General Linear Methods for Ordinary Differential Equations Book

Preface xiii

1 Differential Equations and Systems 1

1.1 The initial value problem 1

1.2 Examples of differential equations and systems 3

1.3 Existence and uniqueness of solutions 9

1.4 Continuous dependence on initial values and the right-hand side 16

1.5 Derivatives with respect to parameters and initial values 22

1.6 Stability theory 27

1.7 Stiff differential equations and systems 37

1.8 Examples of stiff differential equations and systems 50

2 Introduction to General Linear Methods 59

2.1 Representation of general linear methods 59

2.2 Preconsistency, consistency, stage-consistency, and zero-stability 69

2.3 Convergence 74

2.4 Order and stage order conditions 81

2.5 Local discretization error of methods of high stage order 87

2.6 Linear stability theory of general linear methods 90

2.7 Types of general linear methods 95

2.8 Illustrative examples of general linear methods 100

2.8.1 Type 1: p = r = s = 2 and q = 1 or 2 100

2.8.2 Type 2: p = r = s = 2 and q = 1 or 2 104

2.8.3 Type 3: p = r = s = 2 and q = 1 or 2 106

2.8.4 Type 4: p = r = s = 2 and q = 1 or 2 109

2.9 Algebraic stability of general linear methods 112

2.10 Underlying one-step method 123

2.11 Starting procedures 125

2.12 Codes based on general linear methods 126

3 Diagonally Implicit Multistage Integration Methods 131

3.1 Representation of DIMSIMs 131

3.2 Representation formulas for the coefficient matrix B 133

3.3 A transformation for the analysis of DIMSIMs 139

3.4 Construction of DIMSIMs of type 1 143

3.5 Construction of DIMSIMs of type 2 148

3.6 Construction of DIMSIMs of type 3 152

3.7 Construction of DIMSIMs of type 4 157

3.8 Fourierseries approach to the construction of DIMSIMs of high order 168

3.9 Least-squares minimization 173

3.10 Examples of DIMSIMs of types 1 and 2 179

3.11 Nordsieck representation of DIMSIMs 181

3.12 Representation formulas for coefficient matrices P and G 185

3.13 Examples of DIMSIMs in Nordsieck form 189

3.14 Regularity properties of DIMSIMs 191

4 Implementation of DIMSIMs 201

4.1 Variable step size formulation of DIMSIMs 201

4.2 Local error estimation 204

4.3 Local error estimation for large step sizes 211

4.4 Construction of continuous interpolants 215

4.5 Step size and order changing strategy 217

4.6 Updating the vector of external approximations 220

4.7 Step-control stability of DIMSIMs 221

4.8 Simplified Newton iterations for implicit methods 227

4.9 Numerical experiments with type 1 DIMSIMs 229

4.10 Numerical experiments with type 2 DIMSIMs 233

5 Two-Step Runge-Kutta Methods 237

5.1 Representation of two-step Runge-Kutta methods 237

5.2 Order conditions for TSRK methods 239

5.3 Derivation of order conditions up to order 6 250

5.4 Analysis of TSRK methods with one stage 253

5.4.1 Explicit TSRK methods: s = 1, p = 2 or 3 253

5.4.2 Implicit TSRK methods: s = 1, p = 2 or 3 257

5.5 Analysis of TSRK methods with two stages 262

5.5.1 Explicit TSRK methods: s = 2, p = 2, q = 1 or 2 262

5.5.2 Implicit TSRK methods: s = 2, p = 2, q = 1 or 2 265

5.5.3 Explicit TSRK methods: s = 2, p = 4, or 5 267

5.5.4 Implicit TSRK methods: s = 2, p = 4, or 5 272

5.6 Analysis of TSRK methods with three stages 275

5.6.1 Explicit TSRK methods: s = 3, p = 3, q = 2 or 3 275

5.6.2 Implicit TSRK methods: s = 3, p = 3, q = 2 or 3 279

5.7 Two-step collocation methods 281

5.8 Linear stability analysis of two-step collocation methods 286

5.9 Two-step collocation methods with one stage 288

5.10 Two-step collocation methods with two stages 292

5.11 TSRK methods with quadratic stability functions 296

5.12 Construction of TSRK methods with inherent quadratic stability 303

5.13 Examples of highly stable quadratic polynomials and TSRK methods 307

6 Implementation of TSRK Methods 315

6.1 Variable step size formulation of TSRK methods 315

6.2 Starting procedures for TSRK methods 317

6.3 Error propagation, order conditions, and error constant 321

6.4 Computation of approximations to the Nordsieck vector and local error estimation 324

6.5 Computation of approximations to the solution and stage values between grid points 327

6.6 Construction of TSRK methods with a given error constant and assessment of local error estimation 328

6.7 Continuous extensions of TSRK methods 331

6.8 Numerical experiments 336

6.9 Local error estimation of two-step collocation methods 340

6.10 Recent work on two-step collocation methods 344

7 General Linear Methods with Inherent Runge-Kutta Stability 345

7.1 Representation of methods and order conditions 345

7.2 Inherent Runge-Kutta stability 348

7.3 Doubly companion matrices 355

7.4 Transformations between method arrays 364

7.5 Transformations between stability functions 374

7.6 Lower triangular matrices and characterization of matrices with zero spectral radius 381

7.7 Canonical forms of methods 384

7.8 Construction of explicit methods with IRKS and good balance between accuracy and stability 389

7.9 Examples of explicit methods with IRKS 393

7.9.1 Methods with p = q = 1 and s = 2 393

7.9.2 Methods with p = q = 2 and s = 3 395

7.9.3 Methods with p = q = 3 and s = 4 396

7.9.4 Methods with p = q = 4 and s = 5 398

7.9.5 Methods with large intervals of absolute stability on imaginary axis 400

7.10 Construction of A- and L-stable methods with IRKS 404

7.11 Examples of A- and L-stable methods with IRKS 406

7.12 Stiffly accurate methods with IRKS 408

8 Implementation of GLMs with IRKS 417

8.1 Variable step size formulation of GLMs 417

8.2 Starting procedures 419

8.3 Error propagation for GLMs 423

8.4 Estimation of local discretization error and estimation of higher order terms 432

8.5 Computing the input vector of external approximations for the next step 435

8.6 Zero-stability analysis 438

8.7 Testing the reliability of error estimation and estimation of higher order terms 444

8.8 Unconditional stability on nonuniform meshes 448

8.9 Numerical experiments 452

8.10 Local error estimation for stiffly accurate methods 456

8.11 Some remarks on recent work on GLMs 460

References 461

Index 477