NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS
II. Numerical Methods for ODE
II.1. General Idea of the NM for ODE. Euler's Methods
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General idea of the numerical methods for ODE
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Explicit, implicit, improved Euler's methods
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Local approximation error
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A-stability and monotonicity
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II.2. Runge-Kutta Methods
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Idea of the Runge--Kutta methods
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Butcher's tableaux
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Derivation of 1- and 2-stage Runge--Kutta methods
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Study of A-stability and monotonicity of the Runge--Kutta methods
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Adaptive Runge--Kutta methods
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II.3. Adams methods
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Idea of the multi-step methods
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Derivation of Adams--Bashforth methods
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Derivation of Adams--Moulton methods
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Practical questions, related to the implementation of multi-step methods
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Idea of predictor--corrector methods
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II.4 Error and convergence of the methods
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Why do we need practical error estimates and estimates of the order of convergence?
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Runge's method for practical error estimate
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Runge's method for practical estimate of the order of convergence
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Lax theorem for one-step methods for ODE
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III. Finite difference methods for PDE
III.1. Idea of finite difference methods. Finite difference methods for parabolic problems
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Introduction to PDEs; problems, described by PDEs
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Conservation laws, continuity equation
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Diffusion equation---derivation, physical interpretation
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Finite difference schemes---general concepts
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Explicit, strictly implicit scheme, Crank--Nikolson scheme for the diffusion equation---derivation, local approximation error, stability
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Dirichlet, Neumann, Robin boundary conditions; increasing the order of approximation for Neumann and Robin boundary conditions
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III.2. Stability of finite difference schemes
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Definition of stability in a given norm
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Stability with respect to initial conditions, boundary conditions, right-hand side
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Maximum principle and stability in discrete maximum norm
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Method of harmonics and stability in discrete l2-norm
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General form of Lax theorem
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III.3. Finite difference method for hyperbolic equations
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Transport equation--derivation, physical interpretation
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First order accurate schemes for the transport equation---stability (in maximum norm) and arising problems
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Lax-Wendroff scheme for the transport equation with second order of accuracy---study of stability (in l2-norm) and arising problems
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Finite difference methods for the string equation
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