Numerical Methods Contents - SAM

Consider discrete evolution of explicit Euler method (11.2.1) for ODE ẏ = My
Ψ h y = y + hMy ↔ y k+1 = y k + hMy k .
Perform the same transformation as above on the discrete evolution:
1.5
1
0.5
V −1 y k+1 = V −1 y k + hV −1 MV(V −1 y k )
z k :=V −1 y k
⇔
(z k+1 ) i = (z k ) i + hλ i (z k )
} {{ } i .
ˆ= explicit Euler step for ż i = λ i z i
(12.2.5)
Im z
0
−0.5
✁ {z ∈ C: |1 + z| < 1}
Crucial insight:
−1
The explicit Euler method generates uniformly bounded solution sequences (y k ) ∞ k=0 for ẏ =
My with diagonalizable matrix M ∈ R d,d with eigenvalues λ 1 , ...,λ d , if and only if it generates
✗
✖
uniformly bounded sequences for all the scalar ODEs ż = λ i ż, i = 1,...,d.
An analoguous statement is true for all Runge-Kutta methods!
(This is revealed by simple algebraic manipulations of the increment equations.)
So far we conducted the model problem analysis under the premises λ < 0.
However: in Ex. 12.2.1 we have λ 1/2 = 1 2 α ±i √β − 1 4 α2 (complex eigenvalues!). How will explicit
Euler/explicity RK-methods respond to them?
Example 12.2.3 (Explicit Euler method for damped oscillations).
Consider linear model IVP (12.1.1) for λ ∈ C:
Re λ < 0 ⇒ exponentially decaying solution y(t) = y 0 exp(λt) ,
because | exp(λt)| = exp(Re λt).
✔
✕
Ôº½ ½¾º¾
−1.5
−2.5 −2 −1.5 −1 −0.5 0 0.5 1
Re z
Fig. 164
Now we can conjecture what happens in Ex. 12.2.1: the eigenvalue λ 2 = 1 2 α − i √β − 1 4 α2 of M
has a very large (in modulus) negative real part. Since ode45 can be expected to behave as if it
integrates ż = λ 2 z, it faces a severe timestep constraint, if exponential blow-up is to be avoided, see
Ex. 12.1.1. Thus stepsize control must resort to tiny timesteps.
Can we predict this kind of difficulty ?
Example 12.2.4 (Chemical reaction kinetics).
reaction:
k 2
A + B
←−
−→ C
k1
} {{ }
fast reaction
k 4
, A + C
←−
−→ D
k3
} {{ }
slow reaction
Vastly different reaction constants: k 1 ,k 2 ≫ k 3 , k 4
If c A (0) > c B (0) ➢
2nd reaction determines overall long-term reaction dynamics
(12.2.6)
Ôº½ ½¾º¾
✸
Model problem analysis (→ Ex. 12.1.1, Ex. 12.1.3) for explicit Euler method and λ ∈ C:
Sequence generated by explicit Euler method (11.2.1) for model problem (12.1.1):
y k+1 = y k (1 + hλ) . (12.1.2)
lim y k = 0 ⇔ |1 + hλ| < 1 .
k→∞
timestep constraint to get decaying (discrete) solution !
Ôº½ ½¾º¾
Mathematical model: ODE involving concentrations y(t) = (c A (t), c B (t),c C (t), c D (t)) T
⎛ ⎞ ⎛
⎞
c A
−k 1 c A c B + k 2 c C − k 3 c A c C + k 4 c D
ẏ := d ⎜c B
⎟
dt ⎝ cC ⎠ = f(y) := ⎜ −k 1 c A c B + k 2 c C
⎟
⎝ k 1 c A c B − k 2 c C − k 3 c A c C + k 4 c D
⎠ .
c D k 3 c A c C − k 4 c D
MATLAB computation: t 0 = 0, T = 1, k 1 = 10 4 , k 2 = 10 3 , k 3 = 10, k 4 = 1
Ôº¾¼ ½¾º¾