## 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 |

The local truncation errors associated with the Pade approxim | 124 |

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

HYPERBOLIC EQUATIONS | 175 |

ELLIPTIC EQUATIONS AND SYSTEMATIC | 239 |

Comments on the solution of difference equations covering | 257 |

A worked example covering each method | 263 |

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 |

334 | |

A sufficient condition for convergence | 269 |

### Common terms and phrases

2-cyclic analytical solution Assume block tridiagonal boundary conditions calculate central-difference characteristic classical explicit coefficients column vector components consistently ordered constant corresponding Crank-Nicolson equations Crank-Nicolson method curve defined denote derivatives diagonal difference equations difference scheme discontinuity discretization error dU/dt dU/dx eigenvalues eigenvectors elements elimination equa exact solution Example Exercise finite finite-difference equations finite-difference solution follows Gauss-Seidel iteration Gerschgorin's given gives Hence hyperbolic equation implicit initial conditions initial values iterative methods iterative values Jacobi iteration matrix known boundary values L0-stable Laplace's equation matrix form mesh lengths mesh points moduli non-zero numerical solution ordering vector Pade approximant parabolic equation partial differential equation permutation matrix pivotal values Poisson's equation problem proved rate of convergence rounding errors second-order shown shows solution domain solution values solved SOR iteration spectral radius stability Table tends to zero time-level time-step tion tridiagonal matrix truncation error unconditionally stable unknowns written