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
Summary
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 (alternating direction methods) and schemes of splitting into physical processes. Also regionally additive schemes (domain decomposition methods)and unconditionally stable additive schemes of multi-component splitting are considered for evolutionary equations of first and second order as well as for sy
Notes
3.5.3 Peculiarities of hydrodynamic equations written in the primitive variables
