Numerical Methods Contents - SAM

f
10 Numerical Quadrature
Numerical quadrature
∫
= Approximate evaluation of f(x) dx, integration domain Ω ⊂ R d
Ω
Continuous function f : Ω ⊂ R d ↦→ R only available as function y = f(x) (point
evaluation)
Special case d = 1: Ω = [a,b] (interval)
☞ Numerical quadrature methods are key building blocks for methods for the numerical treatment
of partial differential equations.
Time-harmonic excitation:
U(t)
T t
Fig. 108
R 3 R 4
R 1
➀
➃
R b
➂
➄
➁
R
U(t)
L
R e R
I(t)
2
Fig. 109
Integrating power P = UI over period [0,T] yields heat production per period:
∫ T
W therm = U(t)I(t) dt , where I = I(U) .
0
function I = current(U) involves solving non-linear system of equations, see Ex. 3.0.1!
✸
Ôº½ ½¼º¼
Ôº¿ ½¼º½
3
10.1 Quadrature Formulas
2.5
2
1.5
1
0.5
?
Numerical quadrature methods
approximate
∫b
f(t) dt
a
n-point quadrature formula on [a,b]:
(n-point quadrature rule)
∫ b
n∑
f(t) dt ≈ Q n (f) := ωj n f(ξn j ) . (10.1.1)
a
j=1
ω n j : quadrature weights ∈ R (ger.: Quadraturgewichte)
ξ n j : quadrature nodes ∈ [a,b] (ger.: Quadraturknoten)
Remark 10.1.1 (Transformation of quadrature rules).
0
0 0.5 1 1.5 2 2.5 3 3.5 4
t
Fig. 107
Example 10.0.1 (Heating production in electrical circuits).
) n
Given: quadrature formula
(̂ξj , ̂ω j on reference interval [−1, 1]
j=1
Ôº¾ ½¼º¼
Idea: transformation formula for integrals
∫ b
∫ 1
f(t) dt = 1 2 (b − a) a
−1
̂f(τ) dτ , ̂f(τ) := f( 1
2 (1 − τ)a + 1 2 (τ + 1)b) .
Ôº ½¼º½
(10.1.2)