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Additive operator-difference schemes : splitting schemes /
Applied mathematical modeling isconcerned with solving unsteady problems. This bookshows how toconstruct additive difference schemes to solve approximately unsteady multi-dimensional problems for PDEs. Two classes of schemes are highlighted: methods of splitting with respect to spatial variables (al...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Berlin :
De Gruyter,
[2013]
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface; Notation; 1 Introduction; 1.1 Numerical methods; 1.2 Additive operator-difference schemes; 1.3 The main results; 1.4 Contents of the book; 2 Stability of operator-difference schemes; 2.1 The Cauchy problem for an operator-differential equation; 2.1.1 Hilbert spaces; 2.1.2 Linear operators in a finite-dimensional space; 2.1.3 Operators in a finite-dimensional Hilbert space; 2.1.4 The Cauchy problem for an evolutionary equation of first order; 2.1.5 Systems of linear ordinary differential equations; 2.1.6 A boundary value problem for a one-dimensional parabolic equation.
- 2.1.7 Equations of second order2.2 Two-level schemes; 2.2.1 Key concepts; 2.2.2 Stability with respect to the initial data; 2.2.3 Stability with respect to the right-hand side; 2.2.4 Schemes with weights; 2.3 Three-level schemes; 2.3.1 Stability with respect to the initial data; 2.3.2 Reduction to a two-level scheme; 2.3.3 P-stability of three-level schemes; 2.3.4 Estimates in simpler norms; 2.3.5 Stability with respect to the right-hand side; 2.3.6 Schemes with weights for equations of first order; 2.3.7 Schemes with weights for equations of second order.
- 2.4 Stability in finite-dimensional Banach spaces2.4.1 The Cauchy problem for a system of ordinary differential equations; 2.4.2 Scheme with weights; 2.4.3 Difference schemes for a one-dimensional parabolic equation; 2.5 Stability of projection-difference schemes; 2.5.1 Preliminary observations; 2.5.2 Stability of finite element techniques; 2.5.3 Stability of projection-difference schemes; 2.5.4 Conditions for -stability of projection-difference schemes; 2.5.5 Schemes with weights; 2.5.6 Stability with respect to the right-hand side.
- 2.5.7 Stability of three-level schemes with respect to the initial data2.5.8 Stability with respect to the right-hand side; 2.5.9 Schemes for an equation of first order; 3 Operator splitting; 3.1 Time-dependent problems of convection-diffusion; 3.1.1 Differential problem; 3.1.2 Semi-discrete problem; 3.1.3 Two-level schemes; 3.2 Splitting operators in convection-diffusion problems; 3.2.1 Splitting with respect to spatial variables; 3.2.2 Splitting with respect to physical processes; 3.2.3 Schemes for problems with an operator semibounded from below; 3.3 Domain decomposition methods.
- 3.3.1 Preliminaries3.3.2 Model boundary value problems; 3.3.3 Standard finite difference approximations; 3.3.4 Domain decomposition; 3.3.5 Problems with non-self-adjoint operators; 3.4 Difference schemes for time-dependent vector problems; 3.4.1 Preliminary discussions; 3.4.2 Statement of the problem; 3.4.3 Estimates for the solution of differential problems; 3.4.4 Approximation in space; 3.4.5 Schemes with weights; 3.4.6 Alternating triangle method; 3.5 Problems of hydrodynamics of an incompressible fluid; 3.5.1 Differential problem; 3.5.2 Discretization in space.