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· 분류 : 외국도서 > 과학/수학/생태 > 수학 > 수리분석
· ISBN : 9783031313424
· 쪽수 : 392쪽
목차
1 Ordinary differential equations
1.1 Initial value problems
1.2 Well-posedness
1.3 Discontinuous ODEs
1.4 Dissipative problems
1.5 Conservative problems
1.6 Stability of solutions
1.7 Exercises
2 Discretization of the problem
2.1 Domain discretization
2.2 Difference equations: the discrete counterpart of differential equations
2.2.1 Linear difference equations
2.2.2 Homogeneous case
2.2.3 Inhomogeneous case
2.3 Step-by-step schemes
2.4 A theory of one-step methods
2.4.1 Consistency
2.4.2 Zero-stability
2.4.3 Convergence
2.5 Handling implicitness
2.6 Exercises
3 Linear Multistep Methods
3.1 The principle of multistep numerical integration
3.2 Handling implicitness by fixed point iterations
3.3 Consistency and order conditions
3.4 Zero-stability
3.5 Convergence
3.6 Exercises
4 Runge-Kutta methods
4.1 Genesis and formulation of Runge-Kutta methods
4.2 Butcher theory of order
4.2.1 Rooted trees
4.2.2 Elementary differentials
4.2.3 B-series
4.2.4 Elementary weights
4.2.5 Order conditions
4.3 Explicit methods
4.4 Fully implicit methods
4.4.1 Gauss methods
4.4.2 Radau methods
4.4.3 Lobatto methods
4.5 Collocation methods
4.6 Exercises
5 Multivalue methods
5.1 Multivalue numerical dynamics
5.2 General linear methods representation
5.3 Convergence analysis
5.4 Two-step Runge-Kutta Methods
5.5 Dense output multivalue methods
5.6 Exercises
6 Linear stability
6.1 Dahlquist test equation
6.2 Absolute stability of linear multistep methods
6.3 Absolute stability of Runge-Kutta methods
6.4 Absolute stability of multivalue methods
6.5 Boundary locus
6.6 Unbounded stability regions
6.6.1 A-stability
6.6.2 Pade approximations
6.6.3 L-stability
6.7 Order stars
6.8 Exercises
7 Stiff problems
7.1 Looking for a definition
7.2 Prothero-Robinson analysis
7.3 Order reduction of Runge-Kutta methods
7.4 Discretizations free from order reduction
7.4.1 Two-step collocation methods
7.4.2 Almost collocation methods
7.4.3 Multivalue collocation methods free from order reduction
7.5 Stiffly-stable methods: backward differentiation formulae
7.6 Principles of adaptive integration
7.6.1 Predictor-corrector schemes
7.6.2 Stepsize control strategies
7.6.3 Error estimation for Runge-Kutta methods
7.6.4 Newton iterations for fully implicit Runge-Kutta methods
7.7 Exercises
8 Geometric numerical integration
8.1 Historical overview
8.2 Principles of nonlinear stability for Runge-Kutta methods
8.3 Preservation of linear and quadratic invariants
8.4 Symplectic methods
8.5 Symmetric methods
8.6 Backward error analysis
8.6.1 Modified differential equations
8.6.2 Truncated modified differential equations
8.6.3 Long-term analysis of symplectic methods
8.7 Long-term analysis of multivalue methods
8.7.1 Modified differential equations
8.7.2 Bounds on the parasitic components
8.7.3 Long-time conservation for Hamiltonian systems
8.8 Exercises
9 Numerical methods for stochastic differential equations
9.1 Discretization of the Brownian motion
9.2 Ito and Stratonovich integrals
9.3 Stochastic differential equations
9.4 One-step methods
9.4.1 Euler-Maruyama and Milstein methods
9.4.2 Stochastic ?-methods
9.4.3 Stochastic perturbation of Runge-Kutta methods
9.5 Accuracy analysis
9.6 Linear stability analysis
9.6.1 Mean-square stability
9.6.2 Mean-square stability of stochastic ?-methods
9.6.3 A-stability preserving SRK methods
9.7 Principles of stochastic geometric numerical integration
9.7.1 Nonlinear stability analysis: exponential mean-square contractivity
9.7.2 Mean-square contractivity of stochastic ?-methods
9.7.3 Nonlinear stability of stochastic Runge-Kutta methods
9.7.4 A glance to the numerics for stochastic Hamiltonian problems
9.8 Exercises
A Summary of test problems
Bibliography
