ut+cux = 0 0< x < b, t >0 u(x,0) = sin (x)
u(0, t) = 0 u(b, t) = 0
Skema FTBS Eksplisit:
un+1j −un j ∆t +c
un j −u
n j−1
∆x = 0
un+1j −unj + c∆t
∆x u n j −u
n j−1
= 0
Misalkan c∆t∆x =s,maka
un+1j −u n j +s u
n j −u
n j−1
= 0
un+1j = (1−s)unj +su n
j−1 (1)
Analisis Kestabilan, misalkan un
j = ρneiaj, i =
√
−1, a ∈ R. Substitusi pemisalan pada persamaan (1)
ρn+1eiaj= (1−s)ρneiaj+sρneia(j−1) (2) ρ= (1−s) +se−ia
ρ = (1−s) +s[cosa−isina] = (1−s) +scosa−issina = [1 + (cosa−1)s] +i[−ssina]
Skema numerik (1) akan stabil jika terdapat|ρ| ≤1
|ρ| ≤ 1
|[1 + (cosa−1)s] +i[−ssina]| ≤ 1
q
[1 + (cosa−1)s]2+ [−ssina]2 ≤ 1 [1 + (cosa−1)s]2+ [−ssina]2 ≤ 1 1 + 2 (cosa−1)s+ (cosa−1)2s2+s2sin2a ≤ 1 1 + 2 (cosa−1)s+ cos2a−2 cosa+ 1
s2+s2sin2a ≤ 1 1 + 2 (cosa−1)s−2s2cosa+s2+s2 ≤ 1 1 + 2s(cosa−1)−2s2(cosa−1) ≤ 1
1 + 2s−2s2
(cosa−1) ≤ 1 2s(1−s) (cosa−1) ≤ 0 (1−s) (cosa−1) ≤ 0
Karena
−2≤(cosa−1)≤0
−1≤cosa≤1
sehingga
−2 (1−s)≤(1−s) (cosa−1)≤0 (1−s)
−2 + 2s≤(1−s) (cosa−1)≤0
−2 + 2s ≤ 0 2s ≤ 2 s ≤ 1
Jadi skema (1) stabil dengan syarats=c∆t ∆x ≤1.
ut+cux = 0 0< x < b, t >0 u(x,0) = sin (x)
u(0, t) = 0 u(b, t) = 0
FTCS Implisist,
un+1j −unj ∆t +c
un+1j+1 −un+1j−1
2∆x = 0
un+1j + c∆t 2∆x u
n+1 j+1 −un+1j−1
= un j un+1j +s un+1j+1 −un+1j−1
=un
j (3)
ρn+1eiaj+sρn+1eia(j+1)−ρn+1eia(j−1) = ρneiaj ρ+s ρeia−ρe−ia = 1 ρ
1 +s eia−e−ia
= 1 ρ[1 +s(cosa+isina−[cosa−isina])] = 1 ρ[1 +i2ssina] = 1
ρ = 1
1 +i2ssina
Syarat kestabilan,
Skema numerik FTCS implisit untuk persamaan transport stabil tanpa syarat.
Konsistensi, truncation term, error estimate, accuracy metode FTBS eksplisit untuk persamaan transport
un+1j −un
Substitusikan (5) dan (6) pada persamaan (4) shg diperoleh
∆t ut|nj +
Error pemotongan pertama =∆t 2 utt−
c∆x 2 uxx
Orde error: O(∆t,∆x)
Skema (4) dikatakan konsisten jika
epp= ∆t 2 utt−
c∆x
2 uxx (7)
ut+cux = 0 ut = −cux
utt = (ut)t = (−cux)t = −cuxt = −cutx = −c(ut)x = −c(−cux)x = c2uxx
epp = ∆t 2 c
2
uxx−c∆x2 uxx
= c
2∆t 2 uxx−
c∆x 2 uxx =
c2∆t
2 −
c∆x 2
uxx
Kita ingin
c2∆t
2 −
c∆x 2
uxx= 0
maka
c2∆t
2 −
c∆x 2
= 0
c2∆t = c∆x c∆t = ∆x c∆t
∆x = 1
