, u) K x, s ds, 0 x, s The two point boundary conditions associated are given by: Page 4

5 u 0, u 0 where 0, are finite constants. Many authors have discussed various techniques for numerical integration and methods for approximate solution of two point boundary value problems and their applications to various physical models. The presented variable mesh approximation is directly applicable to the integro-differential equations with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of difference scheme for the diffusion convection equation is discussed in details. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the methods discussed in Chapters 2, 3 in which the internal grid points are strictly odd in number. Although the proposed variable mesh method involves more algebra and hence greater computational time is involved, but applicable to the solution space having both odd and even number of internal grid points. The numerical results indicate that the proposed method is computationally nearly equal to the method discussed in Chapter 2 and is applicable to the solution space with all internal grid points. Some examples of the application to problems with Dirichlet boundary conditions are presented and results obtained are compared with another new variable mesh method of lower order accuracy described in this chapter for the given integro-differential equation. In Chapter 5, we report a new nine-point compact discretization of order two in y- and order four in x-directions, based on cubic spline approximation, for the solution of two dimensional quasi-linear elliptic partial differential equations of the form defined in the domain with boundary, where and in. The corresponding Dirichlet boundary conditions are prescribed by, The main spline relations are presented and incorporated into solution procedures for elliptic partial differential equations. Available numerical methods based on cubic spline approximations for the numerical solution of quasi-linear elliptic equations are of 2 2 O( y x ) accurate. Although 9-point finite difference approximations of O( y y x x ) accurate for the solution of nonlinear and quasi-linear elliptic differential equations have been discussed in past, but these methods require five evaluations of the function f. In this chapter, using the same number of grid points and three evaluations of the function f, we have derived a new stable cubic spline method of O( y y x x ) accuracy for the solution of given quasi-linear elliptic equation. However, for a fixed parameter, the proposed method behaves like a fourth order method. The accuracy of the proposed method is exhibited from the computed results. The Page 5

6 proposed method is applicable to Poisson s equation in polar coordinates and two dimensional Burgers equation, which is main highlight of the work. The convergence analysis of the proposed cubic spline approximation for the nonlinear elliptic equation is discussed in details and we have shown under appropriate conditions the proposed method converges. In Chapter 6, using nine-point compact stencil, we discuss a new Numerov type stable method of order two in y- and order three in x-directions on a variable mesh based on cubic spline approximations for the solution of two-dimensional nonlinear elliptic boundary value problems The corresponding Dirichlet boundary conditions are prescribed by, We use cubic spline approximations in -direction and second order finite difference approximations in -direction. It has been experienced in the past that for problems in polar coordinates the solution for high order methods usually deteriorates in the vicinity of the singularities. We have overcome this difficulty by modifying our method in such a way that the solution retains its order and accuracy everywhere in the solution region even in the vicinity of the singularity. Convergence analysis of the method is briefly discussed. The available numerical methods for the solution of two dimensional nonlinear elliptic boundary value problems on a non-uniform mesh are of first order accurate only and from application point of view in most cases the method is unstable. In this chapter, we have developed a new stable high order nine point compact scheme of ( l l ) O k k h h based on cubic spline approximations for the solution of two dimensional nonlinear elliptic boundary value problems. The method is successfully applied to Poisson s equation in cylindrical polar coordinates and two-dimensional Burgers equation with high Reynolds number. The numerical results confirm that the proposed method produces oscillation free solutions for high Reynolds number, whereas the corresponding lower method becomes unstable. The last chapter presents conclusions of the results presented in the thesis and few problems for future research work. Page 6