CCM, Part III (4)

CCM, Part III (4)

Daniele Boffi

Dipartimento di Matematica “F. Casorati”

Universit` a di Pavia boffi@dimat.unipv.it

http://www-dimat.unipv.it/boffi/

March 2004

Parabolic problems

Heat equation

∂u(t)

∂t − ∆u(t) = f (t)

Variational formulation: for each t, find u(t) ∈ V = H ₀ ¹ (Ω) s.t.

 ∂u(t)

∂t , v



+ a(u(t), v) = (f (t), v) ∀v ∈ V

Space semidiscretization. Take V _h ⊂ V and, for each t, look for u _h (t) ∈ V _h such that

 ∂u _h (t)

∂t , v _h



+ a(u _h (t), v _h ) = (f (t), v _h ) ∀v _h ∈ V _h

Parabolic problems (cont’ed)

Fully discretized problem

I Explicit Euler

u ⁿ⁺¹ _h − u ⁿ _h

∆t , v _h

!

+ a(u ⁿ _h , v _h ) = (f ⁿ , v _h ) ∀v _h ∈ V _h

I Implicit Euler

u ⁿ⁺¹ _h − u ⁿ _h

∆t , v _h

!

+ a(u ⁿ⁺¹ _h , v _h ) = (f ⁿ⁺¹ , v _h ) ∀v _h ∈ V _h

Parabolic problems (cont’ed)

θ-method (somewhat inbetween explicit and implicit) 0 ≤ θ ≤ 1

u ⁿ⁺¹ _h − u ⁿ _h

∆t , v _h

!

+(1 − θ)a(u ⁿ _h , v _h ) + θ a(u ⁿ⁺¹ _h , v _h ) =

(1 − θ)(f ⁿ , v _h ) + θ(f ⁿ⁺¹ , v _h ) ∀v _h ∈ V _h In one space dimension (finite differences, and f = 0)

u ⁿ⁺¹ _i − u ⁿ _i

∆t = 1

h ² (1 − θ)(u ⁿ _i+1 − 2u ⁿ _i + u ⁿ _i−1 )+

θ (u ⁿ⁺¹ _i+1 − 2u ⁿ⁺¹ _i + u ⁿ⁺¹ _i−1 )

Parabolic problems (cont’ed)

If u(0) = sin(πx) then the solution in [0, π] with homogeneous Dirichlet boundary conditions is

u(t) = sin(πx) exp(−π ² t)

In particular, it goes to zero as t → +∞

Study of discrete (absolute) stability Discrete solution has the form

u ⁿ _i = α ⁿ sin(πih)

Stability condition |α| ≤ 1

Parabolic problems (cont’ed)

u ⁿ⁺¹ _i − u ⁿ _i = k

h ² (1 − θ)(u ⁿ _i+1 − 2u ⁿ _i + u ⁿ _i−1 )+

θ(u ⁿ⁺¹ _i+1 − 2u ⁿ⁺¹ _i + u ⁿ⁺¹ _i−1 ) u ⁿ _i = α ⁿ sin(πih)

Some trigonometry

sin π(i + 1)h − 2 sin(πih) + sin π(i − 1)h = 2 sin(πih) cos(πh) − 2 sin(πih) =

sin(πih)(−4 sin ² (πh/2)) Hence

α − 1 = k

h ² ((1 − θ) + θα)(−4 sin ² (πh/2))

Parabolic problems (cont’ed)

Finally

α = 1 − (1 − θ)w

1 + θw = 1 − w 1 + θw with w = 4 _h ^k

sin ² (πh/2) ≥ 0

Condition |α| ≤ 1 equivalent to

w(1 − 2θ) ≤ 2

I 1/2 ≤ θ ≤ 1 inconditionally stable

I 0 ≤ θ < 1/2 stability condition k

h ² ≤ 1

2(1 − 2θ)

End of Part IV