Numerical Methods Contents - SAM

Partial derivative:
dG
dz (0,z) = I
Implicit function theorem: for sufficiently small |h| the equation G(h,z) = 0 defines a continuous
function z = z(h).
Recall the interpretation of the y k as approximations of y(t k ):
Ψ(h,y) ≈ Φ h y , (11.2.6)
where Φ is the evolution operator (→ Def. 11.1.6) for ẏ = f(y).
The Euler methods provide approximations for evolution operator for ODEs
How to interpret the sequence (y k ) N k=0 from (11.2.1)?
△
This is what every single step method does: it tries to approximate the evolution operator Φ for an
ODE by a mapping of the type (11.2.5).
➙ mapping Ψ from (11.2.5) is called discrete evolution.
By “geometric insight” we expect: y k ≈ y(t k )
Vice versa:
a mapping Ψ as in (11.2.5) defines a single step method.
(Throughout, we use the notation y(t) for the exact solution of an IVP.)
If we are merely interested in the final state y(T), then the explicit Euler method will give us the
answer y N .
Ôº ½½º¾
☞ In a sense, a single step method defined through its associated discrete evolution does not
approximate and initial value problem, but tries to approximate an ODE.
Ôº ½½º¾
If we are interested in an approximate solution y h (t) ≈ y(t) as a function [t 0 , T] ↦→ R d , we have to
do
Definition 11.2.1 (Single step method (for autonomous ODE)).
Given a discrete evolution Ψ : Ω ⊂ R × D ↦→ R d , an initial state y 0 , and a temporal mesh
M := {t 0 < t 1 < · · · < t N = T } the recursion
post-processing = reconstruction of a function from y k , k = 0,...,N
y k+1 := Ψ(t k+1 − t k ,y k ) , k = 0,...,N − 1 , (11.2.7)
Technique: interpolation, see Ch. 8
defines a single step method (SSM, ger.: Einschrittverfahren) for the autonomous IVP ẏ = f(y),
y(0) = y 0 .
Simplest option: piecewise linear interpolation (→ Sect. 9.2.1) ➙ Euler polygon, see Fig. 134.
Procedural view of discrete evolutions:
Abstract single step methods
Ψ h y ←→ function y1 = esvstep(h,y0) .
( function y1 = esvstep(@(y) rhs(y),h,y0) )
Recall Euler methods for autonomous ODE ẏ = f(y):
explicit Euler: y k+1 = y k + h k f(y k ) ,
implicit Euler: y k+1 : y k+1 = y k + h k f(y k+1 ) .
Both formulas provide a mapping
Ôº ½½º¾
(y k , h k ) ↦→ Ψ(h,y k ) := y k+1 . (11.2.5)
✎ Notation: Ψ h y := Ψ(h,y)
Concept of single step method according to Def. 11.2.1 can be generalized to non-autonomous ODEs,
which leads to recursions of the form:
y k+1 := Ψ(t k ,t k+1 ,y k ) , k = 0,...,N − 1 ,
Ôº ½½º¾