Mapped Tent Pitching Method for Hyperbolic Conservation Laws - PowerPoint PPT Presentation

Inverse transformation G − 1 ( ˆ U ) ˆ U ≡ G (ˆ u ) := ˆ u − f (ˆ u ) ∇ ϕ (3) Convection equation b ∈ R 2 Flux function: f ( u ) := bu , ˆ U ˆ U = (1 − b · ∇ ϕ )ˆ ⇔ u = ˆ u 1 − b · ∇ ϕ 1 solvable if |∇ ϕ | < | b | 11

Inverse transformation G − 1 ( ˆ U ) ˆ U ≡ G (ˆ u ) := ˆ u − f (ˆ u ) ∇ ϕ (3) Convection equation b ∈ R 2 Flux function: f ( u ) := bu , ˆ U ˆ U = (1 − b · ∇ ϕ )ˆ ⇔ u = ˆ u 1 − b · ∇ ϕ 1 solvable if |∇ ϕ | < | b | (known CFL-condition) 11

Inverse transformation G − 1 ( ˆ U ) ˆ U ≡ G (ˆ u ) := ˆ u − f (ˆ u ) ∇ ϕ (3) Convection equation b ∈ R 2 Flux function: f ( u ) := bu , ˆ U ˆ U = (1 − b · ∇ ϕ )ˆ ⇔ u = ˆ u 1 − b · ∇ ϕ 1 solvable if |∇ ϕ | < | b | (known CFL-condition) Theorem If there holds |∇ ϕ | < 1 c , then (3) has a unique solution ˆ u . c . . . maximal speed 11

Inverse transformation G − 1 ( ˆ U ) Find ( ρ, m , E ) : Ω × (0 , T ] → R × R N × R s.t.     ρ m    1  ∂ t  + div ρ m ⊗ m + pI  = 0 m   m E ρ ( E + p ) 12

Inverse transformation G − 1 ( ˆ U ) Find ( ρ, m , E ) : Ω × (0 , T ] → R × R N × R s.t.     ρ m    1  ∂ t  + div ρ m ⊗ m + pI  = 0 m   m E ρ ( E + p ) ˆ U ≡ G (ˆ u ) := ˆ u − f (ˆ u ) ∇ ϕ (3) 12

Inverse transformation G − 1 ( ˆ U ) Find ( ρ, m , E ) : Ω × (0 , T ] → R × R N × R s.t.     ρ m    1  ∂ t  + div ρ m ⊗ m + pI  = 0 m   m E ρ ( E + p ) ˆ U ≡ G (ˆ u ) := ˆ u − f (ˆ u ) ∇ ϕ (3) m , ˆ u = (ˆ ˆ ρ, ˆ E ) U = ( ˆ ˆ R , ˆ M , ˆ F ) m , ˆ E ) = ˆ G − 1 ( ˆ R , ˆ M , ˆ (ˆ ρ, ˆ F ) 12

Inverse transformation G − 1 ( ˆ U ) Find ( ρ, m , E ) : Ω × (0 , T ] → R × R N × R s.t.     ρ m    1  ∂ t  + div ρ m ⊗ m + pI  = 0 m   m E ρ ( E + p ) ˆ R 2 ˆ ρ = a 1 − 2 d |∇ ϕ | 2 a 3 m = ˆ ρ M + 2 ( ˆ ˆ d a 3 ∇ ϕ ) ˆ R E = ˆ ρ F + 2 a 3 ˆ ( ˆ ρ ∇ ϕ · ˆ m ) ˆ d ˆ R where a 2 a 1 = ˆ R − ˆ a 2 = 2 ˆ F ˆ R − | ˆ M | 2 , M · ∇ ϕ, a 3 = . � 1 − 4( d +1) a 2 |∇ ϕ | 2 a 2 a 1 + d 2 12

Conservation ˆ ˆ K i K i t t ν i ˆ U ( x , 1) Φ ν i − 1 ˆ U ( x , 0) x x 13

Conservation ˆ ˆ K i K i t t ν i ˆ U ( x , 1) Φ ν i − 1 ˆ U ( x , 0) x x Parametrizations γ i − 1 : x �→ ( x , τ i − 1 ( x )) , γ i : x �→ ( x , τ i ( x )) and space-time unit normal vectors ν i − 1 , ν i . 13

Conservation ˆ ˆ K i K i t t ν i ˆ U ( x , 1) Φ ν i − 1 ˆ U ( x , 0) x x Parametrizations γ i − 1 : x �→ ( x , τ i − 1 ( x )) , γ i : x �→ ( x , τ i ( x )) and space-time unit normal vectors ν i − 1 , ν i . 13

Conservation ˆ ˆ K i K i t t ν i ˆ U ( x , 1) Φ ν i − 1 ˆ U ( x , 0) x x Parametrizations γ i − 1 : x �→ ( x , τ i − 1 ( x )) , γ i : x �→ ( x , τ i ( x )) and space-time unit normal vectors ν i − 1 , ν i . Conservation After time step ˆ U ( x , 0) → ˆ U ( x , 1) there holds � � � � � � f ( u ) f ( u ) · ν i d s = · ν i − 1 d s . u u γ i γ i − 1 13

Conservation Conservation After time step ˆ U ( x , 0) → ˆ U ( x , 1) there holds � � � � � � f ( u ) f ( u ) · ν i d s = · ν i − 1 d s . u u γ i γ i − 1 14

Conservation Conservation After time step ˆ U ( x , 0) → ˆ U ( x , 1) there holds � � � � � � f ( u ) f ( u ) · ν i d s = · ν i − 1 d s . u u γ i γ i − 1 � � � � −∇ τ i ( x ) −∇ τ i − 1 ( x ) ν i ≈ , ν i − 1 ≈ 1 1 14

Conservation Conservation After time step ˆ U ( x , 0) → ˆ U ( x , 1) there holds � � � � � � f ( u ) f ( u ) · ν i d s = · ν i − 1 d s . u u γ i γ i − 1 � � � � −∇ τ i ( x ) 0 ν i ≈ , ν i − 1 = 1 1 ν i ν i − 1 14

Conservation Conservation After time step ˆ U ( x , 0) → ˆ U ( x , 1) there holds � � � � f ( u ) · ν i d s = u d s . u γ i γ i − 1 � � � � −∇ τ i ( x ) 0 ν i ≈ , ν i − 1 = 1 1 ν i ν i − 1 14

Conservation Conservation After time step ˆ U ( x , 0) → ˆ U ( x , 1) there holds � � � � � � f ( u ) f ( u ) · ν i d s = · ν i − 1 d s . u u γ i γ i − 1 � � � � 0 −∇ τ i − 1 ( x ) ν i = , ν i − 1 ≈ 1 1 ν i ν i − 1 14

Conservation Conservation After time step ˆ U ( x , 0) → ˆ U ( x , 1) there holds � � � � f ( u ) u d s = · ν i − 1 d s . u γ i γ i − 1 � � � � 0 −∇ τ i − 1 ( x ) ν i = , ν i − 1 ≈ 1 1 ν i ν i − 1 14

Tent pitching algorithm in 1D t Advancing front τ ∇ ϕ = 0 x ∇ ϕ = 0 ϕ ( x , ˆ t ) := (1 − ˆ t ) τ i − 1 ( x ) + ˆ t τ i ( x ) 15

Tent pitching algorithm in 1D t Advancing front τ ∇ ϕ = 0 x ∇ ϕ = 0 ϕ ( x , ˆ t ) := (1 − ˆ t ) τ i − 1 ( x ) + ˆ t τ i ( x ) ˆ U = ˆ u − f (ˆ u ) ∇ ϕ 15

Tent pitching algorithm in 1D t Advancing front τ ∇ ϕ = 0 x ∇ ϕ = 0 ϕ ( x , ˆ t ) := (1 − ˆ t ) τ i − 1 ( x ) + ˆ t τ i ( x ) ∇ ϕ =0 ˆ ˆ U = ˆ u − f (ˆ u ) ∇ ϕ = ⇒ U = ˆ u 15

The wave equation Find ψ : Ω × (0 , T ] → R s.t. ∂ tt ψ − div( ∇ ψ ) = 0 in Ω × (0 , T ] . 16

The wave equation Find ψ : Ω × (0 , T ] → R s.t. ∂ tt ψ − div( ∇ ψ ) = 0 in Ω × (0 , T ] . With � � � � −∇ ψ q := , µ ∂ t ψ 16

The wave equation Find ψ : Ω × (0 , T ] → R s.t. ∂ tt ψ − div( ∇ ψ ) = 0 in Ω × (0 , T ] . With � � � � −∇ ψ q := , µ ∂ t ψ we obtain � � � � q I µ + div = 0 . ∂ t q ⊤ µ 16

The wave equation Find ψ : Ω × (0 , T ] → R s.t. ∂ tt ψ − div( ∇ ψ ) = 0 in Ω × (0 , T ] . With � � � � −∇ ψ q := , µ ∂ t ψ we obtain � � � � q I µ + div = 0 . ∂ t q ⊤ µ Mapping to space-time cylinder leads to �� � � � � � � �� q ˆ I ˆ µ I ˆ µ ∂ ˆ − ∇ ϕ + div δ = 0 . t q ⊤ q ⊤ ˆ ˆ ˆ µ 16

The wave equation Find ψ : Ω × (0 , T ] → R s.t. ∂ tt ψ − div( ∇ ψ ) = 0 in Ω × (0 , T ] . With � � � � q −∇ ψ := , µ ∂ t ψ we obtain � � � � q I µ ∂ t + div = 0 . q ⊤ µ Mapping to space-time cylinder leads to �� � � �� � � I −∇ ϕ q ˆ I δ ˆ µ ∂ ˆ + div = 0 . t −∇ ϕ ⊤ q ⊤ 1 µ ˆ δ ˆ 16

The wave equation �� � � �� � � I −∇ ϕ q ˆ I δ ˆ µ ∂ ˆ + div = 0 . (4) t −∇ ϕ ⊤ q ⊤ 1 ˆ δ ˆ µ 17

The wave equation �� � � �� � � I −∇ ϕ q ˆ I δ ˆ µ ∂ ˆ + div = 0 . (4) t −∇ ϕ ⊤ q ⊤ 1 ˆ δ ˆ µ Space-discretization by DG leads to time-dependent mass matrix ∂ ˆ t M ˆ u + A ˆ u = 0 . 17

The wave equation �� � � �� � � I −∇ ϕ q ˆ I δ ˆ µ ∂ ˆ + div = 0 . (4) t −∇ ϕ ⊤ q ⊤ 1 ˆ δ ˆ µ Space-discretization by DG leads to time-dependent mass matrix ∂ ˆ t M ˆ u + A ˆ u = 0 . Introduce a new variable y = M ˆ u and discretize transformed system t y + AM − 1 y = 0 ∂ ˆ by a Runge-Kutta method. 17

The wave equation Find ψ : Ω × (0 , T ] → R s.t. ∂ tt ψ − div( ∇ ψ ) = 0 in Ω × (0 , T ] . With � � � � q −∇ ψ := , µ ∂ t ψ we obtain � � � � q I µ ∂ t + div = 0 . q ⊤ µ √ Domain Ω = [0 , π ] 2 , T = 2 π , √ 1 ψ ( x , t ) = √ cos( x 1 ) cos( x 2 ) sin( 2 t ) 2 18

The wave equation, 2+1 dimensions Figure 1: Convergence rates in two space dimensions with RK2 for various spatial polynomial degrees p of approximation, with e 2 = � q ( · , T ) − q h � 2 L 2 (Ω) + � µ ( · , T ) − µ h � 2 L 2 (Ω) . p = 1 10 − 1 p = 2 p = 3 10 − 2 p = 4 10 − 3 O ( h ) O ( h 2 ) 10 − 4 e O ( h 3 ) 10 − 5 O ( h 4 ) 10 − 6 10 − 7 10 2 10 3 ndofs 19

The wave equation Instead of t M ˆ u + A ˆ u = 0 , ∂ ˆ 20

The wave equation Instead of t M ˆ u + A ˆ u = 0 , ∂ ˆ consider the system t ˆ U + A ˆ u = 0 , (5a) ∂ ˆ ˆ U = M ˆ u . (5b) 20

The wave equation Instead of t M ˆ u + A ˆ u = 0 , ∂ ˆ consider the system t ˆ U + A ˆ u = 0 , (5a) ∂ ˆ ˆ U = M ˆ u . (5b) i ˆ i ˆ u = � U = � u i and ˆ i ˆ i ˆ Expansion of ˆ t t U i , with 20

The wave equation Instead of t M ˆ u + A ˆ u = 0 , ∂ ˆ consider the system t ˆ U + A ˆ u = 0 , (5a) ∂ ˆ ˆ U = M ˆ u . (5b) i ˆ i ˆ u = � U = � u i and ˆ i ˆ i ˆ Expansion of ˆ t t U i , with 1 ˆ U n +1 = n + 1 A ˆ u n , U n +1 − M ′ ˆ u n +1 = ˆ M ˆ u n . 20

The wave equation, 2+1 dimensions Figure 2: Convergence rates in two space dimensions with 2 Taylor steps for various spatial polynomial degrees p of approximation, with e 2 = � q ( · , T ) − q h � 2 L 2 (Ω) + � µ ( · , T ) − µ h � 2 L 2 (Ω) . p = 1 10 − 1 p = 2 p = 3 10 − 2 p = 4 10 − 3 O ( h ) O ( h 2 ) 10 − 4 e O ( h 3 ) 10 − 5 O ( h 4 ) 10 − 6 10 − 7 10 2 10 3 ndofs 21

The wave equation, 2+1 dimensions Figure 3: Convergence rates in two space dimensions with 4 Taylor steps for various spatial polynomial degrees p of approximation, with e 2 = � q ( · , T ) − q h � 2 L 2 (Ω) + � µ ( · , T ) − µ h � 2 L 2 (Ω) . p = 1 10 − 1 p = 2 p = 3 p = 4 10 − 3 O ( h ) O ( h 2 ) e 10 − 5 O ( h 3 ) O ( h 4 ) 10 − 7 10 − 9 10 2 10 3 ndofs 22

The wave equation Find ψ : Ω × (0 , T ] → R s.t. ∂ tt ψ − div( α ∇ ψ ) = 0 in Ω × (0 , T ] . With � � � � q −∇ ψ := , µ ∂ t ψ we obtain � � � � q I µ + div = 0 ∂ t q ⊤ µ Domain Ω = [0 , π ] 3 , T = 2 π 3 , √ √ 1 ψ ( x , t ) = √ cos( x 1 ) cos( x 2 ) cos( x 3 ) sin( 3 t ) 3 23

The wave equation, 3+1 dimensions Figure 4: Convergence rates in three space dimensions for various spatial polynomial degrees p of approximation and p Taylor steps, with e 2 = � q ( · , T ) − q h � 2 L 2 (Ω) + � µ ( · , T ) − µ h � 2 L 2 (Ω) . p = 2 10 − 2 p = 3 p = 4 10 − 3 O ( h 2 ) 10 − 4 O ( h 3 ) O ( h 4 ) 10 − 5 e 10 − 6 10 − 7 10 − 8 10 − 9 10 4 10 5 10 6 ndofs 24

The Maxwell equations The Maxwell equations � � � � curl H ε E = ∂ t µ H − curl E 25

The Maxwell equations The Maxwell equations � � � � curl H ε E = ∂ t µ H − curl E can be written as � � � � − skew H ε E + div = 0 , ∂ t µ H skew E with (skew E ) ij := ε ijk E k . 25

The Maxwell equations Figure 5: Resonator, 489k curved elements, largest to smallest element: 5:1 26

The Maxwell equations Figure 6: H y at t=260, 260 time slabs, 148k tents per slab, p 2 local Taylor time-steps Shared memory server, 4 E7-8867 CPUs with 16 cores each. 27

The Maxwell equations Figure 6: H y at t=260, 260 time slabs, 148k tents per slab, p 2 local Taylor time-steps Shared memory server, 4 E7-8867 CPUs with 16 cores each. p=2: 29 374 980 dofs, 20 min 27

The Maxwell equations Figure 6: H y at t=260, 260 time slabs, 148k tents per slab, p 2 local Taylor time-steps Shared memory server, 4 E7-8867 CPUs with 16 cores each. p=2: 29 374 980 dofs, 20 min p=3: 58 751 160 dofs, 3 h 33 min 27

The Maxwell equations Figure 7: Resonator with sharp edges, 224k curved elements, largest to smallest element: 10:1 28

The Maxwell equations Figure 8: H y at t=260, 260 time slabs, 66k tents per slab, p 2 local Taylor time-steps Shared memory server, 4 E7-8867 CPUs with 16 cores each. 29

The Maxwell equations Figure 8: H y at t=260, 260 time slabs, 66k tents per slab, p 2 local Taylor time-steps Shared memory server, 4 E7-8867 CPUs with 16 cores each. p=2: 13 452 000 dofs, 8 min 29

The Maxwell equations Figure 8: H y at t=260, 260 time slabs, 66k tents per slab, p 2 local Taylor time-steps Shared memory server, 4 E7-8867 CPUs with 16 cores each. p=2: 13 452 000 dofs, 8 min p=3: 26 904 000 dofs, 1 h 27 min 29

Euler equations Find ( ρ, m , E ) : Ω × (0 , T ] → R × R N × R s.t.     ρ m    1  ∂ t  + div ρ m ⊗ m + pI  = 0 m   m E ρ ( E + p ) 30

Entropy admissibility condition Entropy admissibility condition E ( u ) ∈ R . . . entropy, F ( u ) ∈ R N . . . entropy flux 31

Entropy admissibility condition Entropy admissibility condition E ( u ) ∈ R . . . entropy, F ( u ) ∈ R N . . . entropy flux ⇒ ∂ t E ( u ) + div F ( u ) ≤ 0 31

Entropy admissibility condition Entropy admissibility condition E ( u ) ∈ R . . . entropy, F ( u ) ∈ R N . . . entropy flux ⇒ ∂ t E ( u ) + div F ( u ) ≤ 0 The pair ( E , F ) is called the entropy pair. 31

Entropy admissibility condition Entropy admissibility condition E ( u ) ∈ R . . . entropy, F ( u ) ∈ R N . . . entropy flux ⇒ ∂ t E ( u ) + div F ( u ) ≤ 0 The pair ( E , F ) is called the entropy pair. ˆ E ( w ) = E ( w ) − F ( w ) ∇ ϕ, ˆ F ( w ) = δ F ( w ) . 31

Entropy admissibility condition Entropy admissibility condition E ( u ) ∈ R . . . entropy, F ( u ) ∈ R N . . . entropy flux ⇒ ∂ t E ( u ) + div F ( u ) ≤ 0 The pair ( E , F ) is called the entropy pair. ˆ E ( w ) = E ( w ) − F ( w ) ∇ ϕ, ˆ F ( w ) = δ F ( w ) . Mapped entropy admissibility condition t ˆ u ) + div ˆ ∂ ˆ E (ˆ F (ˆ u ) = δ ( ∂ t E ( u ) + div F ( u )) ◦ Φ ≤ 0 31