## Numerical Solution of Partial Differential Equations: Finite Difference MethodsSubstantially revised, this authoritative study covers the standard finite difference methods of parabolic, hyperbolic, and elliptic equations, and includes the concomitant theoretical work on consistency, stability, and convergence. The new edition includes revised and greatly expanded sections on stability based on the Lax-Richtmeyer definition, the application of Pade approximants to systems of ordinary differential equations for parabolic and hyperbolic equations, and a considerably improved presentation of iterative methods. A fast-paced introduction to numerical methods, this will be a useful volume for students of mathematics and engineering, and for postgraduates and professionals who need a clear, concise grounding in this discipline. |

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### Contents

FINITE | 11 |

ALTERNATIVE | 111 |

ii The eigenvalueeigenvector solution of dVdt AV | 134 |

iv Richardsons deferred approach to the limit | 141 |

A comparison of results for methods i ii and iii for | 147 |

Introduction to the analytical solution of homogeneous differ | 153 |

HYPERBOLIC EQUATIONS | 175 |

difference equations | 197 |

A worked example covering each method | 263 |

A sufficient condition for convergence | 269 |

Eigenvalues of the Jacobi and SOR iteration matrices and | 275 |

Theoretical determination of the optimum relaxation parame | 282 |

Introduction to 2cyclic matrices and consistent ordering | 288 |

The ordering vector for a block tridiagonal matrix | 294 |

Stones strongly implicit iterative method | 302 |

A recent direct method | 309 |

### Common terms and phrases

2-cyclic accuracy accurate analytical solution approximation associated Assume block boundary conditions boundary values calculate Chapter characteristic clearly coefficients column components Consider consistently ordered constant continuous convergence corresponding Crank-Nicolson curve defined definition denote dependent derivatives diagonal difference equations directions discontinuity domain easily eigenvalues eigenvectors elements elimination equa equal Example Exercise explicit expressed finite finite-difference follows formula function given gives Hence hyperbolic implicit increases independent initial conditions initial values iteration matrix Jacobi iteration known leads length linear lines mesh points method norm obtained ordering vector Padé approximant partial differential equation particular positive problem proved reference replaced respect result roots satisfies scheme shown shows side Similarly solved square stability Substitution symmetric TABLE temperature theorem time-level tion tridiagonal truncation error Ujj+1 unknowns usually variables written