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 : u tt − u xx = 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 ∈ V h ( M ) = � K ∈M P p ( K ) Notation: ♦ u = ( u x , − u t ) and ∇ u = ( u x , u t ) . i.b.p. � � − ( ∇ · ♦ u , v ) K d x = ( f , v ) K d x ⇒ K ∈M K ∈M DG-magic � � � ( ♦ u , ∇ v ) K − � ♦ u · n , v � ∂ K = ( f , v ) K ⇒ K ∈M K ∈M K ∈M Anders Skajaa Space-time DG methods for the wave equation

� � ( ♦ u , ∇ v ) K − �{ ♦ u } , [ v ] � E − �{ ♦ v } , [ u ] � E = ( f , v ) K K ∈M K ∈M Notation: { v } = ( v + + v − ) / 2 and [ v ] = v + n + + v − n − . Numerical scheme is then: ∀ K ∈ M , seek u h ∈ V h such that a K ( u h , v h ) = ℓ K ( v h ) , for all v h ∈ P p ( K ) where ℓ K ( v ) = ( f , v ) K and a K ( u , v ) = ( ♦ u , ∇ v ) K −�{ ♦ u } , [ v ] � ∂ K −�{ ♦ v } , [ u ] � ∂ K + � α [ u ] , [ v ] � ∂ K Choose basis, and plug in: local b.f. � � � � a K ( µ i b i , q j b j ) = ℓ K ( q j b j ) , for all v h = q j b j ⇒ i j j j   j + 1 � − 1 � µ ( n + 1 ) A ( n + 1 ) ℓ − A ( n − 1 ) µ ( n − 1 ) A ( n ) µ ( n )  � � � = � − �  j j j j i i i = j − 1 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 (n+1) K j (n) (n) ∆ t K K j−1 j+1 (n) K j (n−1) K j ∆ x Determined entries in matrices analytically in Maple Then implemented method in Matlab Anders Skajaa Space-time DG methods for the wave equation

Unfortunately: Constant κ and α 10 7 10 6 Maximal eigenvalue (abs) 10 5 10 4 3 10 2 10 1 10 0 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 γ 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