Alexei Yurievich Vinogradov 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
AlexeiVinogradov@yandex.ru,
+7(963)991-05-10, +7(977)810-55-23 (WhatsApp, Viber)
Table of contents
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Table of contents. |
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2 |
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Introduction. |
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4 |
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Chapter 1. Known formulas of the theory of matrices for ordinary differential equations. |
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10 |
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Chapter 2. Improvement of S.K.Godunov’s method of orthogonal sweep for solving boundary value problems with stiff ordinary differential equations. |
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12 |
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2.1. The formula for the beginning of the calculation by S.K.Godunov’s sweep method. |
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12 |
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2.2. The second algorithm for the beginning of the calculation by S.K.Godunov’s sweep method. |
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16 |
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2.3. The replacement of the Runge-Kutta’s numerical integration method in S.K.Godunov’s sweep method. |
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17 |
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2.4 Matrix-block realizations of algorithms for starting calculation by S.K.Godunov’s sweep method. |
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17 |
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2.5. Conjugation of parts of the integration interval for S.K.Godunov’s sweep method. |
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20 |
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2.6. Properties of the transfer of boundary value conditions in S.K.Godunov’s sweep method. |
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22 |
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2.7. Modification of S.K.Godunov’s sweep method. |
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23 |
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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. |
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25 |
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Chapter 4. The method of "additional boundary value conditions" for solving boundary value problems with non-stiff ordinary differential equations. |
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26 |
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Chapter 5. The method of "half of the constants" for solving boundary value problems with non-stiff ordinary differential equations. |
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29 |
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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. |
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31 |
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6.1. The method of "transfer of boundary value conditions" to any point of the interval of integration. |
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31 |
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6.2. The case of "stiff" differential equations. |
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33 |
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6.3. Formulas for computing the vector of a particular solution of inhomogeneous system of differential equations. |
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35 |
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6.4. Applicable formulas for orthonormalization. |
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39 |
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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. |
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41 |
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Chapter 8. Calculation of shells of composite and with frames by the simplest method of "conjugation of sections of the integration interval". |
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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. |
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43 |
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8.2. Composite shells of rotation. |
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44 |
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8.3. Frame, expressed not by differential, but algebraic equations. |
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47 |
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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. |
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51 |
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Appendix. Computational experiments (a C++ program). |
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55 |
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List of published works. |
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64 |
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