## Solving Differential Equations by Multistep Initial and Boundary Value MethodsThe numerical approximation of solutions of differential equations has been, and continues to be, one of the principal concerns of numerical analysis and is an active area of research. The new generation of parallel computers have provoked a reconsideration of numerical methods. This book aims to generalize classical multistep methods for both initial and boundary value problems; to present a self-contained theory which embraces and generalizes the classical Dahlquist theory; to treat nonclassical problems, such as Hamiltonian problems and the mesh selection; and to select appropriate methods for a general purpose software capable of solving a wide range of problems efficiently, even on parallel computers. |

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### Contents

Differential Equations | 1 |

Linear Difference Equations with Constant Coefficients | 15 |

Polynomials and Toeplitz Matrices | 51 |

Notes | 77 |

Generalized Backward Differentiation Formulae | 121 |

Notes | 140 |

Notes | 157 |

Symmetric Schemes | 159 |

Hamiltonian Matrices | 358 |

Symplectic Matrices | 360 |

B Answers to the Exercises | 363 |

Chapter 2 | 364 |

Chapter 3 | 370 |

Chapter 4 | 373 |

Chapter 5 | 380 |

Chapter 6 | 382 |

Notes | 183 |

Boundary Value Problems | 213 |

Mesh Selection Strategies | 237 |

Block BVMs | 279 |

Parallel Implementation of B2VMs | 301 |

Extensions and Applications to Special Problems | 325 |

Functions of matrices | 349 |

Mmatrices | 353 |

The Kronecker Product | 354 |

1 Use of Kronecker Product for Solving Matrix Equations | 357 |

Chapter 7 | 384 |

Chapter 8 | 387 |

Chapter 9 | 390 |

Chapter 11 | 391 |

Chapter 12 | 392 |

Appendix A | 394 |

399 | |

413 | |

### Common terms and phrases

A-stable additional methods algorithm asymptotically stable B2VMs behavior block boundary conditions boundary locus boundary value problems BVMs BVPs Chapter characteristic polynomial coefficients complex plane computed solution condition number Consequently constant continuous problem continuous solution continuously invertible convergence corresponding defined definition denote difference equation discrete problem discrete solution eigenvalues entries ETR2s example Exercise fact Figure formula function GAMs GBDF of order Hamiltonian imaginary axis initial conditions initial value problems inversive IVPs k-step Lemma linear system mesh midpoint method Moreover Neumann polynomial nonsingular numerical methods obtained parallel parameters permutation matrix perturbation polynomial p(z processors Proof regular Jordan curve root of unit round-off errors satisfy scalar Schur polynomial Section solution of problem solved stability region stepsize h stiff Suppose symmetric schemes symplectic T-matrices test equation Theorem Toeplitz Toeplitz matrix trapezoidal rule unit circumference unit disk unit modulus vector verifies zero