Numerical methods of solving stiff and non-stiff

Boundary value problems

 

 

 

Propositions: Improvement of S.K.Godunov’s method of orthogonal sweep, 3 methods for non-stiff cases of boundary value problems, 2 methods for stiff cases of boundary value problems, 1 method for calculating composite shells and with frames, a C++ program for the best method proposed.

 

Monograph

 

 

2019 Moscow, Russia

[email protected],

+7(963)991-05-10, +7(977)810-55-23 (WhatsApp, Viber)

 

Table of contents

Table of contents.		2
Introduction.		4
Chapter 1. Known formulas of the theory of matrices for ordinary differential equations.		10
Chapter 2. Improvement of S.K.Godunov’s method of orthogonal sweep for solving boundary value problems with stiff ordinary differential equations.		12
2.1. The formula for the beginning of the calculation by S.K.Godunov’s sweep method.		12
2.2. The second algorithm for the beginning of the calculation by S.K.Godunov’s sweep method.		16
2.3. The replacement of the Runge-Kutta’s numerical integration method in S.K.Godunov’s sweep method.		17
2.4 Matrix-block realizations of algorithms for starting calculation by S.K.Godunov’s sweep method.		17
2.5. Conjugation of parts of the integration interval for S.K.Godunov’s sweep method.		20
2.6. Properties of the transfer of boundary value conditions in S.K.Godunov’s sweep method.		22
2.7. Modification of S.K.Godunov’s sweep method.		23
Chapter 3. The method of "transferring of boundary value conditions" (the direct version of the method) for solving boundary value problems with non-stiff ordinary differential equations.		25
Chapter 4. The method of "additional boundary value conditions" for solving boundary value problems with non-stiff ordinary differential equations.		26
Chapter 5. The method of "half of the constants" for solving boundary value problems with non-stiff ordinary differential equations.		29
Chapter 6. The method of "transferring of boundary conditions" (step-by-step version of the method) for solving boundary value problems with stiff ordinary differential equations.		31
6.1. The method of "transfer of boundary value conditions" to any point of the interval of integration.		31
6.2. The case of "stiff" differential equations.		33
6.3. Formulas for computing the vector of a particular solution of inhomogeneous system of differential equations.		35
6.4. Applicable formulas for orthonormalization.		39
Chapter 7. The simplest method for solving boundary value problems with stiff ordinary differential equations without orthonormalization - the method of "conjugation of sections of the integration interval", which are expressed by matrix exponents.		41
Chapter 8. Calculation of shells of composite and with frames by the simplest method of "conjugation of sections of the integration interval".
8.1. The variant of recording of the method for solving stiff boundary value problems without orthonormalization - the method of "conjugation of sections, expressed by matrix exponents "- with positive directions of matrix formulas of integration of differential equations.		43
8.2. Composite shells of rotation.		44
8.3. Frame, expressed not by differential, but algebraic equations.		47
8.4. The case where the equations (of shells and frames) are expressed not with abstract vectors, but with vectors, consisting of specific physical parameters.		51
Appendix. Computational experiments (a C++ program).		55
List of published works.		64

Introduction.

Relevance of the problem:

The solution to the problem of weight reduction of structures is associated with their complication and the use of thin-walled elements. Even the simplest variant method of constructive optimization requires parametric studies on a computer using numerical methods for solving boundary value problems. The most famous among them are:

- finite-difference methods for constructing approximate solutions of differential equations using finite-difference approximations of derivatives;

- various modifications of the finite element method, the Bubnov-Galerkin method, the Rayleigh-Ritz method, which are based on approximations of solutions of differential equations by finite linear combinations of given functions: polynomials, trigonometric functions, etc .;

- methods for the numerical determination of the integrals of the ordinary differential equations Runge-Kutta, Volterra, Picard, etc.

The main success of the methods of finite differences and finite elements is that on their basis universal algorithms are constructed and packages of applied programs for calculating complex structures are created. The constructed computing facilities are able to detect the flow of forces in the structure and, therefore, the most strained elements of it. Nevertheless, they require unjustifiably high costs of the programmer's efforts and powerful computing tools when the task is to determine the stresses in the places of their concentration.

The most obvious effectiveness of the methods of numerical integration of ordinary differential equations consists in calculating individual parts of complex spatial structures and their individual thin-walled elements with the refinement of the stress-strain state in the places of its rapid change. Efficiency is determined by the small expenses of the programmer's work, by the small expenditure of computer time and the computer's operational memory.

Thus, increasing the effectiveness of known numerical methods, constructing their modifications and constructing new methods, is an urgent research task.

The proposed scientific novelty consists in the following:

1. S.K.Godunov’s method of orthogonal sweep has been improved,

2. A method of "transferring of boundary value conditions" (a direct version of the method) is proposed for solving boundary value problems with non-stiff ordinary differential equations,

3. A method of "additional boundary value conditions" is proposed for solving boundary value problems with non-stiff ordinary differential equations,

4. A "half of constants" method is proposed for solving boundary value problems with non-stiff ordinary differential equations,

5. A method of "transferring of boundary value conditions" (step-by-step version of the method) is proposed for solving boundary value problems with stiff ordinary differential equations,

6. The simplest method for solving boundary value problems with stiff ordinary differential equations without orthonormalization is proposed - the method of "conjugation of sections of the integration interval", which are expressed by matrix exponentials,

7. The simplest method for calculating the shells of composite and with frames is proposed.

 

 

Some of the works on which the methods are based are published jointly with Dr.Sc. Professor Yu.I.Vinogradov.

Contribution of Dr.Sc. Professor Yu.I. Vinogradov in these joint publications consisted either of 1) in the discussion of the results of verification calculations of those formulas and methods proposed by A.Yu. Vinogradov, or that 2) in addition to A.Yu.Vinogradov’s methods Yu.I.Vinogradov proposed statement that the Cauchy matrix can be computed not only in the form of matrix exponentials, but in addition there is the possibility of calculating them in the sense of Cauchy-Krylov functions, using for this purpose the analytical solutions of the systems of differential equations of the structural mechanics of plates and shells obtained by someone, that 3) Yu.I. Vinogradov proposed his own, different from A.Yu.Vinogradov’s formula, the formula for computing the vector of the particular solution of an inhomogeneous system of ordinary differential equations, which, however, looks much more complicated than the simple A.Yu.Vinogradov’s formula.

Also, in the co-authors of some articles, Yu.A.Gusev and Yu.I.Klyuyev. Their contribution to the publication material consisted in performing multivariate verification calculations in accordance with the formulas, algorithms and methods proposed by A.Yu. Vinogradov in his Ph.D. thesis. The Ph.D. thesis was defended in 1996.

In addition, we can say that on the basis of the material from A.Yu.Vinogradov’s Ph.D. thesis completed two more candidate's physical and mathematical dissertations under the direction of Yu.I.Vinogradov, whose material consists mainly of a multivariate application and verification by calculations of what was proposed by A.Yu.Vinogradov in his Ph.D. thesis, in application to various concrete problems of construction mechanics of thin-walled shells with the identification and analysis of properties of formulas, algorithms and methods from A.Yu.Vinogradov’s Ph.D. thesis.

 

Here are the data of these two dissertations:

Year: 2008 Petrov, Vitaliy Igorevich "Reduction of boundary value problems to initial problems and investigation of stress concentration in thin-walled constructions by the multiplicative method"

Scientific degree: Candidate of Physical and Mathematical Sciences

Specialty code of VAC: 05.13.18 Specialty: Mathematical modeling, numerical methods and program complexes.

 

Year: 2003 Gusev, Yuri Alekseevich "Multiplicative algorithms for the transfer of boundary conditions in problems of the mechanics of shell deformation"

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