Questions for the final test
----------------------------



General notions and definitions
===============================


Numerical differentiation, three ways to derive formulas:
interpolation, Taylor series approach, method of undetermined
coefficients

Newton interpolating polynomial with equispaced points.
Divided differences, forward differences, backward differences

Stable and asymptoticaly stable solutions of differential equation

Conditions of stability for the system y'=Ay

Conditions of stability for higher order linear ODE
with constant coefficients

Difference operator for one step methods

Local truncation error (d_n) and the order of an algorithm

Local error (l_n), relation to d_n

Gloabal convergence

0-stability, region of absolute stability, A-stability for a one step method

The use of a homogeneous and of an inhomogeneous test equation (3.28)
for determining absolute stability and stiff decay properties of algorithms.

Stiff equations and methods with a stiff decay property

Rough problems

Taylor methods

General formulation of Runge-Kutta methods: formulation withs Y_i and
with K_i

Implicit Runge-Kutta methods based on collocation.
(Gauss, Lobatto, and Radau nmethods).

Applications of pairs of methods (such as Fehlberg, Dormand-Prince, etc)

Step doubling


Multistep linear (MSL) methods:
Stable and asymptotically stable difference equations, root condition.

Difference operator and 0-stability for a multistep method.

All types of stability for MSL methods (stable, asymptotically stable,
weakly/strongly stable)



Things to derive or to prove
============================

Taylor methods
--------------

Thm 1.1 p.9, (no proof) AP (Ascher & Petzold)
Thm 3.1 (prove it)

Derive implicit trapezoidal method (p.60), check for order, absolute stability,
stiff decay.

Derive implicit and explicit midpoint
methods, check order, absolute stability;
show that these methods have (or do not --- whichever is correct) the stiff
decay property.

Analyze the explicit trapezoid method (p.78)

Know what methods (both RK and MLS) and what families
are A-stable, what methods/families have the stiff decay properties,
and which methods do not...


One step methods
----------------

General formulation of Runge-Kutta methods

Derive the formula for the regions of absolute
stability for RK methods (Equation (4.19) and the one above it).

Derive necessary order conditions for Runge-Kutta
methods, make conclusion that p cannot exceed s.

Explain the use of pairs of methods (such as Fehlberg or
Dormand-Prince) and derive an expression for the desired step
size to achieve a given local error.

Step doubling (how and what for)

Define collocaation equation(s) and show that it leads to a RK method.
Be able to design lower order methods of these
families by hand, given collocation points.

Know properties of these families of methods as
compared to each other (i.e. which one is more
accurate, more stable, has or does not have
stiff decay etc, without proof).


Multu-Step Linear methods (MSL methods)
---------------------------------------

Adams-Moulton, Adams-Bashforth, and BDF.
Know basic idea, be able to derive
lower-order methods of these families.

Derive the formula for the BDF method, p. 129.

Characteristic polynomials of an MSL method

Derive order conditions  (Section 5.2.1)

Prove that for a MSL method 0-stability and consistency
imply convergence  (p. 137).

Derive general solution to a homgeneous difference
equation (in the simplest case of distinct roots).

Root conditions for stability of an MSL method
(Section 5.2.2). Explain how this condition is related
to stability of the difference equations.


Predictor/corrector methods, different
versions: describe the idea and the use of
these methods (p.144).

Examples 5.4, 5.5, 5.6, 5.7, 5.8, 5.9.


General notions and skills (some overlap with the topics above)
===============================================================

Consistency of the differential equation. Know the definitions,
Know how to check it using Taylor series and the formulas for the MSL case.

0-stability.  Know the definitions both for the single step
methods and multistep methods.  Know how to check 0-stability
for MSL's (root condition) and know the definitions of strong
stability and weak stability.  Be clear on how consistency
plus 0-stability imply convergence.

Regions of absolute stability.  Know how to check if a value
of z=h*lambda is in the region of absolute stability for any
of the methods.

Stiffness: What is it, how do you recognize it. Be able to
give examples.

Be ready to choose an appropriate method for solving an IVP;
justify your choice based on stiffness, accuracy, and
smoothness, etc.