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

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 ˇ 1 Seminar for Applied Mathematics, ETH-Zurich Switzerland 2 Applied and Computational Mathematics Princeton University NJ, USA Eleventh International Conference on Hyperbolic Problems Theory, Numerics, Applications ENS Lyon, France, July 17-21, 2006 M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 1/ 26

Review Compact Third Order Reconstruction Numerical Experiment Some Definitions Standard finite volume update: d dt ¯ L i (ˆ u i = u i ) 1 u ( − ) u (+) u ( − ) u (+) � F (ˆ 2 , ˆ 2 ) − F (ˆ 2 , ˆ � = 2 ) i − 1 i − 1 i + 1 i + 1 ∆ x Left and right limits: u ( ± ) ˆ ) ± ˆ 2 = lim u ( x ) i + 1 x → ( x i + 1 2 Slopes and its ratio ("local smoothness measurement"): δ i − 1 2 = ¯ u i + 1 − ¯ 2 δ i + 1 u i and θ i = δ i + 1 2 M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 2/ 26

Review Compact Third Order Reconstruction Numerical Experiment Semi-discrete Method Seperation of time- and space-discretization using Godunov’s flux and Heun: u i + ∆ x u i + 1 u ( − ) ˆ ¯ 2 σ i = ¯ = 2 φ ( θ i ) δ i + 1 i + 1 2 2 u i − ∆ x u i − 1 u (+) ˆ ¯ 2 σ i = ¯ = 2 φ ( θ i ) δ i + 1 i − 1 2 2 Slope-limiter in terms of flux limiting: δ i + 1 σ i = ( ∆ x ) φ ( θ i ) 2 M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 3/ 26

Review Compact Third Order Reconstruction Numerical Experiment Standard Second Order TVD Limiters classical 2. order TVD limiter 2 1.8 1.6 1.4 1.2 φ ( θ ) 1 0.8 0.6 0.4 Minmod MC 0.2 van − Leer Superbee 0 0 1 2 3 4 5 6 θ M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 4/ 26

Review Compact Third Order Reconstruction Numerical Experiment Standard Second Order TVD Limiters Accuracy Test: advection profile with classical 2. order TVD limiter 1.2 exact Minmod Superbee 1 MC van Leer 0.8 0.6 u 0.4 0.2 0 − 0.2 − 1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1 x M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 5/ 26

Review Compact Third Order Reconstruction Numerical Experiment Extension to the classical 2 nd 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 (∆ x 3 ) ) Keep it simple and efficient M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 6/ 26

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 (∆ x 3 ) Handles discontinuity as well as local extrema r 0 ( x ) ≈ A + B log ( x + C ) + D log ( x + D ) M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 7/ 26

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: u ( − ) u i + Φ ( − ) (¯ ˆ ¯ = u i , δ i − 1 2 , δ i + 1 2 ) i + 1 2 u (+) u i − Φ (+) (¯ ˆ ¯ = 2 ) u i , δ i − 1 2 , δ i + 1 i − 1 2 M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 8/ 26

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: u ( − ) u i + Φ ( − ) (¯ ˆ ¯ = u i , δ i − 1 2 , δ i + 1 2 ) i + 1 2 u (+) u i − Φ (+) (¯ ˆ ¯ = 2 ) u i , δ i − 1 2 , δ i + 1 i − 1 2 M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 8/ 26

Review Compact Third Order Reconstruction Numerical Experiment Logarithmic Limiter We can write the reconstructed values: u i + 1 u i − 1 u ( − ) u (+) 2 φ ( θ − 1 ˆ 2 = ¯ ˆ = ¯ 2 φ ( θ i ) δ i + 1 and ) δ i − 1 i + 1 i − 1 i 2 2 2 with the limiter: φ ( θ ) = 2 p (( p 2 − 2 p θ + 1 ) log ( p ) − ( 1 − θ )( p 2 − 1 )) ( p 2 − 1 )( p − 1 ) 2 with | θ | q p = p ( θ ) = 2 1 + | θ | 2 q . M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 9/ 26

Review Compact Third Order Reconstruction Numerical Experiment Logarithmic Limiter Logarithmic Limiter 3.5 φ ( θ ,1.0) 3 φ ( θ ,1.4) 2.5 φ ( θ ,2.0) (2+ θ )/3 2 1.5 φ ( θ ) 1 0.5 0 − 0.5 − 1 − 1.5 − 6 − 4 − 2 0 2 4 6 8 θ M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 10/ 26

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 ( ξ 4 ) and φ ( − 1 + ξ ) = 1 3 + ξ 3 + O ( ξ 4 ) lim θ → 0 φ ( θ ) → 0 and lim θ →±∞ φ ( θ ) → 0 M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 11/ 26

Review Compact Third Order Reconstruction Numerical Experiment Simplified Limiter Simplified Limiter 2 " ( ! ,1.4) limiter 1.5 1 " ( ! ) 0.5 0 ! 0.5 ! 1 ! 6 ! 4 ! 2 0 2 4 6 8 ! � 2 + θ � �� 2 | θ | 3 0 . 6 + | θ | , 5 φ ( θ ) = max −| θ | , min 3 , | θ | M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 12/ 26

Review Compact Third Order Reconstruction Numerical Experiment Time Integration Explicit 3. order Runge-Kutta-method [Gottlieb and Shu 98]: u ( 1 ) u n u n ) ¯ ¯ i + ∆ tL i (¯ = i 3 i + 1 u ( 2 ) u ( 1 ) u n u ( 1 ) )) ¯ 4 ¯ 4 (¯ + ∆ tL i (¯ = i i 1 i + 2 u ( 2 ) u n + 1 u n u ( 2 ) )) ¯ 3 ¯ 3 (¯ + ∆ tL i (¯ = i i M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 13/ 26

Review Compact Third Order Reconstruction Numerical Experiment Stability Region of 3. Order Runge-Kutta Stability region: CFL ≤ 1.5 M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 14/ 26

Review Linear Advection Equation Compact Third Order Reconstruction Non-linear Convervation Laws Numerical Experiment Convergence Test u t + u x = 0 , smooth and periodic Error Order (slope) 0 10 4 3 ! 5 L 1 10 L 1 2 1 ! 10 10 0 1 2 3 4 10 10 10 10 0 500 1000 1500 2000 2500 3000 N N Error Order (slope) 0 10 3 2 ! 5 L ! 10 L ! 1 ! 10 10 1 2 3 4 0 10 10 10 10 0 500 1000 1500 2000 2500 3000 N N M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 15/ 26

Review Linear Advection Equation Compact Third Order Reconstruction Non-linear Convervation Laws Numerical Experiment Accuracy Test 1.2 exact LIM3 ENO3 1 LHHR LDLR 0.8 0.6 u 0.4 0.2 0 − 0.2 − 1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1 x M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 16/ 26

Review Linear Advection Equation Compact Third Order Reconstruction Non-linear Convervation Laws Numerical Experiment Accuracy Test 0.62 1 0.82 0.6 1 0.8 0.58 0.78 0.98 0.76 0.95 0.56 0.74 0.96 0.54 u 0.72 0.52 0.7 0.94 0.9 0.68 0.5 0.92 0.66 0.48 0.64 0.85 0.46 0.9 − 0.75 − 0.7 − 0.65 − 0.35 − 0.3 − 0.25 0.05 0.1 0.15 0.45 0.5 0.55 x M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 17/ 26

Review Linear Advection Equation Compact Third Order Reconstruction Non-linear Convervation Laws Numerical Experiment Euler system in one dimension Conservation of mass ∂ t ρ + ∂ x ( ρ v ) = 0 Conservation of momentum ∂ t ( ρ v ) + ∂ x ( ρ v 2 + p ) = 0 Conservation of energy ∂ t E + ∂ x v ( E + p ) = 0 M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 18/ 26

Review Linear Advection Equation Compact Third Order Reconstruction Non-linear Convervation Laws Numerical Experiment Shu-Osher Shock Acoustic Problem 5 4.5 4 3.5 3 ρ 2.5 2 1.5 1 0.5 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 x M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 19/ 26

Review Linear Advection Equation Compact Third Order Reconstruction Non-linear Convervation Laws Numerical Experiment Shu-Osher Shock Acoustic Problem exact 4.6 LIM3 LHHR 4.4 LDLR 4.2 4 ρ 3.8 3.6 3.4 3.2 3 2.8 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 x M. ˇ Cada, M. Torrilhon, R. Jeltsch Compact Third Order Logarithmic Limiting 20/ 26