1=Implicitâexplicit multistep methods for nonlinear parabolic ... - ACMAC
Implicit–explicit multistep methods for nonlinear parabolic equations Modern Techniques in the Numerical Solution of PDE’s, Heraklion, September 19–23, 2011
Georgios Akrivis
Computer Science Department University of Ioannina Greece G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
1 / 31
Outline 1
Nonlinear parabolic equations (notation)
2
Implicit–explicit multistep schemes Three implicit–explicit multistep schemes Stability conditions and stability constants
3
A priori error analysis Consistency Local stability Error estimates
4
More on the stability constants
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
2 / 31
1. Nonlinear parabolic equations Seek u : [0, T ] → D(A) s.t. ( 0 u (t) + Au(t) = B(t, u(t)), 0 < t < T, u(0) = u0 , with H, (·, ·) Hilbert space, A : D(A) → H positive definite, self-adjoint, linear operator, D(A) dense in H, B(t, ·) : D(A) → H, u0
t ∈ [0, T ], “small”,
∈ H.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
3 / 31
Notation:
| · | norm of H V := D(A1/2 ) k · k norm of V, kvk = |A1/2 v| Identify H with its dual and denote by V 0 the dual of V k · k? norm of V 0 , kvk? = |A−1/2 v| (·, ·) inner product in H and duality pairing between V 0 and V Then kvk = (Av, v)1/2 ,
G. Akrivis ([email protected])
kvk? = (v, A−1 v)1/2 .
Multistep schemes for parabolic equations
September 19–23, 2011
4 / 31
2. Implicit–explicit multistep schemes
(α, β) an implicit q−step scheme (α, γ) an explicit q−step scheme, α(ζ) =
q X
i
αi ζ ,
i=0
β(ζ) =
q X
i
βi ζ ,
γ(ζ) =
i=0
q−1 X
γi ζ i .
i=0
Let N ∈ N, k := T /N, and tn := nk, n = 0, . . . , N. Let U 0 , . . . , U q−1 be given starting approximations. We define approximations U m to um := u(tm ), m = q, . . . , N, in three different ways:
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
5 / 31
2. Implicit–explicit multistep schemes
(α, β) an implicit q−step scheme (α, γ) an explicit q−step scheme, α(ζ) =
q X
i
αi ζ ,
i=0
β(ζ) =
q X
i
βi ζ ,
γ(ζ) =
i=0
q−1 X
γi ζ i .
i=0
Let N ∈ N, k := T /N, and tn := nk, n = 0, . . . , N. Let U 0 , . . . , U q−1 be given starting approximations. We define approximations U m to um := u(tm ), m = q, . . . , N, in three different ways:
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
5 / 31
i) By the implicit (α, β) scheme q X
(αi I + kβi A)U n+i = k
i=0
q X
βi B(tn+i , U n+i ),
n = 0, . . . , N − q.
i=0
Assumption: The scheme (α, β) is A(ϑ)−stable, i.e., for z ∈ Sϑ , χ(z; ·) = α(·) − zβ(·) satisfies the root condition. y
Sϑ
ϑ ϑ
Advantage: Good stability properties Drawback: Difficult to implement G. Akrivis ([email protected])
Multistep schemes for parabolic equations
x
[(α, γ) is unstable] September 19–23, 2011
6 / 31
Let K(α,β) := sup max x>0 ζ∈K
|xβ(ζ)| , |α(ζ) + xβ(ζ)|
with K the unit circle, K := {ζ ∈ C : |ζ| = 1}. Then K(α,β) =
1 . sin ϑ
Local Lipschitz condition: kB(t, v) − B(t, w)k? ≤ λkv − wk + µ|v − w|
∀v, w ∈ Tu ,
with the stability constant λ < 1, in a tube Tu := {v ∈ V : min kv − u(t)k ≤ 1}. 0≤t≤T
Local stability:
for K(α,β) λ < 1
Instability in general: for K(α,β) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
7 / 31
Let K(α,β) := sup max x>0 ζ∈K
|xβ(ζ)| , |α(ζ) + xβ(ζ)|
with K the unit circle, K := {ζ ∈ C : |ζ| = 1}. Then K(α,β) =
1 . sin ϑ
Local Lipschitz condition: kB(t, v) − B(t, w)k? ≤ λkv − wk + µ|v − w|
∀v, w ∈ Tu ,
with the stability constant λ < 1, in a tube Tu := {v ∈ V : min kv − u(t)k ≤ 1}. 0≤t≤T
Local stability:
for K(α,β) λ < 1
Instability in general: for K(α,β) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
7 / 31
Let K(α,β) := sup max x>0 ζ∈K
|xβ(ζ)| , |α(ζ) + xβ(ζ)|
with K the unit circle, K := {ζ ∈ C : |ζ| = 1}. Then K(α,β) =
1 . sin ϑ
Local Lipschitz condition: kB(t, v) − B(t, w)k? ≤ λkv − wk + µ|v − w|
∀v, w ∈ Tu ,
with the stability constant λ < 1, in a tube Tu := {v ∈ V : min kv − u(t)k ≤ 1}. 0≤t≤T
Local stability:
for K(α,β) λ < 1
Instability in general: for K(α,β) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
7 / 31
ii) By the implicit–explicit (α, β, γ) scheme q X
(αi I + kβi A)U
n+i
=k
q−1 X
i=0
γi B(tn+i , U n+i ),
n = 0, . . . , N − q.
i=0
Assumption: The scheme (α, β) is strongly A(0)−stable, i.e., A(0)−stable and s.t. the roots of β are (strictly) less than 1 in modulus. Advantage: Easy to implement Drawback: Not so good stability properties Let K(α,β,γ) := sup max x>0 ζ∈K
Local stability:
|xγ(ζ)| . |α(ζ) + xβ(ζ)|
for K(α,β,γ) λ < 1
Instability in general: for K(α,β,γ) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
8 / 31
ii) By the implicit–explicit (α, β, γ) scheme q X
(αi I + kβi A)U
n+i
=k
q−1 X
i=0
γi B(tn+i , U n+i ),
n = 0, . . . , N − q.
i=0
Assumption: The scheme (α, β) is strongly A(0)−stable, i.e., A(0)−stable and s.t. the roots of β are (strictly) less than 1 in modulus. Advantage: Easy to implement Drawback: Not so good stability properties Let K(α,β,γ) := sup max x>0 ζ∈K
Local stability:
|xγ(ζ)| . |α(ζ) + xβ(ζ)|
for K(α,β,γ) λ < 1
Instability in general: for K(α,β,γ) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
8 / 31
ii) By the implicit–explicit (α, β, γ) scheme q X
(αi I + kβi A)U
n+i
=k
q−1 X
i=0
γi B(tn+i , U n+i ),
n = 0, . . . , N − q.
i=0
Assumption: The scheme (α, β) is strongly A(0)−stable, i.e., A(0)−stable and s.t. the roots of β are (strictly) less than 1 in modulus. Advantage: Easy to implement Drawback: Not so good stability properties Let K(α,β,γ) := sup max x>0 ζ∈K
Local stability:
|xγ(ζ)| . |α(ζ) + xβ(ζ)|
for K(α,β,γ) λ < 1
Instability in general: for K(α,β,γ) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
8 / 31
ii) By the implicit–explicit (α, β, γ) scheme q X
(αi I + kβi A)U
n+i
=k
q−1 X
i=0
γi B(tn+i , U n+i ),
n = 0, . . . , N − q.
i=0
Assumption: The scheme (α, β) is strongly A(0)−stable, i.e., A(0)−stable and s.t. the roots of β are (strictly) less than 1 in modulus. Advantage: Easy to implement Drawback: Not so good stability properties Let K(α,β,γ) := sup max x>0 ζ∈K
Local stability:
|xγ(ζ)| . |α(ζ) + xβ(ζ)|
for K(α,β,γ) λ < 1
Instability in general: for K(α,β,γ) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
8 / 31
BDF schemes α(ζ) =
q X 1 j=1
j
ζ q−j (ζ − 1)j ,
K(α,β,γ) = 2q − 1,
β(ζ) = ζ q , K(α,β) =
γ(ζ) = ζ q − (ζ − 1)q .
1 , sin ϑq
q = 1, . . . , 6.
q
K(α,β,γ)
K(α,β)
K(α,β,γ) /K(α,β)
λ(α,β)
1 2 3 4 5 6
1 3 7 15 31 63
1 1 1.0024 1.0437 1.2718 3.2641
1 3 6.9832 14.3711 24.3750 19.3007
1 1 0.9976 0.9581 0.7863 0.3064
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
9 / 31
References
1
Crouzeix: Numer. Math. (1980)
2
A., Crouzeix, Makridakis: Math. Comp. (1998)
3
A., Crouzeix, Makridakis: Numer. Math. (1999)
4
A., Crouzeix: Math. Comp. (2003)
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
10 / 31
Motivation for an intermediate alternative
1
u0 (t) + Au(t) = B(t, u(t)) with A : D(A) → H positive definite, self-adjoint, time-independent, linear operator
2
K(α,β,γ) /K(α,β) may be large
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
11 / 31
iii) An intermediate alternative Decompose the nonlinear part B into two parts, B(t, u) = B1 (t, u) + B2 (t, u), u0 (t) + Au(t) − B1 (t, u(t)) = B2 (t, u(t)) Interesting example: u0 (t) + A(t)u(t) = B(t, u(t)) Choose: A := 12 [A(0) + A(0)? ] and B1 (t, u(t)) := A − A(t). q q−1 X X n+i n+i n+i n+i αi U +kβi AU −B1 (t , U ) = k γi B2 (tn+i , U n+i ), i=0
i=0
n = 0, . . . , N − q.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
12 / 31
iii) An intermediate alternative Decompose the nonlinear part B into two parts, B(t, u) = B1 (t, u) + B2 (t, u), u0 (t) + Au(t) − B1 (t, u(t)) = B2 (t, u(t)) Interesting example: u0 (t) + A(t)u(t) = B(t, u(t)) Choose: A := 12 [A(0) + A(0)? ] and B1 (t, u(t)) := A − A(t). q q−1 X X n+i n+i n+i n+i αi U +kβi AU −B1 (t , U ) = k γi B2 (tn+i , U n+i ), i=0
i=0
n = 0, . . . , N − q.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
12 / 31
iii) An intermediate alternative Decompose the nonlinear part B into two parts, B(t, u) = B1 (t, u) + B2 (t, u), u0 (t) + Au(t) − B1 (t, u(t)) = B2 (t, u(t)) Interesting example: u0 (t) + A(t)u(t) = B(t, u(t)) Choose: A := 12 [A(0) + A(0)? ] and B1 (t, u(t)) := A − A(t). q q−1 X X n+i n+i n+i n+i αi U +kβi AU −B1 (t , U ) = k γi B2 (tn+i , U n+i ), i=0
i=0
n = 0, . . . , N − q.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
12 / 31
Local Lipschitz conditions: ∀v, w ∈ Tu
kB1 (t, v) − B1 (t, w)k? ≤ λ1 kv − wk + µ1 |v − w|,
∀v, w ∈ Tu
kB2 (t, v) − B2 (t, w)k? ≤ λ2 kv − wk + µ2 |v − w|
Local stability:
for K(α,β) λ1 + K(α,β,γ) λ2 < 1
Instability in general: for sup max x>0 ζ∈K
λ1 |xβ(ζ)| + λ2 |xγ(ζ)| >1 |α(ζ) + xβ(ζ)|
Advantage: Easier to implement than the implicit scheme (α, β), more expensive than the implicit–explicit scheme (α, β, γ) Drawback: Better stability properties than the (α, β, γ) scheme, but worse than the (α, β) scheme. G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
13 / 31
Local Lipschitz conditions: ∀v, w ∈ Tu
kB1 (t, v) − B1 (t, w)k? ≤ λ1 kv − wk + µ1 |v − w|,
∀v, w ∈ Tu
kB2 (t, v) − B2 (t, w)k? ≤ λ2 kv − wk + µ2 |v − w|
Local stability:
for K(α,β) λ1 + K(α,β,γ) λ2 < 1
Instability in general: for sup max x>0 ζ∈K
λ1 |xβ(ζ)| + λ2 |xγ(ζ)| >1 |α(ζ) + xβ(ζ)|
Advantage: Easier to implement than the implicit scheme (α, β), more expensive than the implicit–explicit scheme (α, β, γ) Drawback: Better stability properties than the (α, β, γ) scheme, but worse than the (α, β) scheme. G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
13 / 31
Local Lipschitz conditions: ∀v, w ∈ Tu
kB1 (t, v) − B1 (t, w)k? ≤ λ1 kv − wk + µ1 |v − w|,
∀v, w ∈ Tu
kB2 (t, v) − B2 (t, w)k? ≤ λ2 kv − wk + µ2 |v − w|
Local stability:
for K(α,β) λ1 + K(α,β,γ) λ2 < 1
Instability in general: for sup max x>0 ζ∈K
λ1 |xβ(ζ)| + λ2 |xγ(ζ)| >1 |α(ζ) + xβ(ζ)|
Advantage: Easier to implement than the implicit scheme (α, β), more expensive than the implicit–explicit scheme (α, β, γ) Drawback: Better stability properties than the (α, β, γ) scheme, but worse than the (α, β) scheme. G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
13 / 31
3. Error Analysis — i) Consistency E n consistency error for the solution u: kE n =
q−1 q X X γi B2 (tn+i , un+i ), αi un+i +kβi Aun+i −B1 (tn+i , un+i ) −k i=0
i=0
the amount by which the exact solution misses being an approximate solution. With E1n
=
q X
n+i
αi u
0
n+i
− kβi u (t
i=0
q X n ) , E2 = k (βi − γi )B2 (tn+i , un+i ) i=0
we have kE n = E1n + E2n .
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
14 / 31
Assumption:
q X
i ` αi = `
i=0
q X
i`−1 βi = `
i=0
q−1 X
i`−1 γi , ` = 0, 1, . . . , p,
i=0
i.e., the order of both schemes (α, β) and (α, γ) is p. Taylor expansion around tn yields q Z n+i 1 X t n E1 = (tn+i − s)p−1 αi (tn+i − s) − pkβi u(p+1) (s)ds p! n i=0 t Z tn+i q X k dp n E2 = (βi − γi ) (tn+i − s)p−1 p B2 s, u(s) ds. (p − 1)! dt tn i=0
Thus, under obvious regularity assumptions, max
0≤n≤N −q
G. Akrivis ([email protected])
kE n k? ≤ Ck p .
Multistep schemes for parabolic equations
September 19–23, 2011
15 / 31
ii) Local stability Let U m , V m ∈ Tu be s.t., for n = 0, . . . , N − q, q X
αi U n+i + kβi AU n+i − B1 (tn+i , U n+i )
=k
q−1 X
i=0
i=0
q X
q−1 X
αi V n+i + kβi AV n+i − B1 (tn+i , V n+i )
=k
i=0
γi B2 (tn+i , U n+i ), γi B2 (tn+i , V n+i ).
i=0
With ϑn := U n − V n , bni := Bi (tn , U n ) − Bi (tn , V n ), i = 1, 2, we have q X
(αi I + kβi A)ϑ
i=0
n+i
=k
q X i=0
βi bn+i 1
+k
q−1 X
γi b2n+i ,
i=0
n = 0, . . . , N − q. G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
16 / 31
Let the rational functions f (`, ·) and f˜(`, ·) be defined through the expansions ∞ X γ(ζ) = f (`, x)ζ −` , α(ζ) + xβ(ζ) `=1
∞ X β(ζ) = f˜(`, x)ζ −` , α(ζ) + xβ(ζ) `=0
for |ζ| ≥ 1.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
17 / 31
With n X n ϑ1 := k f˜(n − `, kA)b`1 , n = 0, . . . , N, `=0 n−1 X 0 n ϑ := 0, ϑ := k f (n − `, kA)b`2 , n = 1, . . . , N, 2 2 `=0 n ϑ3 := ϑn − ϑn1 − ϑn2 , n = 0, . . . , N, we have q X (αi I + kβi A)ϑn+i j i=0
G. Akrivis ([email protected])
q X k βi bn+i 1 , j = 1, i=0 q−1 X = γi bn+i 2 , j = 2, k i=0 0, j = 3.
Multistep schemes for parabolic equations
September 19–23, 2011
18 / 31
Lemma For n = 0, . . . , N, the following estimates are valid n n−1 X X ` 2 2 k kϑ k ≤ K k kb`1 k2? , 1 (α,β) `=0 `=0 n n−1 X X 2 k kϑ`2 k2 ≤ K(α,β,γ) k kb`2 k2? , `=0 `=0 q−1 n X j X kϑ`3 k2 ≤ c |ϑ3 |2 + kkϑj3 k2 . k `=0
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j=0
Multistep schemes for parabolic equations
September 19–23, 2011
19 / 31
Proof. We shall prove the second estimate. It suffices to let b`2 = 0, for ` ≥ n, and replace n by ∞. Let ϑˆ2 (t) :=
∞ X
ˆb2 (t) :=
ϑ`2 e2 i π`t ,
`=0
∞ X
b`2 e2 i π`t .
`=0
Then n h i−1 o −1 ϑˆ2 (t) = kAγ e−2 i πt α e−2 i πt + kAβ e−2 i πt A ˆb2 (t), whence kϑˆ2 (t)k ≤ K(α,β,γ) kˆb2 (t)k? . Therefore, in view of Parseval’s identity, with K := K(α,β,γ) , ∞ X `=0
kϑ`2 k2
Z
1
= 0
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kϑˆ2 (t)k2 dt ≤ K 2
Z
1
kˆb2 (t)k2? dt = K 2
0 Multistep schemes for parabolic equations
∞ X
kb`2 k2? .
`=0 September 19–23, 2011
20 / 31
Proposition (Local stability) For n = q, . . . , N, the following estimate is valid n 2
|ϑ | + k
n X `=0
G. Akrivis ([email protected])
` 2
kϑ k ≤ C
q−1 X
|ϑj |2 + kkϑj k2 .
j=0
Multistep schemes for parabolic equations
September 19–23, 2011
21 / 31
iii) Error estimates
Theorem (Error estimate) Let the stability conditions be satisfied, the order of both methods be p and the solution u be sufficiently smooth. Assume for the starting approximations that max |u(tj ) − U j | + k 1/2 ku(tj ) − U j k ≤ ck p . 0≤j≤q−1
Then, we have the error estimate max |u(tn ) − U n | ≤ Ck p .
0≤n≤N
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Multistep schemes for parabolic equations
September 19–23, 2011
22 / 31
4. More on the constants K(α,β) and K(α,β,γ) α(ζ) , ζ ∈ K, s.t. β(ζ) 6= 0, β(ζ) denote the points of the root locus curve Let d(ζ) :=
Let K+ := {ζ ∈ K : Re d(ζ) ≥ 0}, K− := {ζ ∈ K : Re d(ζ) < 0}. Let k(x, ζ) :=
xγ(ζ) 1 γ(ζ) = . −1 α(ζ) + xβ(ζ) 1 + x d(ζ) β(ζ)
Then, ∀ζ ∈ K+
sup |k(x, ζ)| = x>0
and ∀ζ ∈ K−
sup |k(x, ζ)| = x>0
G. Akrivis ([email protected])
|γ(ζ)| |β(ζ)|
|d(ζ)| |γ(ζ)| . | Im d(ζ)| |β(ζ)|
Multistep schemes for parabolic equations
September 19–23, 2011
23 / 31
4. More on the constants K(α,β) and K(α,β,γ) α(ζ) , ζ ∈ K, s.t. β(ζ) 6= 0, β(ζ) denote the points of the root locus curve Let d(ζ) :=
Let K+ := {ζ ∈ K : Re d(ζ) ≥ 0}, K− := {ζ ∈ K : Re d(ζ) < 0}. Let k(x, ζ) :=
xγ(ζ) 1 γ(ζ) = . −1 α(ζ) + xβ(ζ) 1 + x d(ζ) β(ζ)
Then, ∀ζ ∈ K+
sup |k(x, ζ)| = x>0
and ∀ζ ∈ K−
sup |k(x, ζ)| = x>0
G. Akrivis ([email protected])
|γ(ζ)| |β(ζ)|
|d(ζ)| |γ(ζ)| . | Im d(ζ)| |β(ζ)|
Multistep schemes for parabolic equations
September 19–23, 2011
23 / 31
4. More on the constants K(α,β) and K(α,β,γ) α(ζ) , ζ ∈ K, s.t. β(ζ) 6= 0, β(ζ) denote the points of the root locus curve Let d(ζ) :=
Let K+ := {ζ ∈ K : Re d(ζ) ≥ 0}, K− := {ζ ∈ K : Re d(ζ) < 0}. Let k(x, ζ) :=
xγ(ζ) 1 γ(ζ) = . −1 α(ζ) + xβ(ζ) 1 + x d(ζ) β(ζ)
Then, ∀ζ ∈ K+
sup |k(x, ζ)| = x>0
and ∀ζ ∈ K−
sup |k(x, ζ)| = x>0
G. Akrivis ([email protected])
|γ(ζ)| |β(ζ)|
|d(ζ)| |γ(ζ)| . | Im d(ζ)| |β(ζ)|
Multistep schemes for parabolic equations
September 19–23, 2011
23 / 31
Therefore, K(α,β,γ) = max
n
max
|d(ζ)| |γ(ζ)| o |γ(ζ)| , max , |β(ζ)| ζ∈K− | Im d(ζ)| |β(ζ)|
max
|γ(ζ)| o |γ(ζ)| 1 , max , |β(ζ)| ζ∈K− | sin ϕ(ζ)| |β(ζ)|
ζ∈K+
or, equivalently, K(α,β,γ) = max
n
ζ∈K+
y d(ζ) ϕ(ζ)
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
x
September 19–23, 2011
24 / 31
Assumption: The scheme (α, β) is A−stable Then, K+ = K and K(α,β,γ) = max ζ∈K
|γ(ζ)| . |β(ζ)|
In particular, K(α,β) = 1. Assumption: The scheme (α, β) is A(ϑ)−stable Then, K(α,β) = max
ζ∈K−
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1 1 = . | sin ϕ(ζ)| sin ϑ
Multistep schemes for parabolic equations
September 19–23, 2011
25 / 31
Assumption: The scheme (α, β) is A−stable Then, K+ = K and K(α,β,γ) = max ζ∈K
|γ(ζ)| . |β(ζ)|
In particular, K(α,β) = 1. Assumption: The scheme (α, β) is A(ϑ)−stable Then, K(α,β) = max
ζ∈K−
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1 1 = . | sin ϕ(ζ)| sin ϑ
Multistep schemes for parabolic equations
September 19–23, 2011
25 / 31
Discrepancy between necessary and sufficient conditions Sufficient condition:
K(α,β) λ1 + K(α,β,γ) λ2 < 1
Necessary condition: sup max x>0 ζ∈K
λ1 |xβ(ζ)| + λ2 |xγ(ζ)| ≤1 |α(ζ) + xβ(ζ)|
The necessary condition can be equivalenty written as λ1 + λ2 max
ζ∈K+
|γ(ζ)| ≤ 1, |β(ζ)|
h |d(ζ)| |γ(ζ)| i λ1 + λ2 ≤ 1. |β(ζ)| ζ∈K− | Im d(ζ)| sup
A more practical, but weaker in general, necessary condition is λ1 + K(α,β,γ) λ2 ≤ 1,
K(α,β) λ1 + c?(β,γ) λ2 ≤ 1,
with c?(β,γ) := |γ(ζ ? )|/|β(ζ ? )|, if K(α,β) is attained at ζ ? ∈ K− . G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
26 / 31
y
y
1
1
1 K(α,β,γ)
1 K(α,β,γ)
1 K(α,β) c? (β,γ)
T
S 1 K(α,β)
x
1 K(α,β)
1
x 1
y 1
1 K(α,β,γ)
R
1 K(α,β)
G. Akrivis ([email protected])
x 1
Multistep schemes for parabolic equations
September 19–23, 2011
27 / 31
BDF methods: q
|T |
|T |/|S|
|T |/|R|
c?(β,γ)
3 4 5 6
1.70 × 10−4 1.32 × 10−3 2.69 × 10−3 1.62 × 10−3
2.39 × 10−3 4.13 × 10−2 2.13 × 10−1 6.68 × 10−1
2.78 × 10−3 4.17 × 10−2 2.13 × 10−1 6.68 × 10−1
1.73 3.49 0.71 1.00
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
28 / 31
On the necessary stability condition Necessary condition: sup max x>0 ζ∈K
λ1 |xβ(ζ)| + λ2 |xγ(ζ)| ≤1 |α(ζ) + xβ(ζ)|
Assume this condition is not satisfied. Then, with an appropriate Θ ∈ [0, 2π), for the function k, k(x, ζ) :=
λ1 x ei Θ β(ζ) + λ2 xγ(ζ) , α(ζ) + xβ(ζ)
x > 0, |ζ| ≥ 1.
we have |k(x, z)| > 1 for appropriate z ∈ K, x > 0. Since lim |k(x, ζ)| = λ1
|ζ|→∞
xβq < 1, αq + xβq
there exists a ζ ? ∈ C with |ζ ? | > 1 s.t. |k(x, ζ ? )| = 1, i.e., λ1 x ei Θ β(ζ ? ) + λ2 xγ(ζ ? ) = e− i ϕ , α(ζ ? ) + xβ(ζ ? ) G. Akrivis ([email protected])
for a ϕ ∈ [0, 2π).
Multistep schemes for parabolic equations
September 19–23, 2011
29 / 31
Therefore, α(ζ ? ) + xβ(ζ ? ) − λ1 x ei(Θ+ϕ) β(ζ ? ) − λ2 x ei ϕ γ(ζ ? ) = 0. For B1 (t, ·) := λ1 ei(Θ+ϕ) A and B2 (t, ·) := λ2 ei ϕ A, the Lipschitz conditions are satisfied. According to the von Neumann criterion, a necessary stability condition is that, if ν is an eigenvalue of A, the solutions of q X αi + kν βi − λ1 ei(Θ+ϕ) βi − λ2 ei ϕ γi v n+i = 0 i=0
are bounded; for kν = x this is not the case, since the root condition is not satisfied. Therefore, the scheme is not unconditionally stable.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
30 / 31
Thank you very much!
Georgios Akrivis
Computer Science Department University of Ioannina Greece G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
1 / 31
Outline 1
Nonlinear parabolic equations (notation)
2
Implicit–explicit multistep schemes Three implicit–explicit multistep schemes Stability conditions and stability constants
3
A priori error analysis Consistency Local stability Error estimates
4
More on the stability constants
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
2 / 31
1. Nonlinear parabolic equations Seek u : [0, T ] → D(A) s.t. ( 0 u (t) + Au(t) = B(t, u(t)), 0 < t < T, u(0) = u0 , with H, (·, ·) Hilbert space, A : D(A) → H positive definite, self-adjoint, linear operator, D(A) dense in H, B(t, ·) : D(A) → H, u0
t ∈ [0, T ], “small”,
∈ H.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
3 / 31
Notation:
| · | norm of H V := D(A1/2 ) k · k norm of V, kvk = |A1/2 v| Identify H with its dual and denote by V 0 the dual of V k · k? norm of V 0 , kvk? = |A−1/2 v| (·, ·) inner product in H and duality pairing between V 0 and V Then kvk = (Av, v)1/2 ,
G. Akrivis ([email protected])
kvk? = (v, A−1 v)1/2 .
Multistep schemes for parabolic equations
September 19–23, 2011
4 / 31
2. Implicit–explicit multistep schemes
(α, β) an implicit q−step scheme (α, γ) an explicit q−step scheme, α(ζ) =
q X
i
αi ζ ,
i=0
β(ζ) =
q X
i
βi ζ ,
γ(ζ) =
i=0
q−1 X
γi ζ i .
i=0
Let N ∈ N, k := T /N, and tn := nk, n = 0, . . . , N. Let U 0 , . . . , U q−1 be given starting approximations. We define approximations U m to um := u(tm ), m = q, . . . , N, in three different ways:
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
5 / 31
2. Implicit–explicit multistep schemes
(α, β) an implicit q−step scheme (α, γ) an explicit q−step scheme, α(ζ) =
q X
i
αi ζ ,
i=0
β(ζ) =
q X
i
βi ζ ,
γ(ζ) =
i=0
q−1 X
γi ζ i .
i=0
Let N ∈ N, k := T /N, and tn := nk, n = 0, . . . , N. Let U 0 , . . . , U q−1 be given starting approximations. We define approximations U m to um := u(tm ), m = q, . . . , N, in three different ways:
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
5 / 31
i) By the implicit (α, β) scheme q X
(αi I + kβi A)U n+i = k
i=0
q X
βi B(tn+i , U n+i ),
n = 0, . . . , N − q.
i=0
Assumption: The scheme (α, β) is A(ϑ)−stable, i.e., for z ∈ Sϑ , χ(z; ·) = α(·) − zβ(·) satisfies the root condition. y
Sϑ
ϑ ϑ
Advantage: Good stability properties Drawback: Difficult to implement G. Akrivis ([email protected])
Multistep schemes for parabolic equations
x
[(α, γ) is unstable] September 19–23, 2011
6 / 31
Let K(α,β) := sup max x>0 ζ∈K
|xβ(ζ)| , |α(ζ) + xβ(ζ)|
with K the unit circle, K := {ζ ∈ C : |ζ| = 1}. Then K(α,β) =
1 . sin ϑ
Local Lipschitz condition: kB(t, v) − B(t, w)k? ≤ λkv − wk + µ|v − w|
∀v, w ∈ Tu ,
with the stability constant λ < 1, in a tube Tu := {v ∈ V : min kv − u(t)k ≤ 1}. 0≤t≤T
Local stability:
for K(α,β) λ < 1
Instability in general: for K(α,β) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
7 / 31
Let K(α,β) := sup max x>0 ζ∈K
|xβ(ζ)| , |α(ζ) + xβ(ζ)|
with K the unit circle, K := {ζ ∈ C : |ζ| = 1}. Then K(α,β) =
1 . sin ϑ
Local Lipschitz condition: kB(t, v) − B(t, w)k? ≤ λkv − wk + µ|v − w|
∀v, w ∈ Tu ,
with the stability constant λ < 1, in a tube Tu := {v ∈ V : min kv − u(t)k ≤ 1}. 0≤t≤T
Local stability:
for K(α,β) λ < 1
Instability in general: for K(α,β) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
7 / 31
Let K(α,β) := sup max x>0 ζ∈K
|xβ(ζ)| , |α(ζ) + xβ(ζ)|
with K the unit circle, K := {ζ ∈ C : |ζ| = 1}. Then K(α,β) =
1 . sin ϑ
Local Lipschitz condition: kB(t, v) − B(t, w)k? ≤ λkv − wk + µ|v − w|
∀v, w ∈ Tu ,
with the stability constant λ < 1, in a tube Tu := {v ∈ V : min kv − u(t)k ≤ 1}. 0≤t≤T
Local stability:
for K(α,β) λ < 1
Instability in general: for K(α,β) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
7 / 31
ii) By the implicit–explicit (α, β, γ) scheme q X
(αi I + kβi A)U
n+i
=k
q−1 X
i=0
γi B(tn+i , U n+i ),
n = 0, . . . , N − q.
i=0
Assumption: The scheme (α, β) is strongly A(0)−stable, i.e., A(0)−stable and s.t. the roots of β are (strictly) less than 1 in modulus. Advantage: Easy to implement Drawback: Not so good stability properties Let K(α,β,γ) := sup max x>0 ζ∈K
Local stability:
|xγ(ζ)| . |α(ζ) + xβ(ζ)|
for K(α,β,γ) λ < 1
Instability in general: for K(α,β,γ) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
8 / 31
ii) By the implicit–explicit (α, β, γ) scheme q X
(αi I + kβi A)U
n+i
=k
q−1 X
i=0
γi B(tn+i , U n+i ),
n = 0, . . . , N − q.
i=0
Assumption: The scheme (α, β) is strongly A(0)−stable, i.e., A(0)−stable and s.t. the roots of β are (strictly) less than 1 in modulus. Advantage: Easy to implement Drawback: Not so good stability properties Let K(α,β,γ) := sup max x>0 ζ∈K
Local stability:
|xγ(ζ)| . |α(ζ) + xβ(ζ)|
for K(α,β,γ) λ < 1
Instability in general: for K(α,β,γ) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
8 / 31
ii) By the implicit–explicit (α, β, γ) scheme q X
(αi I + kβi A)U
n+i
=k
q−1 X
i=0
γi B(tn+i , U n+i ),
n = 0, . . . , N − q.
i=0
Assumption: The scheme (α, β) is strongly A(0)−stable, i.e., A(0)−stable and s.t. the roots of β are (strictly) less than 1 in modulus. Advantage: Easy to implement Drawback: Not so good stability properties Let K(α,β,γ) := sup max x>0 ζ∈K
Local stability:
|xγ(ζ)| . |α(ζ) + xβ(ζ)|
for K(α,β,γ) λ < 1
Instability in general: for K(α,β,γ) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
8 / 31
ii) By the implicit–explicit (α, β, γ) scheme q X
(αi I + kβi A)U
n+i
=k
q−1 X
i=0
γi B(tn+i , U n+i ),
n = 0, . . . , N − q.
i=0
Assumption: The scheme (α, β) is strongly A(0)−stable, i.e., A(0)−stable and s.t. the roots of β are (strictly) less than 1 in modulus. Advantage: Easy to implement Drawback: Not so good stability properties Let K(α,β,γ) := sup max x>0 ζ∈K
Local stability:
|xγ(ζ)| . |α(ζ) + xβ(ζ)|
for K(α,β,γ) λ < 1
Instability in general: for K(α,β,γ) λ > 1 G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
8 / 31
BDF schemes α(ζ) =
q X 1 j=1
j
ζ q−j (ζ − 1)j ,
K(α,β,γ) = 2q − 1,
β(ζ) = ζ q , K(α,β) =
γ(ζ) = ζ q − (ζ − 1)q .
1 , sin ϑq
q = 1, . . . , 6.
q
K(α,β,γ)
K(α,β)
K(α,β,γ) /K(α,β)
λ(α,β)
1 2 3 4 5 6
1 3 7 15 31 63
1 1 1.0024 1.0437 1.2718 3.2641
1 3 6.9832 14.3711 24.3750 19.3007
1 1 0.9976 0.9581 0.7863 0.3064
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
9 / 31
References
1
Crouzeix: Numer. Math. (1980)
2
A., Crouzeix, Makridakis: Math. Comp. (1998)
3
A., Crouzeix, Makridakis: Numer. Math. (1999)
4
A., Crouzeix: Math. Comp. (2003)
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
10 / 31
Motivation for an intermediate alternative
1
u0 (t) + Au(t) = B(t, u(t)) with A : D(A) → H positive definite, self-adjoint, time-independent, linear operator
2
K(α,β,γ) /K(α,β) may be large
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
11 / 31
iii) An intermediate alternative Decompose the nonlinear part B into two parts, B(t, u) = B1 (t, u) + B2 (t, u), u0 (t) + Au(t) − B1 (t, u(t)) = B2 (t, u(t)) Interesting example: u0 (t) + A(t)u(t) = B(t, u(t)) Choose: A := 12 [A(0) + A(0)? ] and B1 (t, u(t)) := A − A(t). q q−1 X X n+i n+i n+i n+i αi U +kβi AU −B1 (t , U ) = k γi B2 (tn+i , U n+i ), i=0
i=0
n = 0, . . . , N − q.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
12 / 31
iii) An intermediate alternative Decompose the nonlinear part B into two parts, B(t, u) = B1 (t, u) + B2 (t, u), u0 (t) + Au(t) − B1 (t, u(t)) = B2 (t, u(t)) Interesting example: u0 (t) + A(t)u(t) = B(t, u(t)) Choose: A := 12 [A(0) + A(0)? ] and B1 (t, u(t)) := A − A(t). q q−1 X X n+i n+i n+i n+i αi U +kβi AU −B1 (t , U ) = k γi B2 (tn+i , U n+i ), i=0
i=0
n = 0, . . . , N − q.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
12 / 31
iii) An intermediate alternative Decompose the nonlinear part B into two parts, B(t, u) = B1 (t, u) + B2 (t, u), u0 (t) + Au(t) − B1 (t, u(t)) = B2 (t, u(t)) Interesting example: u0 (t) + A(t)u(t) = B(t, u(t)) Choose: A := 12 [A(0) + A(0)? ] and B1 (t, u(t)) := A − A(t). q q−1 X X n+i n+i n+i n+i αi U +kβi AU −B1 (t , U ) = k γi B2 (tn+i , U n+i ), i=0
i=0
n = 0, . . . , N − q.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
12 / 31
Local Lipschitz conditions: ∀v, w ∈ Tu
kB1 (t, v) − B1 (t, w)k? ≤ λ1 kv − wk + µ1 |v − w|,
∀v, w ∈ Tu
kB2 (t, v) − B2 (t, w)k? ≤ λ2 kv − wk + µ2 |v − w|
Local stability:
for K(α,β) λ1 + K(α,β,γ) λ2 < 1
Instability in general: for sup max x>0 ζ∈K
λ1 |xβ(ζ)| + λ2 |xγ(ζ)| >1 |α(ζ) + xβ(ζ)|
Advantage: Easier to implement than the implicit scheme (α, β), more expensive than the implicit–explicit scheme (α, β, γ) Drawback: Better stability properties than the (α, β, γ) scheme, but worse than the (α, β) scheme. G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
13 / 31
Local Lipschitz conditions: ∀v, w ∈ Tu
kB1 (t, v) − B1 (t, w)k? ≤ λ1 kv − wk + µ1 |v − w|,
∀v, w ∈ Tu
kB2 (t, v) − B2 (t, w)k? ≤ λ2 kv − wk + µ2 |v − w|
Local stability:
for K(α,β) λ1 + K(α,β,γ) λ2 < 1
Instability in general: for sup max x>0 ζ∈K
λ1 |xβ(ζ)| + λ2 |xγ(ζ)| >1 |α(ζ) + xβ(ζ)|
Advantage: Easier to implement than the implicit scheme (α, β), more expensive than the implicit–explicit scheme (α, β, γ) Drawback: Better stability properties than the (α, β, γ) scheme, but worse than the (α, β) scheme. G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
13 / 31
Local Lipschitz conditions: ∀v, w ∈ Tu
kB1 (t, v) − B1 (t, w)k? ≤ λ1 kv − wk + µ1 |v − w|,
∀v, w ∈ Tu
kB2 (t, v) − B2 (t, w)k? ≤ λ2 kv − wk + µ2 |v − w|
Local stability:
for K(α,β) λ1 + K(α,β,γ) λ2 < 1
Instability in general: for sup max x>0 ζ∈K
λ1 |xβ(ζ)| + λ2 |xγ(ζ)| >1 |α(ζ) + xβ(ζ)|
Advantage: Easier to implement than the implicit scheme (α, β), more expensive than the implicit–explicit scheme (α, β, γ) Drawback: Better stability properties than the (α, β, γ) scheme, but worse than the (α, β) scheme. G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
13 / 31
3. Error Analysis — i) Consistency E n consistency error for the solution u: kE n =
q−1 q X X γi B2 (tn+i , un+i ), αi un+i +kβi Aun+i −B1 (tn+i , un+i ) −k i=0
i=0
the amount by which the exact solution misses being an approximate solution. With E1n
=
q X
n+i
αi u
0
n+i
− kβi u (t
i=0
q X n ) , E2 = k (βi − γi )B2 (tn+i , un+i ) i=0
we have kE n = E1n + E2n .
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
14 / 31
Assumption:
q X
i ` αi = `
i=0
q X
i`−1 βi = `
i=0
q−1 X
i`−1 γi , ` = 0, 1, . . . , p,
i=0
i.e., the order of both schemes (α, β) and (α, γ) is p. Taylor expansion around tn yields q Z n+i 1 X t n E1 = (tn+i − s)p−1 αi (tn+i − s) − pkβi u(p+1) (s)ds p! n i=0 t Z tn+i q X k dp n E2 = (βi − γi ) (tn+i − s)p−1 p B2 s, u(s) ds. (p − 1)! dt tn i=0
Thus, under obvious regularity assumptions, max
0≤n≤N −q
G. Akrivis ([email protected])
kE n k? ≤ Ck p .
Multistep schemes for parabolic equations
September 19–23, 2011
15 / 31
ii) Local stability Let U m , V m ∈ Tu be s.t., for n = 0, . . . , N − q, q X
αi U n+i + kβi AU n+i − B1 (tn+i , U n+i )
=k
q−1 X
i=0
i=0
q X
q−1 X
αi V n+i + kβi AV n+i − B1 (tn+i , V n+i )
=k
i=0
γi B2 (tn+i , U n+i ), γi B2 (tn+i , V n+i ).
i=0
With ϑn := U n − V n , bni := Bi (tn , U n ) − Bi (tn , V n ), i = 1, 2, we have q X
(αi I + kβi A)ϑ
i=0
n+i
=k
q X i=0
βi bn+i 1
+k
q−1 X
γi b2n+i ,
i=0
n = 0, . . . , N − q. G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
16 / 31
Let the rational functions f (`, ·) and f˜(`, ·) be defined through the expansions ∞ X γ(ζ) = f (`, x)ζ −` , α(ζ) + xβ(ζ) `=1
∞ X β(ζ) = f˜(`, x)ζ −` , α(ζ) + xβ(ζ) `=0
for |ζ| ≥ 1.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
17 / 31
With n X n ϑ1 := k f˜(n − `, kA)b`1 , n = 0, . . . , N, `=0 n−1 X 0 n ϑ := 0, ϑ := k f (n − `, kA)b`2 , n = 1, . . . , N, 2 2 `=0 n ϑ3 := ϑn − ϑn1 − ϑn2 , n = 0, . . . , N, we have q X (αi I + kβi A)ϑn+i j i=0
G. Akrivis ([email protected])
q X k βi bn+i 1 , j = 1, i=0 q−1 X = γi bn+i 2 , j = 2, k i=0 0, j = 3.
Multistep schemes for parabolic equations
September 19–23, 2011
18 / 31
Lemma For n = 0, . . . , N, the following estimates are valid n n−1 X X ` 2 2 k kϑ k ≤ K k kb`1 k2? , 1 (α,β) `=0 `=0 n n−1 X X 2 k kϑ`2 k2 ≤ K(α,β,γ) k kb`2 k2? , `=0 `=0 q−1 n X j X kϑ`3 k2 ≤ c |ϑ3 |2 + kkϑj3 k2 . k `=0
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j=0
Multistep schemes for parabolic equations
September 19–23, 2011
19 / 31
Proof. We shall prove the second estimate. It suffices to let b`2 = 0, for ` ≥ n, and replace n by ∞. Let ϑˆ2 (t) :=
∞ X
ˆb2 (t) :=
ϑ`2 e2 i π`t ,
`=0
∞ X
b`2 e2 i π`t .
`=0
Then n h i−1 o −1 ϑˆ2 (t) = kAγ e−2 i πt α e−2 i πt + kAβ e−2 i πt A ˆb2 (t), whence kϑˆ2 (t)k ≤ K(α,β,γ) kˆb2 (t)k? . Therefore, in view of Parseval’s identity, with K := K(α,β,γ) , ∞ X `=0
kϑ`2 k2
Z
1
= 0
G. Akrivis ([email protected])
kϑˆ2 (t)k2 dt ≤ K 2
Z
1
kˆb2 (t)k2? dt = K 2
0 Multistep schemes for parabolic equations
∞ X
kb`2 k2? .
`=0 September 19–23, 2011
20 / 31
Proposition (Local stability) For n = q, . . . , N, the following estimate is valid n 2
|ϑ | + k
n X `=0
G. Akrivis ([email protected])
` 2
kϑ k ≤ C
q−1 X
|ϑj |2 + kkϑj k2 .
j=0
Multistep schemes for parabolic equations
September 19–23, 2011
21 / 31
iii) Error estimates
Theorem (Error estimate) Let the stability conditions be satisfied, the order of both methods be p and the solution u be sufficiently smooth. Assume for the starting approximations that max |u(tj ) − U j | + k 1/2 ku(tj ) − U j k ≤ ck p . 0≤j≤q−1
Then, we have the error estimate max |u(tn ) − U n | ≤ Ck p .
0≤n≤N
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
22 / 31
4. More on the constants K(α,β) and K(α,β,γ) α(ζ) , ζ ∈ K, s.t. β(ζ) 6= 0, β(ζ) denote the points of the root locus curve Let d(ζ) :=
Let K+ := {ζ ∈ K : Re d(ζ) ≥ 0}, K− := {ζ ∈ K : Re d(ζ) < 0}. Let k(x, ζ) :=
xγ(ζ) 1 γ(ζ) = . −1 α(ζ) + xβ(ζ) 1 + x d(ζ) β(ζ)
Then, ∀ζ ∈ K+
sup |k(x, ζ)| = x>0
and ∀ζ ∈ K−
sup |k(x, ζ)| = x>0
G. Akrivis ([email protected])
|γ(ζ)| |β(ζ)|
|d(ζ)| |γ(ζ)| . | Im d(ζ)| |β(ζ)|
Multistep schemes for parabolic equations
September 19–23, 2011
23 / 31
4. More on the constants K(α,β) and K(α,β,γ) α(ζ) , ζ ∈ K, s.t. β(ζ) 6= 0, β(ζ) denote the points of the root locus curve Let d(ζ) :=
Let K+ := {ζ ∈ K : Re d(ζ) ≥ 0}, K− := {ζ ∈ K : Re d(ζ) < 0}. Let k(x, ζ) :=
xγ(ζ) 1 γ(ζ) = . −1 α(ζ) + xβ(ζ) 1 + x d(ζ) β(ζ)
Then, ∀ζ ∈ K+
sup |k(x, ζ)| = x>0
and ∀ζ ∈ K−
sup |k(x, ζ)| = x>0
G. Akrivis ([email protected])
|γ(ζ)| |β(ζ)|
|d(ζ)| |γ(ζ)| . | Im d(ζ)| |β(ζ)|
Multistep schemes for parabolic equations
September 19–23, 2011
23 / 31
4. More on the constants K(α,β) and K(α,β,γ) α(ζ) , ζ ∈ K, s.t. β(ζ) 6= 0, β(ζ) denote the points of the root locus curve Let d(ζ) :=
Let K+ := {ζ ∈ K : Re d(ζ) ≥ 0}, K− := {ζ ∈ K : Re d(ζ) < 0}. Let k(x, ζ) :=
xγ(ζ) 1 γ(ζ) = . −1 α(ζ) + xβ(ζ) 1 + x d(ζ) β(ζ)
Then, ∀ζ ∈ K+
sup |k(x, ζ)| = x>0
and ∀ζ ∈ K−
sup |k(x, ζ)| = x>0
G. Akrivis ([email protected])
|γ(ζ)| |β(ζ)|
|d(ζ)| |γ(ζ)| . | Im d(ζ)| |β(ζ)|
Multistep schemes for parabolic equations
September 19–23, 2011
23 / 31
Therefore, K(α,β,γ) = max
n
max
|d(ζ)| |γ(ζ)| o |γ(ζ)| , max , |β(ζ)| ζ∈K− | Im d(ζ)| |β(ζ)|
max
|γ(ζ)| o |γ(ζ)| 1 , max , |β(ζ)| ζ∈K− | sin ϕ(ζ)| |β(ζ)|
ζ∈K+
or, equivalently, K(α,β,γ) = max
n
ζ∈K+
y d(ζ) ϕ(ζ)
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
x
September 19–23, 2011
24 / 31
Assumption: The scheme (α, β) is A−stable Then, K+ = K and K(α,β,γ) = max ζ∈K
|γ(ζ)| . |β(ζ)|
In particular, K(α,β) = 1. Assumption: The scheme (α, β) is A(ϑ)−stable Then, K(α,β) = max
ζ∈K−
G. Akrivis ([email protected])
1 1 = . | sin ϕ(ζ)| sin ϑ
Multistep schemes for parabolic equations
September 19–23, 2011
25 / 31
Assumption: The scheme (α, β) is A−stable Then, K+ = K and K(α,β,γ) = max ζ∈K
|γ(ζ)| . |β(ζ)|
In particular, K(α,β) = 1. Assumption: The scheme (α, β) is A(ϑ)−stable Then, K(α,β) = max
ζ∈K−
G. Akrivis ([email protected])
1 1 = . | sin ϕ(ζ)| sin ϑ
Multistep schemes for parabolic equations
September 19–23, 2011
25 / 31
Discrepancy between necessary and sufficient conditions Sufficient condition:
K(α,β) λ1 + K(α,β,γ) λ2 < 1
Necessary condition: sup max x>0 ζ∈K
λ1 |xβ(ζ)| + λ2 |xγ(ζ)| ≤1 |α(ζ) + xβ(ζ)|
The necessary condition can be equivalenty written as λ1 + λ2 max
ζ∈K+
|γ(ζ)| ≤ 1, |β(ζ)|
h |d(ζ)| |γ(ζ)| i λ1 + λ2 ≤ 1. |β(ζ)| ζ∈K− | Im d(ζ)| sup
A more practical, but weaker in general, necessary condition is λ1 + K(α,β,γ) λ2 ≤ 1,
K(α,β) λ1 + c?(β,γ) λ2 ≤ 1,
with c?(β,γ) := |γ(ζ ? )|/|β(ζ ? )|, if K(α,β) is attained at ζ ? ∈ K− . G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
26 / 31
y
y
1
1
1 K(α,β,γ)
1 K(α,β,γ)
1 K(α,β) c? (β,γ)
T
S 1 K(α,β)
x
1 K(α,β)
1
x 1
y 1
1 K(α,β,γ)
R
1 K(α,β)
G. Akrivis ([email protected])
x 1
Multistep schemes for parabolic equations
September 19–23, 2011
27 / 31
BDF methods: q
|T |
|T |/|S|
|T |/|R|
c?(β,γ)
3 4 5 6
1.70 × 10−4 1.32 × 10−3 2.69 × 10−3 1.62 × 10−3
2.39 × 10−3 4.13 × 10−2 2.13 × 10−1 6.68 × 10−1
2.78 × 10−3 4.17 × 10−2 2.13 × 10−1 6.68 × 10−1
1.73 3.49 0.71 1.00
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
28 / 31
On the necessary stability condition Necessary condition: sup max x>0 ζ∈K
λ1 |xβ(ζ)| + λ2 |xγ(ζ)| ≤1 |α(ζ) + xβ(ζ)|
Assume this condition is not satisfied. Then, with an appropriate Θ ∈ [0, 2π), for the function k, k(x, ζ) :=
λ1 x ei Θ β(ζ) + λ2 xγ(ζ) , α(ζ) + xβ(ζ)
x > 0, |ζ| ≥ 1.
we have |k(x, z)| > 1 for appropriate z ∈ K, x > 0. Since lim |k(x, ζ)| = λ1
|ζ|→∞
xβq < 1, αq + xβq
there exists a ζ ? ∈ C with |ζ ? | > 1 s.t. |k(x, ζ ? )| = 1, i.e., λ1 x ei Θ β(ζ ? ) + λ2 xγ(ζ ? ) = e− i ϕ , α(ζ ? ) + xβ(ζ ? ) G. Akrivis ([email protected])
for a ϕ ∈ [0, 2π).
Multistep schemes for parabolic equations
September 19–23, 2011
29 / 31
Therefore, α(ζ ? ) + xβ(ζ ? ) − λ1 x ei(Θ+ϕ) β(ζ ? ) − λ2 x ei ϕ γ(ζ ? ) = 0. For B1 (t, ·) := λ1 ei(Θ+ϕ) A and B2 (t, ·) := λ2 ei ϕ A, the Lipschitz conditions are satisfied. According to the von Neumann criterion, a necessary stability condition is that, if ν is an eigenvalue of A, the solutions of q X αi + kν βi − λ1 ei(Θ+ϕ) βi − λ2 ei ϕ γi v n+i = 0 i=0
are bounded; for kν = x this is not the case, since the root condition is not satisfied. Therefore, the scheme is not unconditionally stable.
G. Akrivis ([email protected])
Multistep schemes for parabolic equations
September 19–23, 2011
30 / 31
Thank you very much!
