## Computational methods in partial differential equations |

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### Contents

Basic linear algebra | 1 |

Parabolic equations | 17 |

Elliptic equations | 100 |

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### Common terms and phrases

amplification matrix arbitrary biharmonic equation boundary value problem calculation computation constant convergence difference approximation difference equation difference replacement difference scheme Dirichlet domain of dependence Douglas-Rachford eigenvalues eigenvectors elimination elliptic equations equation 14 Example Exercise explicit difference formula figure finite difference methods flow formula 26 function grid spacing heat conduction hyperbolic equations implicit formula initial boundary value initial condition intermediate boundary values internal grid points iteration parameter iterative method L.O.D. methods L2 norm Laplace's equation Lax-Wendroff method leads linear lines parallel locally one-dimensional matrix of order method for solving method of solution Neumann problem non-linear norm obtained parabolic equation parameter sequence partial differential equations Peaceman-Rachford method plane rectangular region replacement of equation respectively result satisfied Show sin2 solving equation symmetric symmetric matrix theoretical solution totality of equations tridiagonal truncation error Un+1 unconditionally stable unit square variable coefficients vector wave equation written