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Compact Third Order Logarithmic Limiting for Non-Linear Hyperbolic - - PowerPoint PPT Presentation

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Review Compact Third Order Reconstruction Numerical Experiment Compact Third Order Logarithmic Limiting for Non-Linear Hyperbolic Conservation Laws Cada 1 , Manuel Torrilhon 2 and Rolf Jeltsch 1 Miroslav 1 Seminar for Applied Mathematics,

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SLIDE 1

Review Compact Third Order Reconstruction Numerical Experiment

Compact Third Order Logarithmic Limiting for Non-Linear Hyperbolic Conservation Laws

Miroslav ˇ Cada1, Manuel Torrilhon2 and Rolf Jeltsch1

1Seminar for Applied Mathematics,

ETH-Zurich Switzerland

2Applied and Computational Mathematics

Princeton University NJ, USA

Eleventh International Conference on Hyperbolic Problems Theory, Numerics, Applications

ENS Lyon, France, July 17-21, 2006

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SLIDE 2

Review Compact Third Order Reconstruction Numerical Experiment

Some Definitions

Standard finite volume update: d dt ¯ ui = Li(ˆ ui) = 1 ∆x

  • F(ˆ

u(−)

i− 1

2 , ˆ

u(+)

i− 1

2 ) − F(ˆ

u(−)

i+ 1

2 , ˆ

u(+)

i+ 1

2 )

  • Left and right limits:

ˆ u(±)

i+ 1

2 =

lim

x→(xi+ 1

2

)± ˆ

u(x) Slopes and its ratio ("local smoothness measurement"): δi+ 1

2 = ¯

ui+1 − ¯ ui and θi = δi− 1

2

δi+ 1

2

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Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 2/ 26

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SLIDE 3

Review Compact Third Order Reconstruction Numerical Experiment

Semi-discrete Method

Seperation of time- and space-discretization using Godunov’s flux and Heun: ˆ u(−)

i+ 1

2

= ¯ ui + ∆x 2 σi = ¯ ui + 1 2φ(θi)δi+ 1

2

ˆ u(+)

i− 1

2

= ¯ ui − ∆x 2 σi = ¯ ui − 1 2φ(θi)δi+ 1

2

Slope-limiter in terms of flux limiting: σi = ( δi+ 1

2

∆x )φ(θi)

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SLIDE 4

Review Compact Third Order Reconstruction Numerical Experiment

Standard Second Order TVD Limiters

1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 θ φ(θ) classical 2. order TVD limiter Minmod MC van−Leer Superbee

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SLIDE 5

Review Compact Third Order Reconstruction Numerical Experiment

Standard Second Order TVD Limiters

Accuracy Test:

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −0.2 0.2 0.4 0.6 0.8 1 1.2 x u advection profile with classical 2. order TVD limiter exact Minmod Superbee MC van Leer

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Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 5/ 26

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SLIDE 6

Review Compact Third Order Reconstruction Numerical Experiment

Extension to the classical 2nd Order TVD Limiters

Aim: Capture smooth Extrema better (φ(θ) = 0 for θ < 0) Use the compact stencil (5 point) more efficiently (piecewise quadratic reconstruction ⇒ O(∆x3)) Keep it simple and efficient

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SLIDE 7

Review Compact Third Order Reconstruction Numerical Experiment

Local Double Logarithmic Reconstruction (LDLR)

Based on the idea of non-linear and non-polynomial, i.e. hyperbolic [Marquina 94] and logaritmic reconstruction [Artebrant and Schroll 05]. Essentially O(∆x3) Handles discontinuity as well as local extrema r0(x) ≈ A + B log(x + C) + D log(x + D)

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Review Compact Third Order Reconstruction Numerical Experiment

LDLR

Some questions: Can this method be formulated in “standard” limitation context? What will the limiter function look like? Can we avoid the evaluation of the log-function? Some answers: We can expect the final representation: ˆ u(−)

i+ 1

2

= ¯ ui + Φ(−)(¯ ui, δi− 1

2 , δi+ 1 2 )

ˆ u(+)

i− 1

2

= ¯ ui − Φ(+)(¯ ui, δi− 1

2 , δi+ 1 2 )

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SLIDE 9

Review Compact Third Order Reconstruction Numerical Experiment

LDLR

Some questions: Can this method be formulated in “standard” limitation context? What will the limiter function look like? Can we avoid the evaluation of the log-function? Some answers: We can expect the final representation: ˆ u(−)

i+ 1

2

= ¯ ui + Φ(−)(¯ ui, δi− 1

2 , δi+ 1 2 )

ˆ u(+)

i− 1

2

= ¯ ui − Φ(+)(¯ ui, δi− 1

2 , δi+ 1 2 )

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Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 8/ 26

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SLIDE 10

Review Compact Third Order Reconstruction Numerical Experiment

Logarithmic Limiter

We can write the reconstructed values: ˆ u(−)

i+ 1

2 = ¯

ui + 1 2φ(θi)δi+ 1

2

and ˆ u(+)

i− 1

2

= ¯ ui − 1 2φ(θ−1

i

)δi− 1

2

with the limiter: φ(θ) = 2p((p2 − 2pθ + 1) log(p) − (1 − θ)(p2 − 1)) (p2 − 1)(p − 1)2 with p = p(θ) = 2 |θ|q 1 + |θ|2q .

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SLIDE 11

Review Compact Third Order Reconstruction Numerical Experiment

Logarithmic Limiter

−6 −4 −2 2 4 6 8 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 θ φ(θ) Logarithmic Limiter φ(θ,1.0) φ(θ,1.4) φ(θ,2.0) (2+θ)/3

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SLIDE 12

Review Compact Third Order Reconstruction Numerical Experiment

Logarithmic Limiter

Full third order reconstruction: φ(θ) = 2+θ

3 , results in a standard third order interpolation

Essential characteristics of the log-limiter: φ(1 + ξ) = 1 + ξ

3 + O(ξ4) and φ(−1 + ξ) = 1 3 + ξ 3 + O(ξ4)

limθ→0 φ(θ) → 0 and limθ→±∞ φ(θ) → 0

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SLIDE 13

Review Compact Third Order Reconstruction Numerical Experiment

Simplified Limiter

!6 !4 !2 2 4 6 8 !1 !0.5 0.5 1 1.5 2 ! "(!) Simplified Limiter "(!,1.4) limiter

φ(θ) = max

  • 3

−|θ|, min

2+θ

3 , 2|θ| 0.6+|θ|, 5 |θ|

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SLIDE 14

Review Compact Third Order Reconstruction Numerical Experiment

Time Integration

Explicit 3. order Runge-Kutta-method [Gottlieb and Shu 98]: ¯ u(1)

i

= ¯ un

i + ∆tLi(¯

un) ¯ u(2)

i

= 3 4 ¯ un

i + 1

4(¯ u(1)

i

+ ∆tLi(¯ u(1))) ¯ un+1

i

= 1 3 ¯ un

i + 2

3(¯ u(2)

i

+ ∆tLi(¯ u(2)))

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Review Compact Third Order Reconstruction Numerical Experiment

Stability Region of 3. Order Runge-Kutta

Stability region: CFL ≤ 1.5

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SLIDE 16

Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Convergence Test

ut + ux = 0, smooth and periodic

10

1

10

2

10

3

10

4

10

!10

10

!5

10 Error N L1 10

1

10

2

10

3

10

4

10

!10

10

!5

10 Error N L! 500 1000 1500 2000 2500 3000 1 2 3 4 Order (slope) N L1 500 1000 1500 2000 2500 3000 1 2 3 Order (slope) N L!

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SLIDE 17

Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Accuracy Test

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −0.2 0.2 0.4 0.6 0.8 1 1.2 x u exact LIM3 ENO3 LHHR LDLR

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Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Accuracy Test

−0.75 −0.7 −0.65 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 x u −0.35−0.3−0.25 0.9 0.92 0.94 0.96 0.98 1 0.05 0.1 0.15 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.45 0.5 0.55 0.85 0.9 0.95 1

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SLIDE 19

Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Euler system in one dimension

Conservation of mass ∂tρ + ∂x(ρv) = 0 Conservation of momentum ∂t(ρv) + ∂x(ρv2 + p) = 0 Conservation of energy ∂tE + ∂xv(E + p) = 0

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Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Shu-Osher Shock Acoustic Problem

−5 −4 −3 −2 −1 1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x ρ

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Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Shu-Osher Shock Acoustic Problem

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 x ρ exact LIM3 LHHR LDLR

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Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Shock-blast Interaction

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 x ! LDLR 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 LHHR 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7

  • simpl. Limiter

ref calcul.

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Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Shock-blast Interaction

0.72 0.74 0.76 0.78 0.8 0.82 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5 x ! LDLR 0.72 0.74 0.76 0.78 0.8 0.82 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5 6.6 LHHR 0.74 0.76 0.78 0.8 0.82 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5 6.6

  • simpl. Limiter

ref calcul.

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SLIDE 24

Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Shock-blast Interaction

0.75 0.8 0.85 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x ! LDLR 0.8 0.85 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 LHHR 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

  • simpl. Limiter

ref calcul.

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Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Sod-Problem

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x ! Sod problem with different CFL!num. ref. CFL=0.8 CFL=1.5

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Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Sod-Problem

0.5 0.55 0.6 0.65 0.7 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 x ! Sod problem with different CFL!num. ref. CFL=0.8 CFL=1.5

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SLIDE 27

Review Compact Third Order Reconstruction Numerical Experiment Linear Advection Equation Non-linear Convervation Laws

Sod-Problem

0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 x ! Sod problem with different CFL!num. ref. CFL=0.8 CFL=1.5

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