Goals: Devise space-time DG-method for the wave equation : u tt u xx - - PowerPoint PPT Presentation

Goals: Devise space-time DG-method for the wave equation: utt − uxx = f in Q := Ω×]0, T[ Implement method and test it Investigate stabilty of method Weak formulation: M = {K} is a mesh that covers the space-time domain Q Testfunction v ∈ Vh(M) =

K∈M Pp(K)

Notation: ♦u = (ux, −ut) and ∇u = (ux, ut). −

• K∈M

(∇ · ♦u, v)K dx =

• K∈M

(f , v)K dx

i.b.p.

⇒

• K∈M

(♦u, ∇v)K −

• K∈M

♦u · n, v∂K =

• K∈M

(f , v)K

DG-magic

⇒

Anders Skajaa Space-time DG methods for the wave equation

• K∈M

(♦u, ∇v)K − {♦u}, [v]E − {♦v}, [u]E =

• K∈M

(f , v)K Notation: {v} = (v+ + v−)/2 and [v] = v+n+ + v−n−. Numerical scheme is then: ∀ K ∈ M, seek uh ∈ Vh such that aK(uh, vh) = ℓK(vh), for all vh ∈ Pp(K) where ℓK(v) = (f , v)K and aK(u, v) = (♦u, ∇v)K−{♦u}, [v]∂K−{♦v}, [u]∂K+α[u], [v]∂K Choose basis, and plug in: aK(

• i

µibi,

• j

qjbj) = ℓK(

• j

qjbj), for all vh =

• j

qjbj

local b.f.

⇒

• µ(n+1)

j

=

• A(n+1)

j

−1   ℓ − A(n−1)

j

• µ(n−1)

j

−

j+1

• i=j−1

A(n)

i

• µ(n)

i

 

Anders Skajaa Space-time DG methods for the wave equation

Possible if we choose locally supported basis functions Now we have explicit scheme, suited for implementation

∆t ∆x

K

(n−1) j (n) j

K K

(n+1) j

K

(n) j−1

K

(n) j+1

Determined entries in matrices analytically in Maple Then implemented method in Matlab

Anders Skajaa Space-time DG methods for the wave equation

Unfortunately:

0.2 0.4 0.6 0.8 1 1.2 1.4 10 10

1

10

2

10

3

10

4

10

5

10

6

10

7

Constant κ and α γ Maximal eigenvalue (abs)

Numerical experiments: Blow up in solutions Von Neumann analysis showed unconditional instability Possible remedy? Perhaps: Use mesh that avoids vertical edges

Anders Skajaa Space-time DG methods for the wave equation