SLIDE 1 Mapped Tent Pitching Method for Hyperbolic Conservation Laws
Christoph Wintersteiger∗
Institute for Analysis and Scientific Computing, TU Wien
Jay Gopalakrishnan
Portland State University, USA
Joachim Sch¨
Institute for Analysis and Scientific Computing, TU Wien
RICAM, Workshop: Space-Time Methods for PDEs November 8, 2016
SLIDE 2 Outline
- Hyperbolic Conservation Laws
- Mapped Tent Pitching method
- Numerical results
- Summary
2
SLIDE 3 Outline
- Hyperbolic Conservation Laws
- Mapped Tent Pitching method
- Numerical results
- Summary
2
SLIDE 4 Outline
- Hyperbolic Conservation Laws
- Mapped Tent Pitching method
- Numerical results
- Summary
2
SLIDE 5 Outline
- Hyperbolic Conservation Laws
- Mapped Tent Pitching method
- Numerical results
- Summary
2
SLIDE 6
Hyperbolic Conservation Laws
Problem description Let be Ω ⊂ RN. Find u : Ω × (0, T] → Rn such that ∂tu(x, t) + divx f (x, t, u(x, t)) = 0 ∀(x, t) ∈ Ω × (0, T] , u(x, 0) = u0(x) ∀x ∈ Ω , with the given flux function f : Ω × (0, T] × Rn − → Rn×N , (x, t, u(x, t)) − → f (x, t, u(x, t)) .
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SLIDE 7
Hyperbolic Conservation Laws
Problem description Let be Ω ⊂ RN. Find u : Ω × (0, T] → Rn such that ∂tu(x, t) + divx f (x, t, u(x, t)) = 0 ∀(x, t) ∈ Ω × (0, T] , u(x, 0) = u0(x) ∀x ∈ Ω , with the given flux function f : Ω × (0, T] × Rn − → Rn×N , (x, t, u(x, t)) − → f (x, t, u(x, t)) . Hyperbolicity We call the system hyperbolic in the t-direction, if the matrix Du(f ν) has real eigenvalues (characteristic speeds) λ1, . . . λn for all directions ν ∈ SN−1.
3
SLIDE 8
Mapped Tent Pitching (MTP) method
4
SLIDE 9 Mapped Tent Pitching (MTP) method
- construct space-time mesh using a tent pitching algorithm
- map conservation law on each tent to a space-time cylinder
- spatially discretize using a discontinuous Galerkin method
- apply high order time stepping on the cylinder
4
SLIDE 10 Mapped Tent Pitching (MTP) method
- construct space-time mesh using a tent pitching algorithm
- map conservation law on each tent to a space-time cylinder
- spatially discretize using a discontinuous Galerkin method
- apply high order time stepping on the cylinder
4
SLIDE 11 Mapped Tent Pitching (MTP) method
- construct space-time mesh using a tent pitching algorithm
- map conservation law on each tent to a space-time cylinder
- spatially discretize using a discontinuous Galerkin method
- apply high order time stepping on the cylinder
4
SLIDE 12 Mapped Tent Pitching (MTP) method
- construct space-time mesh using a tent pitching algorithm
- map conservation law on each tent to a space-time cylinder
- spatially discretize using a discontinuous Galerkin method
- apply high order time stepping on the cylinder
4
SLIDE 13
Tent pitching algorithm in 1D x t
5
SLIDE 14
Tent pitching algorithm in 1D x t
∝ 1
¯ c , ¯
c . . . maximal characteristic speed
5
SLIDE 15
Tent pitching algorithm in 1D x t
∝ 1
¯ c , ¯
c . . . maximal characteristic speed local CFL-condition |∇τ| < 1
¯ c
Advancing front τ
5
SLIDE 16
Tent pitching algorithm in 1D x t
Advancing front τ
5
SLIDE 17
Tent pitching algorithm in 1D x t
Advancing front τ
5
SLIDE 18
Tent pitching algorithm in 1D x t
Advancing front τ
5
SLIDE 19
Tent pitching algorithm in 1D x t
Advancing front τ
5
SLIDE 20
Tent pitching algorithm in 1D x t
Advancing front τ
5
SLIDE 21
Tent pitching algorithm in 1D x t
Advancing front τ
5
SLIDE 22
Tent pitching algorithm in 1D x t
Advancing front τ
5
SLIDE 23
Tent pitching algorithm in 1D x t
Advancing front τ
5
SLIDE 24
Tent pitching algorithm in 1D x t
Advancing front τ
5
SLIDE 25
Tent pitching algorithm in 1D x t
Advancing front τ
5
SLIDE 26
Tent pitching algorithm in 2D
Gray tents: Level 0 tents, can be solved in parallel
6
SLIDE 27
Tent pitching algorithm in 2D
Gray tents: Level 1 tents, can be solved in parallel
6
SLIDE 28
Tent pitching algorithm in 2D
Gray tents: Level 2 tents, can be solved in parallel
6
SLIDE 29
Tent pitching algorithm in 2D
Gray tents: Level 3 tents, can be solved in parallel
6
SLIDE 30 Tent pitching
- R. S. Falk and G. R. Richter, Explicit finite element methods for
symmetric hyperbolic equations, SIAM J. Numer. Anal., 36 (1999),
- pp. 935–952.
- J. Palaniappan, R. B. Haber, and R. L. Jerrard, A spacetime
discontinuous Galerkin method for scalar conservation laws, Computer Methods in Applied Mechanics and Engineering, 193 (2004),
- pp. 3607–3631.
- P. Monk and G. R. Richter, A discontinuous Galerkin method for
linear symmetric hyperbolic systems in inhomogeneous media, J. Sci. Comput., 22/23 (2005), pp. 443–477.
7
SLIDE 31
Mapping
Space-time cylinder ˆ Ki := Ωv(i) × (0, 1) over the vertex patch Ωv(i)
8
SLIDE 32
Mapping
Space-time cylinder ˆ Ki := Ωv(i) × (0, 1) over the vertex patch Ωv(i) Duffy-like transformation Φ : ˆ Ki − → Ki , (x, ˆ t) − → (x, ϕ(x, ˆ t)) ,
8
SLIDE 33
Mapping
Space-time cylinder ˆ Ki := Ωv(i) × (0, 1) over the vertex patch Ωv(i) Duffy-like transformation Φ : ˆ Ki − → Ki , (x, ˆ t) − → (x, ϕ(x, ˆ t)) , ϕ(x, ˆ t) := (1 − ˆ t) τi−1(x) + ˆ t τi(x) .
8
SLIDE 34
Mapping
Space-time cylinder ˆ Ki := Ωv(i) × (0, 1) over the vertex patch Ωv(i) Duffy-like transformation Φ : ˆ Ki − → Ki , (x, ˆ t) − → (x, ϕ(x, ˆ t)) , ϕ(x, ˆ t) := (1 − ˆ t) τi−1(x) + ˆ t τi(x) .
Ki
x t τi(x) τi−1(x)
ˆ Ki
x ˆ t 1
Φ
8
SLIDE 35
Mapping
Space-time cylinder ˆ Ki := Ωv(i) × (0, 1) over the vertex patch Ωv(i) Duffy-like transformation Φ : ˆ Ki − → Ki , (x, ˆ t) − → (x, ϕ(x, ˆ t)) , ϕ(x, ˆ t) := (1 − ˆ t) τi−1(x) + ˆ t τi(x) .
Ki
x t τi(x) τi−1(x)
ˆ Ki
x ˆ t ˆ t ∗ 1
Φ
8
SLIDE 36
Mapping
Space-time cylinder ˆ Ki := Ωv(i) × (0, 1) over the vertex patch Ωv(i) Duffy-like transformation Φ : ˆ Ki − → Ki , (x, ˆ t) − → (x, ϕ(x, ˆ t)) , ϕ(x, ˆ t) := (1 − ˆ t) τi−1(x) + ˆ t τi(x) .
Ki
x t τi(x) τi−1(x) ϕ(x, ˆ t ∗)
ˆ Ki
x ˆ t ˆ t ∗ 1
Φ
8
SLIDE 37
Mapped conservation laws
F(u) := (f , u) ∈ Rn×(N+1) ∂tu + divx f = 0 ⇔ div(x,t) F(u) = 0 (1)
9
SLIDE 38
Mapped conservation laws
F(u) := (f , u) ∈ Rn×(N+1) ∂tu + divx f = 0 ⇔ div(x,t) F(u) = 0 (1) Piola transformation ˆ Fl = det(ˆ DΦ) [ˆ DΦ]−1 Fl ◦ Φ ∀l ∈ {1, . . . , n}
9
SLIDE 39
Mapped conservation laws
F(u) := (f , u) ∈ Rn×(N+1) ∂tu + divx f = 0 ⇔ div(x,t) F(u) = 0 (1) Piola transformation ˆ Fl = det(ˆ DΦ) [ˆ DΦ]−1 Fl ◦ Φ ∀l ∈ {1, . . . , n} 1 det(ˆ DΦ) div(x,ˆ
t) ˆ
F(u ◦ Φ
=:ˆ u
) = 0 (2)
9
SLIDE 40
Mapped conservation laws
F(u) := (f , u) ∈ Rn×(N+1) ∂tu + divx f = 0 ⇔ div(x,t) F(u) = 0 (1) Piola transformation ˆ Fl = det(ˆ DΦ) [ˆ DΦ]−1 Fl ◦ Φ ∀l ∈ {1, . . . , n} 1 det(ˆ DΦ) div(x,ˆ
t) ˆ
F(u ◦ Φ
=:ˆ u
) = 0 (2) conservation law on the space-time cylinder ˆ Ki
9
SLIDE 41 Mapped conservation laws
Problem description Find ˆ u : ˆ Ki → Rn such that ∂ˆ
t (ˆ
u − f (ˆ u) ∇ϕ)
u)≡ ˆ U
+ divx
u)
in ˆ Ki .
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SLIDE 42 Mapped conservation laws
Problem description Find ˆ u : ˆ Ki → Rn such that ∂ˆ
t (ˆ
u − f (ˆ u) ∇ϕ)
u)≡ ˆ U
+ divx
u)
in ˆ Ki . Problem description Find ˆ U : ˆ Ki → Rn such that ∂ˆ
t ˆ
U + divx
U))
in ˆ Ki , ˆ U(·, 0) = ˆ Ui−1(·, 1) in Ωv(i) .
10
SLIDE 43 Mapped conservation laws
Problem description Find ˆ u : ˆ Ki → Rn such that ∂ˆ
t (ˆ
u − f (ˆ u) ∇ϕ)
u)≡ ˆ U
+ divx
u)
in ˆ Ki . Problem description Find ˆ U : ˆ Ki → Rn such that ∂ˆ
t ˆ
U + divx
U))
in ˆ Ki , ˆ U(·, 0) = ˆ Ui−1(·, 1) in Ωv(i) .
- conservation law in new variable ˆ
U
- inverse transformation G −1( ˆ
U) needed
10
SLIDE 44 Mapped conservation laws
Problem description Find ˆ u : ˆ Ki → Rn such that ∂ˆ
t (ˆ
u − f (ˆ u) ∇ϕ)
u)≡ ˆ U
+ divx
u)
in ˆ Ki . Problem description Find ˆ U : ˆ Ki → Rn such that ∂ˆ
t ˆ
U + divx
U))
in ˆ Ki , ˆ U(·, 0) = ˆ Ui−1(·, 1) in Ωv(i) .
- conservation law in new variable ˆ
U
- inverse transformation G −1( ˆ
U) needed
10
SLIDE 45
Inverse transformation G −1( ˆ U)
ˆ U ≡ G(ˆ u) := ˆ u − f (ˆ u) ∇ϕ (3)
11
SLIDE 46
Inverse transformation G −1( ˆ U)
ˆ U ≡ G(ˆ u) := ˆ u − f (ˆ u) ∇ϕ (3) Convection equation Flux function: f (u) := bu, b ∈ R2 ˆ U = (1 − b · ∇ϕ)ˆ u
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SLIDE 47
Inverse transformation G −1( ˆ U)
ˆ U ≡ G(ˆ u) := ˆ u − f (ˆ u) ∇ϕ (3) Convection equation Flux function: f (u) := bu, b ∈ R2 ˆ U = (1 − b · ∇ϕ)ˆ u ⇔ ˆ u = ˆ U 1 − b · ∇ϕ solvable if |∇ϕ| <
1 |b| 11
SLIDE 48
Inverse transformation G −1( ˆ U)
ˆ U ≡ G(ˆ u) := ˆ u − f (ˆ u) ∇ϕ (3) Convection equation Flux function: f (u) := bu, b ∈ R2 ˆ U = (1 − b · ∇ϕ)ˆ u ⇔ ˆ u = ˆ U 1 − b · ∇ϕ solvable if |∇ϕ| <
1 |b| (known CFL-condition) 11
SLIDE 49
Inverse transformation G −1( ˆ U)
ˆ U ≡ G(ˆ u) := ˆ u − f (ˆ u) ∇ϕ (3) Convection equation Flux function: f (u) := bu, b ∈ R2 ˆ U = (1 − b · ∇ϕ)ˆ u ⇔ ˆ u = ˆ U 1 − b · ∇ϕ solvable if |∇ϕ| <
1 |b| (known CFL-condition)
Theorem If there holds |∇ϕ| < 1
c , then (3) has a unique solution ˆ
u. c . . . maximal speed
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SLIDE 50
Inverse transformation G −1( ˆ U)
Find (ρ, m, E) : Ω × (0, T] → R × RN × R s.t. ∂t ρ m E + div m
1 ρm ⊗ m + pI m ρ (E + p)
= 0
12
SLIDE 51
Inverse transformation G −1( ˆ U)
Find (ρ, m, E) : Ω × (0, T] → R × RN × R s.t. ∂t ρ m E + div m
1 ρm ⊗ m + pI m ρ (E + p)
= 0 ˆ U ≡ G(ˆ u) := ˆ u − f (ˆ u) ∇ϕ (3)
12
SLIDE 52
Inverse transformation G −1( ˆ U)
Find (ρ, m, E) : Ω × (0, T] → R × RN × R s.t. ∂t ρ m E + div m
1 ρm ⊗ m + pI m ρ (E + p)
= 0 ˆ U ≡ G(ˆ u) := ˆ u − f (ˆ u) ∇ϕ (3) ˆ u = (ˆ ρ, ˆ m, ˆ E) ˆ U = ( ˆ R, ˆ M, ˆ F) (ˆ ρ, ˆ m, ˆ E) = ˆ G −1( ˆ R, ˆ M, ˆ F)
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SLIDE 53 Inverse transformation G −1( ˆ U)
Find (ρ, m, E) : Ω × (0, T] → R × RN × R s.t. ∂t ρ m E + div m
1 ρm ⊗ m + pI m ρ (E + p)
= 0 ˆ ρ = ˆ R2 a1 − 2
d |∇ϕ|2a3
ˆ m = ˆ ρ ˆ R ( ˆ M + 2 d a3∇ϕ) ˆ E = ˆ ρ ˆ R ( ˆ F + 2 d a3 ˆ ρ ∇ϕ · ˆ m) where a1 = ˆ R − ˆ M · ∇ϕ, a2 = 2 ˆ F ˆ R − | ˆ M|2, a3 = a2 a1 +
1 − 4(d+1) d2
|∇ϕ|2a2 .
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SLIDE 54
Conservation Ki
x t νi νi−1
ˆ Ki
x ˆ t
ˆ U(x, 0) ˆ U(x, 1)
Φ
13
SLIDE 55
Conservation Ki
x t νi νi−1
ˆ Ki
x ˆ t
ˆ U(x, 0) ˆ U(x, 1)
Φ
Parametrizations γi−1 : x → (x, τi−1(x)) , γi : x → (x, τi(x)) and space-time unit normal vectors νi−1, νi.
13
SLIDE 56
Conservation Ki
x t νi νi−1
ˆ Ki
x ˆ t
ˆ U(x, 0) ˆ U(x, 1)
Φ
Parametrizations γi−1 : x → (x, τi−1(x)) , γi : x → (x, τi(x)) and space-time unit normal vectors νi−1, νi.
13
SLIDE 57 Conservation Ki
x t νi νi−1
ˆ Ki
x ˆ t
ˆ U(x, 0) ˆ U(x, 1)
Φ
Parametrizations γi−1 : x → (x, τi−1(x)) , γi : x → (x, τi(x)) and space-time unit normal vectors νi−1, νi. Conservation After time step ˆ U(x, 0) → ˆ U(x, 1) there holds
u
u
13
SLIDE 58 Conservation
Conservation After time step ˆ U(x, 0) → ˆ U(x, 1) there holds
u
u
14
SLIDE 59 Conservation
Conservation After time step ˆ U(x, 0) → ˆ U(x, 1) there holds
u
u
νi ≈
1
νi−1 ≈
1
SLIDE 60 Conservation
Conservation After time step ˆ U(x, 0) → ˆ U(x, 1) there holds
u
u
νi ≈
1
νi−1 =
νi−1
14
SLIDE 61 Conservation
Conservation After time step ˆ U(x, 0) → ˆ U(x, 1) there holds
u
u ds . νi ≈
1
νi−1 =
νi−1
14
SLIDE 62 Conservation
Conservation After time step ˆ U(x, 0) → ˆ U(x, 1) there holds
u
u
νi =
νi−1 ≈
1
νi−1
14
SLIDE 63 Conservation
Conservation After time step ˆ U(x, 0) → ˆ U(x, 1) there holds
u ds =
u
νi =
νi−1 ≈
1
νi−1
14
SLIDE 64
Tent pitching algorithm in 1D
∇ϕ = 0 ∇ϕ = 0
x t
Advancing front τ ϕ(x, ˆ t) := (1 − ˆ t) τi−1(x) + ˆ t τi(x)
15
SLIDE 65
Tent pitching algorithm in 1D
∇ϕ = 0 ∇ϕ = 0
x t
Advancing front τ ϕ(x, ˆ t) := (1 − ˆ t) τi−1(x) + ˆ t τi(x) ˆ U = ˆ u − f (ˆ u) ∇ϕ
15
SLIDE 66
Tent pitching algorithm in 1D
∇ϕ = 0 ∇ϕ = 0
x t
Advancing front τ ϕ(x, ˆ t) := (1 − ˆ t) τi−1(x) + ˆ t τi(x) ˆ U = ˆ u − f (ˆ u) ∇ϕ
∇ϕ=0
= ⇒ ˆ U = ˆ u
15
SLIDE 67
The wave equation
Find ψ : Ω × (0, T] → R s.t. ∂ttψ − div(∇ψ) = 0 in Ω × (0, T] .
16
SLIDE 68 The wave equation
Find ψ : Ω × (0, T] → R s.t. ∂ttψ − div(∇ψ) = 0 in Ω × (0, T] . With
µ
∂tψ
16
SLIDE 69 The wave equation
Find ψ : Ω × (0, T] → R s.t. ∂ttψ − div(∇ψ) = 0 in Ω × (0, T] . With
µ
∂tψ
we obtain ∂t
µ
q⊤
16
SLIDE 70 The wave equation
Find ψ : Ω × (0, T] → R s.t. ∂ttψ − div(∇ψ) = 0 in Ω × (0, T] . With
µ
∂tψ
we obtain ∂t
µ
q⊤
Mapping to space-time cylinder leads to ∂ˆ
t
q ˆ µ
µ ˆ q⊤
µ ˆ q⊤
16
SLIDE 71 The wave equation
Find ψ : Ω × (0, T] → R s.t. ∂ttψ − div(∇ψ) = 0 in Ω × (0, T] . With
µ
∂tψ
we obtain ∂t
µ
q⊤
Mapping to space-time cylinder leads to ∂ˆ
t
−∇ϕ −∇ϕ⊤ 1 ˆ q ˆ µ
µ δˆ q⊤
16
SLIDE 72 The wave equation
∂ˆ
t
−∇ϕ −∇ϕ⊤ 1 ˆ q ˆ µ
µ δˆ q⊤
(4)
17
SLIDE 73 The wave equation
∂ˆ
t
−∇ϕ −∇ϕ⊤ 1 ˆ q ˆ µ
µ δˆ q⊤
(4) Space-discretization by DG leads to time-dependent mass matrix ∂ˆ
tM ˆ
u + Aˆ u = 0.
17
SLIDE 74 The wave equation
∂ˆ
t
−∇ϕ −∇ϕ⊤ 1 ˆ q ˆ µ
µ δˆ q⊤
(4) Space-discretization by DG leads to time-dependent mass matrix ∂ˆ
tM ˆ
u + Aˆ u = 0. Introduce a new variable y = M ˆ u and discretize transformed system ∂ˆ
ty + AM−1y = 0
by a Runge-Kutta method.
17
SLIDE 75 The wave equation
Find ψ : Ω × (0, T] → R s.t. ∂ttψ − div(∇ψ) = 0 in Ω × (0, T] . With
µ
∂tψ
we obtain ∂t
µ
q⊤
Domain Ω = [0, π]2, T = √ 2π, ψ(x, t) = 1 √ 2 cos(x1) cos(x2) sin( √ 2t)
18
SLIDE 76
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 e2 = q(·, T) − qh2
L2(Ω) + µ(·, T) − µh2 L2(Ω).
102 103 10−7 10−6 10−5 10−4 10−3 10−2 10−1 ndofs e p = 1 p = 2 p = 3 p = 4 O(h) O(h2) O(h3) O(h4)
19
SLIDE 77
The wave equation
Instead of ∂ˆ
tM ˆ
u + Aˆ u = 0,
20
SLIDE 78
The wave equation
Instead of ∂ˆ
tM ˆ
u + Aˆ u = 0, consider the system ∂ˆ
t ˆ
U + Aˆ u = 0, (5a) ˆ U = M ˆ u . (5b)
20
SLIDE 79
The wave equation
Instead of ∂ˆ
tM ˆ
u + Aˆ u = 0, consider the system ∂ˆ
t ˆ
U + Aˆ u = 0, (5a) ˆ U = M ˆ u . (5b) Expansion of ˆ u =
i ˆ
t
i ˆ
ui and ˆ U =
i ˆ
t
i ˆ
Ui, with
20
SLIDE 80
The wave equation
Instead of ∂ˆ
tM ˆ
u + Aˆ u = 0, consider the system ∂ˆ
t ˆ
U + Aˆ u = 0, (5a) ˆ U = M ˆ u . (5b) Expansion of ˆ u =
i ˆ
t
i ˆ
ui and ˆ U =
i ˆ
t
i ˆ
Ui, with ˆ Un+1 = 1 n + 1Aˆ un, M ˆ un+1 = ˆ Un+1 − M′ ˆ un.
20
SLIDE 81
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 e2 = q(·, T) − qh2
L2(Ω) + µ(·, T) − µh2 L2(Ω).
102 103 10−7 10−6 10−5 10−4 10−3 10−2 10−1 ndofs e p = 1 p = 2 p = 3 p = 4 O(h) O(h2) O(h3) O(h4)
21
SLIDE 82
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 e2 = q(·, T) − qh2
L2(Ω) + µ(·, T) − µh2 L2(Ω).
102 103 10−9 10−7 10−5 10−3 10−1 ndofs e p = 1 p = 2 p = 3 p = 4 O(h) O(h2) O(h3) O(h4)
22
SLIDE 83 The wave equation
Find ψ : Ω × (0, T] → R s.t. ∂ttψ − div(α∇ψ) = 0 in Ω × (0, T] . With
µ
∂tψ
we obtain ∂t
µ
q⊤
Domain Ω = [0, π]3, T = 2π
√ 3,
ψ(x, t) = 1 √ 3 cos(x1) cos(x2) cos(x3) sin( √ 3t)
23
SLIDE 84
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 e2 = q(·, T) − qh2
L2(Ω) + µ(·, T) − µh2 L2(Ω).
104 105 106 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 ndofs e p = 2 p = 3 p = 4 O(h2) O(h3) O(h4)
24
SLIDE 85 The Maxwell equations
The Maxwell equations ∂t
µH
− curl E
SLIDE 86 The Maxwell equations
The Maxwell equations ∂t
µH
− curl E
∂t
µH
skew E
with (skew E)ij := εijkEk.
25
SLIDE 87
The Maxwell equations
Figure 5: Resonator, 489k curved elements, largest to smallest element: 5:1
26
SLIDE 88
The Maxwell equations
Figure 6: Hy at t=260, 260 time slabs, 148k tents per slab, p2 local Taylor time-steps
Shared memory server, 4 E7-8867 CPUs with 16 cores each.
27
SLIDE 89
The Maxwell equations
Figure 6: Hy at t=260, 260 time slabs, 148k tents per slab, p2 local Taylor time-steps
Shared memory server, 4 E7-8867 CPUs with 16 cores each. p=2: 29 374 980 dofs, 20 min
27
SLIDE 90
The Maxwell equations
Figure 6: Hy at t=260, 260 time slabs, 148k tents per slab, p2 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
SLIDE 91
The Maxwell equations
Figure 7: Resonator with sharp edges, 224k curved elements, largest to smallest element: 10:1
28
SLIDE 92
The Maxwell equations
Figure 8: Hy at t=260, 260 time slabs, 66k tents per slab, p2 local Taylor time-steps
Shared memory server, 4 E7-8867 CPUs with 16 cores each.
29
SLIDE 93
The Maxwell equations
Figure 8: Hy at t=260, 260 time slabs, 66k tents per slab, p2 local Taylor time-steps
Shared memory server, 4 E7-8867 CPUs with 16 cores each. p=2: 13 452 000 dofs, 8 min
29
SLIDE 94
The Maxwell equations
Figure 8: Hy at t=260, 260 time slabs, 66k tents per slab, p2 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
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SLIDE 95
Euler equations
Find (ρ, m, E) : Ω × (0, T] → R × RN × R s.t. ∂t ρ m E + div m
1 ρm ⊗ m + pI m ρ (E + p)
= 0
30
SLIDE 96
Entropy admissibility condition
Entropy admissibility condition E(u) ∈ R . . . entropy, F(u) ∈ RN . . . entropy flux
31
SLIDE 97
Entropy admissibility condition
Entropy admissibility condition E(u) ∈ R . . . entropy, F(u) ∈ RN . . . entropy flux ⇒ ∂tE(u) + div F(u) ≤ 0
31
SLIDE 98
Entropy admissibility condition
Entropy admissibility condition E(u) ∈ R . . . entropy, F(u) ∈ RN . . . entropy flux ⇒ ∂tE(u) + div F(u) ≤ 0 The pair (E, F) is called the entropy pair.
31
SLIDE 99
Entropy admissibility condition
Entropy admissibility condition E(u) ∈ R . . . entropy, F(u) ∈ RN . . . entropy flux ⇒ ∂tE(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) = δF(w).
31
SLIDE 100
Entropy admissibility condition
Entropy admissibility condition E(u) ∈ R . . . entropy, F(u) ∈ RN . . . entropy flux ⇒ ∂tE(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) = δF(w). Mapped entropy admissibility condition ∂ˆ
t ˆ
E(ˆ u) + div ˆ F(ˆ u) = δ(∂tE(u) + div F(u)) ◦ Φ ≤ 0
31
SLIDE 101
Entropy admissibility condition
Entropy admissibility condition E(u) ∈ R . . . entropy, F(u) ∈ RN . . . entropy flux ⇒ ∂tE(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) = δF(w). Mapped entropy admissibility condition ∂ˆ
t ˆ
E(ˆ u) + div ˆ F(ˆ u) = δ(∂tE(u) + div F(u)) ◦ Φ ≤ 0 = ⇒ entropy viscosity regularization on space-time cylinder
31
SLIDE 102 Entropy viscosity regularization
Problem description Find ˆ U : ˆ Ki → Rn such that ∂ˆ
t ˆ
U + divx
U))
32
SLIDE 103 Entropy viscosity regularization
Problem description Find ˆ U : ˆ Ki → Rn such that ∂ˆ
t ˆ
U + divx
U))
Recall that ˆ U ≡ G(ˆ u) := ˆ u − f (ˆ u) ∇ϕ is discontinuous.
32
SLIDE 104 Entropy viscosity regularization
Problem description Find ˆ U : ˆ Ki → Rn such that ∂ˆ
t ˆ
U + divx
U))
Recall that ˆ U ≡ G(ˆ u) := ˆ u − f (ˆ u) ∇ϕ is discontinuous. = ⇒ artificial viscosity on ˆ u = G −1( ˆ U)
32
SLIDE 105 Entropy viscosity regularization
Problem description Find ˆ U : ˆ Ki → Rn such that ∂ˆ
t ˆ
U + divx
U))
ki divx
U)
U ≡ G(ˆ u) := ˆ u − f (ˆ u) ∇ϕ is discontinuous. = ⇒ artificial viscosity on ˆ u = G −1( ˆ U) νi . . . entropy viscosity coefficient ki = δ(v (i)) . . . tent height
32
SLIDE 106
Euler equations
Find (ρ, m, E) : Ω × (0, T] → R × RN × R s.t. ∂t ρ m E + div m
1 ρm ⊗ m + pI m ρ (E + p)
= 0
33
SLIDE 107
Euler equations
Find (ρ, m, E) : Ω × (0, T] → R × RN × R s.t. ∂t ρ m E + div m
1 ρm ⊗ m + pI m ρ (E + p)
= 0 ρ = 1.4 , m = ρ(3, 0)T , p = 1
33
SLIDE 108 Euler equations
Find (ρ, m, E) : Ω × (0, T] → R × RN × R s.t. ∂t ρ m E + div m
1 ρm ⊗ m + pI m ρ (E + p)
= 0 ρ = 1.4 , m = ρ(3, 0)T , p = 1
0.6 3
x1
0.2 1
x2
inflow
reflecting reflecting 33
SLIDE 109
Euler equations
Figure 9: Tent pitched time slab
34
SLIDE 110
Euler equations
Figure 9: Tent pitched time slab
34
SLIDE 111 Euler equations
Figure 10: Solution of Mach 3 wind tunnel at t = 4, P4 discontinuous finite elements
- n 3951 triangles, 59 265 dofs
Implementation based on NGSolve, NGS-Py
35
