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Notations and terminology for ODE's and systems of ODE's;
reduction of higher order ODE's to 1st order systems of ODE's; the
fundamental existence and uniqueness thm. for ODE's (Lipschitz condition) |
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Introduction of Euler's method, order of Euler's method,
one step methods (introduction, definition, consistency, local truncation
error) |
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Explicit Runge-Kutta (ERK) methods (introduction of the
method in the general case, notations in the general case, derivation of ERK
of second order); Runge-Kutta method of fourth order |
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Examples of implicit methods: trapezoidal rule, midpoint
rule, the theta method, and the implicit Euler's method; computation of
orders for these methods |
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Convergence of one-step methods (the general case; see
also convergence for Euler's method, etc) |
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Asymptotic expansions for the global discretization error
for one step methods, and applications to error estimate |
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Practical implementation of one step methods |
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Linear Multistep methods: examples, derivation using the
Lagrange interpolation polynomial |
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Linear multistep methods: definition and computations of
the local truncation error, order of the method, consistency |
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Implicit and explicit linear multistep methods,
predictor-corrector methods |
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Examples of consistent multistep methods which diverge |
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Linear difference equations: stability (root) condition,
general solution |
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Convergence Thm. for linear multistep methods |
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Order and consistency for linear multistep methods |
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Adaptive methods for one-step and multi-step methods,
error control, Milne device, extrapolation |
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Stiff differential equations, stability and intervals
(regions) of absolute stability, A-stable methods, BDF methods |
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Numerical methods and stability for systems of ODE's |
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Finite difference methods for linear BVP |
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Functional (fixed point) iteration and Newton's iteration
for solving systems of ODE's using an implicit method |
