دانلود کتاب Additive Operator-Difference Schemes: Splitting Schemes

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Preface\nNotation\n1 Introduction\n 1.1 Numerical methods\n 1.2 Additive operator-difference schemes\n 1.3 The main results\n 1.4 Contents of the book\n2 Stability of operator-difference schemes\n 2.1 The Cauchy problem for an operator-differential equation\n 2.1.1 Hilbert spaces\n 2.1.2 Linear operators in a finite-dimensional space\n 2.1.3 Operators in a finite-dimensional Hilbert space\n 2.1.4 The Cauchy problem for an evolutionary equation of first order\n 2.1.5 Systems of linear ordinary differential equations\n 2.1.6 A boundary value problem for a one-dimensional parabolic equation\n 2.1.7 Equations of second order\n 2.2 Two-level schemes\n 2.2.1 Key concepts\n 2.2.2 Stability with respect to the initial data\n 2.2.3 Stability with respect to the right-hand side\n 2.2.4 Schemes with weights\n 2.3 Three-level schemes\n 2.3.1 Stability with respect to the initial data\n 2.3.2 Reduction to a two-level scheme\n 2.3.3 P-stability of three-level schemes\n 2.3.4 Estimates in simpler norms\n 2.3.5 Stability with respect to the right-hand side\n 2.3.6 Schemes with weights for equations of first order\n 2.3.7 Schemes with weights for equations of second order\n 2.4 Stability in finite-dimensional Banach spaces\n 2.4.1 The Cauchy problem for a system of ordinary differential equations\n 2.4.2 Scheme with weights\n 2.4.3 Difference schemes for a one-dimensional parabolic equation\n 2.5 Stability of projection-difference schemes\n 2.5.1 Preliminary observations\n 2.5.2 Stability of finite element techniques\n 2.5.3 Stability of projection-difference schemes\n 2.5.4 Conditions for -stability of projection-difference schemes\n 2.5.5 Schemes with weights\n 2.5.6 Stability with respect to the right-hand side\n 2.5.7 Stability of three-level schemes with respect to the initial data\n 2.5.8 Stability with respect to the right-hand side\n 2.5.9 Schemes for an equation of first order\n3 Operator splitting\n 3.1 Time-dependent problems of convection-diffusion\n 3.1.1 Differential problem\n 3.1.2 Semi-discrete problem\n 3.1.3 Two-level schemes\n 3.2 Splitting operators in convection-diffusion problems\n 3.2.1 Splitting with respect to spatial variables\n 3.2.2 Splitting with respect to physical processes\n 3.2.3 Schemes for problems with an operator semibounded from below\n 3.3 Domain decomposition methods\n 3.3.1 Preliminaries\n 3.3.2 Model boundary value problems\n 3.3.3 Standard finite difference approximations\n 3.3.4 Domain decomposition\n 3.3.5 Problems with non-self-adjoint operators\n 3.4 Difference schemes for time-dependent vector problems\n 3.4.1 Preliminary discussions\n 3.4.2 Statement of the problem\n 3.4.3 Estimates for the solution of differential problems\n 3.4.4 Approximation in space\n 3.4.5 Schemes with weights\n 3.4.6 Alternating triangle method\n 3.5 Problems of hydrodynamics of an incompressible fluid\n 3.5.1 Differential problem\n 3.5.2 Discretization in space\n 3.5.3 Peculiarities of hydrodynamic equations written in the primitive variables\n 3.5.4 A priori estimate for the differential problem\n 3.5.5 Approximation in space\n 3.5.6 Additive difference schemes\n4 Additive schemes of two-component splitting\n 4.1 Alternating direction implicit schemes\n 4.1.1 Problem formulation\n 4.1.2 The Peaceman–Rachford scheme\n 4.1.3 Stability of the Peaceman–Rachford scheme\n 4.1.4 Accuracy of the Peaceman–Rachford scheme\n 4.1.5 Another ADI scheme\n 4.2 Factorized schemes\n 4.2.1 General considerations\n 4.2.2 ADI methods as factorized schemes\n 4.2.3 Stability and accuracy of factorized schemes\n 4.2.4 Regularization principle for constructing factorized schemes\n 4.2.5 Factorized schemes of multicomponent splitting\n 4.3 Alternating triangle method\n 4.3.1 General description of the alternating triangle method\n 4.3.2 Investigation of stability and convergence\n 4.3.3 Three-level additive schemes\n 4.3.4 Problems with non-self-adjoint operators\n 4.4 Equations of second order\n 4.4.1 Model problem\n 4.4.2 Factorized schemes\n 4.4.3 Schemes of the alternating triangle method\n5 Schemes of summarized approximation\n 5.1 Additive formulations of differential problems\n 5.1.1 Model problem\n 5.1.2 Intermediate problems\n 5.1.3 Summarized approximation concept\n 5.1.4 Schemes of the second-order summarized approximation\n 5.2 Investigation of schemes of summarized approximation\n 5.2.1 Schemes of componentwise splitting\n 5.2.2 Estimates for the intermediate problem solutions\n 5.2.3 Stability of componentwise splitting schemes\n 5.2.4 Convergence of componentwise splitting schemes\n 5.2.5 Convergence of additive schemes in Banach spaces\n 5.3 Additively averaged schemes\n 5.3.1 Differential problem\n 5.3.2 Additive schemes\n 5.3.3 Stability of additively averaged schemes\n 5.4 Other variants of componentwise splitting schemes\n 5.4.1 Fully implicit additive schemes\n 5.4.2 ADI methods as additive schemes\n 5.4.3 Additive schemes with second-order accuracy\n 5.4.4 Convergence of higher-order schemes\n6 Regularized additive schemes\n 6.1 Multiplicative regularization of difference schemes\n 6.1.1 Regularization principle for difference schemes\n 6.1.2 Additive regularization\n 6.1.3 Multiplicative regularization\n 6.2 Multiplicative regularization of additive schemes\n 6.2.1 The Cauchy problem for a first-order equation\n 6.2.2 Regularization of additive schemes\n 6.2.3 Stability and convergence\n 6.2.4 Regularized and additively averaged schemes\n 6.3 Schemes of higher-order accuracy\n 6.3.1 Statement of the problem\n 6.3.2 Explicit three-level scheme\n 6.3.3 Regularized schemes\n 6.3.4 Additively averaged scheme\n 6.4 Regularized schemes for equations of second order\n 6.4.1 Model problem\n 6.4.2 Regularized scheme\n 6.4.3 Additively averaged schemes for equations of second order\n 6.5 Regularized schemes with general regularizers\n 6.5.1 General regularizers\n 6.5.2 Additive schemes with a general-form regularizer\n 6.5.3 Factorized additive schemes\n 6.5.4 Generalizations\n7 Schemes based on approximations of a transition operator\n 7.1 Operator-difference schemes\n 7.1.1 Operator-differential problem\n 7.1.2 Difference approximations in time\n 7.1.3 SM-stable schemes for problems with a self-adjoint operator\n 7.1.4 Factorized SM-stable two-level schemes\n 7.1.5 Problems with a skew-symmetric operator\n 7.2 Additive schemes with a multiplicative transition operator\n 7.2.1 Operator-differential problems\n 7.2.2 Componentwise splitting schemes\n 7.3 Splitting schemes with an additive transition operator\n 7.3.1 Additive approximation of a transition operator\n 7.3.2 Additive schemes\n 7.3.3 Regularized additive schemes\n 7.4 Further additive schemes\n 7.4.1 Schemes of the second order\n 7.4.2 Factorized schemes\n 7.4.3 Inhomogeneous approximation of a transition operator\n 7.4.4 Schemes of higher-order approximation\n8 Vector additive schemes\n 8.1 Vector schemes for first-order equations\n 8.1.1 Vector differential problem\n 8.1.2 Stability of vector additive schemes\n 8.1.3 Stability with respect to the right-hand side\n 8.2 Stability of vector additive schemes in Banach spaces\n 8.2.1 Problem formulation\n 8.2.2 Vector additive scheme\n 8.2.3 Study on stability\n 8.3 Schemes of second-order accuracy\n 8.3.1 Statement of the problem\n 8.3.2 Three-level vector schemes\n 8.3.3 Schemes of the alternating triangle method\n 8.4 Vector schemes for equations of second order\n 8.4.1 The Cauchy problem for a second-order equation\n 8.4.2 Vector problem\n 8.4.3 Scheme with weights\n 8.4.4 Additive schemes\n 8.4.5 Stability of additive schemes\n9 Iterative methods\n 9.1 Basics of iterative methods\n 9.1.1 Problem formulation\n 9.1.2 Simple iteration method\n 9.1.3 The Chebyshev iterative method\n 9.1.4 Two-level variation-type methods\n 9.1.5 Conjugate gradient method\n 9.2 Iterative alternating direction method\n 9.2.1 Iterative method with two-component splitting\n 9.2.2 Convergence study\n 9.2.3 Modified iterative method of alternating directions\n 9.2.4 Multicomponent splitting\n 9.3 Iterative alternating triangle method\n 9.3.1 Iterative method\n 9.3.2 Convergence rate\n 9.3.3 Modified iterative method of alternating triangles\n 9.4 Iterative cluster aggregation methods\n 9.4.1 Transition to a system of equations\n 9.4.2 Iterative method\n 9.4.3 Parallel variant\n 9.4.4 Aggregation of unknowns\n10 Splitting of the operator at the time derivative\n 10.1 Schemes with splitting of the operator at the time derivative\n 10.1.1 Preliminary discussions\n 10.1.2 Statement of the problem\n 10.1.3 Vector problem\n 10.1.4 Vector additive schemes\n 10.1.5 Generalizations\n 10.2 General splitting\n 10.2.1 Preliminary discussions\n 10.2.2 Problem formulation\n 10.2.3 Scheme with weights\n 10.2.4 Schemes with a diagonal operator\n 10.2.5 The more general problem\n 10.3 Explicit-implicit splitting schemes\n 10.3.1 Introduction\n 10.3.2 Boundary value problems for systems of equations\n 10.3.3 Schemes with a diagonal operator\n 10.3.4 General case\n11 Equations with pairwise adjoint operators\n 11.1 Splitting schemes for a system of equations\n 11.1.1 Preliminary discussions\n 11.1.2 Statement of the problem\n 11.1.3 A priori estimates\n 11.1.4 Schemes with weights\n 11.1.5 Splitting schemes to find the p-th component of the solution\n 11.1.6 Additive schemes for systems of equations\n 11.2 Additive schemes for a system of first-order equations\n 11.2.1 Statement of the problem\n 11.2.2 Examples\n 11.2.3 Schemes with weights\n 11.2.4 Explicit-implicit schemes\n 11.2.5 Additive schemes of componentwise splitting\n 11.2.6 Regularized additive schemes\n 11.3 Another class of systems of first-order equations\n 11.3.1 Problem formulation\n 11.3.2 Scheme with weights\n 11.3.3 Additive schemes\n 11.3.4 More general problems\n 11.3.5 Problems of hydrodynamics\nBibliography\nIndex