Difference Methods and Their Extrapolations

1 General Properties.- 1.1. The Simplest Example.- 1.2. Expansion Theorems.- 1.3. Acceleration of Convergence.- 1.4. Correction by Higher-Order Differences.- 1.5. Various Extrapolation Methods.- 1.6. The Effects of Computational Errors.- 2 First-Order Ordinary Differential Equations.- 2.1. The Crank-Nicholson Scheme.- 2.2. Explicit Difference Schemes.- 2.3. The Splitting-Up Method for Systems of Equations.- 2.4. Equations with Singularities.- 3 The One-Dimensional Stationary Diffusion Equation.- 3.1. The Dirichlet Problem.- 3.2. Boundary-Value Problems of the Third Kind.- 3.3. Equations with Discontinuous Coefficients.- 3.4. The Sturm-Liouville Problem.- 3.5. Improving the Accuracy of the Finite Elements Method.- 3.6. The Quasilinear Problem.- 4 Elliptic Equations.- 4.1. The Statement of the Problem.- 4.2. Difference Methods for the Dirichlet Problem on a Domain with a Smooth Boundary.- 4.3. The Dirichlet Problem in a Rectangle.- 4.4. A Quasilinear Equation in a Triangular Region.- 4.5. On the Diffraction Problem.- 4.6. On the Separation of Singularities.- 5 Nonstationary Problems.- 5.1. The Simplest Type of Parabolic Equation.- 5.2. Increasing the Accuracy of the Splitting-Up Method.- 5.3. The Two-Dimensional Heat Equation.- 5.4. The Equation of Motion.- 6 Extrapolation for Algebraic Problems and Integral Equations.- 6.1. Regularization of a Singular System of Linear Algebraic Equations.- 6.2. Regularization of a System with a Selfadjoint Matrix.- 6.3. Extrapolation of Solutions Containing Boundary-Layer Functions.- 6.4. The Fredholm Equation of the Second Kind.- 6.5. The Volterra Equation of the Second Kind.- 6.6. The Volterra Equation of the First Kind.- 7 Appendix.- 7.1. Expansion of Difference Relations in the Mesh-Size.- 7.2. On the Solution of Some Special Systems of Equations.- 7.3. Some Results on the Lagrange Interpolation Polynomials.- List of Notation.