VOCATIONAL MATHEMATICS FOR GIRLS
1 The basic properties of Richardson extrapolation|1 11 The introduction of an initial value problem for systems of ODEs|3 12 Numerical treatment of initial value problems for systems of ODEs|6 13 An introduction to Richardson extrapolation|8 14 Accuracy of Richardson extrapolation|9 15 Evaluation of the error|10 16 Major drawbacks and advantages of Richardson extrapolation|11 17 Two implementations of Richardson extrapolation|17 18 Obtaining higher accuracy|20 19 General conclusions related to Chapter 1|21 110 Topics for further research|22 2 Richardson extrapolation for explicit Runge–Kutta methods|23 21 Stability function of one-step methods for solving systems of ODEs|25 22 Stability polynomials of explicit Runge–Kutta methods|30 23 Using Richardson extrapolation together with the scalar test problem|31 24 Impact of Richardson extrapolation on absolute stability properties|32 241 Stability regions related to the first-order one-stage explicit Runge–Kutta method|34 242 Stability regions related to the second-order two-stage explicit Runge–Kutta methods|35 243 Stability regions related to third-order three-stage explicit Runge–Kutta methods|36 244 Stability regions related to fourth-order four-stage explicit Runge–Kutta methods|38 245 On the use of complex arithmetic in the program for drawing the plots|39 25 Preparation of appropriate numerical examples|39 251 Numerical example with a large real eigenvalue|40 252 Numerical example with large complex eigenvalues|40 253 A nonlinear numerical example|42 26 Organization of the computations|44 27 Particular numerical methods used in the experiments|45 28 Numerical results|46 29 The development of methods with enhanced absolute stability properties|55 291 The derivation of two classes of numerical methods with good stability properties|56 292 Selecting particular numerical methods for Case 1: p = 3 and m = 4|60 293 Selecting particular numerical methods for Case 2: p = 4 and m = 6|62 294 Possibilities for further improvement of the results|68 210 General conclusions related to Chapter 2|72 211 Topics for further research|74 3 Linear multistep and predictor-corrector methods|75 31 Linear multistep methods for solving systems of ODEs|77 311 Order conditions|78 312 Basic definitions|78 313 Attainable order of linear multistep methods|80 314 Drawbacks and advantages of linear multistep methods|80 315 Frequently used linear multistep methods|82 32 Variation of the time-step size for linear multistep methods|83 321 Calculation of the coefficients of an LM VSVFM|84 322 Zero-stability properties of an LM VSVFM|85 33 Absolute stability of the linear multistep methods|86 34 Difficulties related to the implementation of Richardson extrapolation|90 35 Introduction of some predictor-corrector schemes|91 36 Local error estimation|93 37 Absolute stability of predictor-corrector schemes|98 38 The application of several different correctors|102 381 An example from the field of environmental modelling|103 382 Some absolute stability considerations related to environmental modelling|105 383 Numerical experiments|108 39 A-stability of the linear multistep methods|110 310 Coefficients of some popular linear multistep methods|111 311 General conclusions related to Chapter 3|111 312 Topics for further research|117 4 Richardson extrapolation for some implicit methods|119 41 Description of the class of ?-methods|120 42 Stability properties of the ?-methods|122 43 Combining the ?-method with Richardson extrapolation|126 44 Stability of Richardson extrapolation combined with ?-method|127 45 The problem with implicitness|134 451 Application of the classical Newton iterative method|135 452 Application of the modified Newton iterative method|138 453 Achieving better efficiency by keeping an old decomposition of the Jacobian matrix|139 454 Selecting stopping criteria|140 455 Richardson extrapolation and the Newton method|144 46 Numerical experiments|146 461 Atmospheric chemical scheme|146 462 Organization of the computations|149 463 Achieving second-order accuracy|152 464 Comparison of the ?-method with ? = 075 and the backward Euler formula|153 465 Comparing the computing times needed to obtain the prescribed accuracy|154 466 Using the trapezoidal rule in the computations|157 47 Using implicit Runge–Kutta methods|159 471 Fully implicit Runge Kutta methods|159 472 Diagonally implicit Runge–Kutta methods|160 473 Evaluating the reduction of the computational cost when DIRK methods are used|161 474 Applying Richardson extrapolation for fully implicit Runge–Kutta methods|163 475 Applying Richardson extrapolation for diagonally implicit Runge–Kutta methods|166 476 Stability results related to active Richardson extrapolation|167 477 Numerical experiments|170 48 General conclusions related to Chapter 4|178 49 Topics for further research|179 5 Richardson extrapolation for splitting techniques|181 51 Richardson extrapolation for sequential splitting|181 52 Derivation of the stability function for the sequential splitting procedure|183 53 Stability properties of the sequential splitting procedure|186 54 Some numerical experiments|192 541 Splitting the atmospheric chemical scheme|193 542 Organization of the computations|193 543 Results obtained when the backward Euler formula is used|195 544 Results obtained when the ?-method with ? = 075 is used|195 545 Results obtained when the trapezoidal rule is used|197 546 Some conclusions from the numerical experiments|197 55 Marchuk–Strang splitting procedure|198 551 Some introductory remarks|198 552 The Marchuk–Strang splitting procedure and Richardson extrapolation|200 553 Stability function of the combined numerical method|201 554 Selection of an appropriate class of Runge–Kutta methods|203 555 Absolute stability regions of the combined numerical methods|205 556 Some numerical results|207 557 Concluding remarks|213 56 General conclusions related to Chapter 5|213 57 Topics for further research|214 6 Richardson extrapolation for advection problems|215 61 The one-dimensional advection problem|216 62 Combining the advection problem with Richardson extrapolation|218 63 Implementation of Richardson extrapolation|220 64 Order of accuracy of the combined numerical method|221 65 Three numerical examples|225 651 Organization of the computations|225 652 Construction of a test problem with steep gradients of the unknown function|227 653 Construction of an oscillatory test problem|230 654 Construction of a test problem with discontinuous derivatives of the unknown function|231 655 Comparison of the four implementations of Richardson extrapolation|233 66 Multi-dimensional advection problem|235 661 Introduction of the multi-dimensional advection equation|235 662 Expanding the unknown function in Taylor series|236 663 Three special cases|241 664 Designing a second-order numerical method for multi-dimensional advection|243 665 Application of Richardson extrapolation|244 67 General conclusions related to Chapter 6|255 68 Topics for further research|256 7 Richardson extrapolation for some other problems|257 71 Acceleration of the speed of convergence for sequences of real numbers|258 711 Improving the accuracy in calculations related to the number ?|259 712 Definitions related to the convergence rate of some iterative processes|259 713 The Aitken scheme and Steffensen’s improvement|261 714 Richardson extrapolation for sequences of real numbers: An example|264 72 Application of Richardson extrapolation to numerical integration|268 73 General conclusions related to Chapter 7|271 74 Topics for further research|271 8 General conclusions|277 References|279