## The operator approach to problems of stability and convergence of solutions of difference equations and the convergence of various iteration procedures |

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### Contents

Stability Analysis for Other Difference Schemes for | 13 |

Convergence of Various Iteration Procedures | 22 |

Stability Analysis of Various Difference Schemes | 31 |

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absolute values alternative proof AM(x Appendix approximation Assume boundary condition 1.3A bounded by lattice Brauer's theorem components conclusion Consider corresponding counterpart Crank-Nicolson difference determinantal equation diagonal elements difference equation 7.3 difference scheme Dirichlet problem discussed eigen eigenvectors equation of heat equations derived error vector expression heat conduction identical implicit difference equation iteration scheme Laplace differential equation line following equation m*+i main diagonal matrix whose elements Minkowski inequality moduli Neumann problem numerically largest eigenvalue numerically smaller obtain points in class previous analysis principal diagonal problem of convergence proving the stability quadratic equation readily seen replace Richardson iteration procedure Richardson process roots Round-off Errors satisfied second member Section Similarly smaller than unity Stability and Convergence system of equations tion true solution truncation error two-dimensional uh+i uk+i ultimately yields un+i upperbound ut+i value of p(x vergence whence ultimately Xr's zero matrix