The shifted box scheme for scalar transport problems Bertil - PowerPoint PPT Presentation

The shifted box scheme for scalar transport problems Bertil Gustafsson Uppsala University and Stanford University Joint work with: Yaser Khalighi, Stanford

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☞ ✒ ✟ ✡ ✟ ✌ ✎ ✝ ✒ ✒ ✏ ✒ ✍ ✎ ✝ ✡ ☞ � ✁ ✂ ☎ � ✞ ✂ ✟ � � ✌ ✎ ✂ ✁ ✂ � ✝ ✞ ✂ � ✝ ☞ ✌ ☎ ✞ ✟ ☞ ✌ ☎ ✡ ✡ ☎ ✠ Scalar transport equation, incompressible flow ✠☛✡ ✄✆☎ ✄✆✟ Velocity components ✄✆✍ ✌✑✏ ✌✑✏ ✄✆✍ Energy conservation: �✔✓ Alternative formulation: ✠☛✡

✕ ✕ ✕ ✕ Requirements on numerical method: 1. Very fast algorithm 2. No stability limit on the time step 3. Energy conservation (no loss of energy) 4. No unphysical oscillations Standard centered finite difference or finite volume method, trapezoidal rule in time: (2), (3) Implicit upwind scheme: (2), (4) Method of characteristics: (1), (2), (3), (4) Method of characteristics on a grid (interpolation): (1), (2), (4) The box scheme?

✌ ✡ ✤ ✏ ✒ ✒ ☞ ☎ ☎ � ✚ ✡ ✝ ☞ ✌ ✍ ✄ ✤ ✤ ✌ � ✓ ✤ ✝ ✏ ✌ ✖ ✄ ✤ ✤ ✝ ✖ ✄ ☎ ✥ ✡ ✓ � � ✝ ✞ ✌ ☞ ✡ ✝ ✞ � ☎ ✂ ✝ ✘ � ☎ ✄ ✄ ✗ ✖ ✂ ✁ ☞ ✍ ✏ ✝ ✖ ✌ ☞ ✄ ✙ ✘ ✏ ✌ ✘ ✡ � ✄ ☞ Model problem 1-D ✄✆✍ ✝✣✢ ✌✑✏ ✄✆✍ ✝✜✛

The box scheme t x

✦ ✦ ☎ ✄ ✪ ✄ ✮ ✭ ✥ ✧ ✬ � ✡ ✧ ✝ ✧ ✩ ★ ✪ ✝ ✧ ✩ ★ ✬ ✬ ✧ ✦ ★ ✚ ✰ � ✩ ★ ✬ ✙ ✡ ✰ ✩ ✝ ✝ ✧ ✬ � ✪ ✝ ✧ ✬ ☎ ✦ ✄ ✂ � ☎ ✄ ✝ ★ ✧ � ✄ ✫ ✖ ✡ ✧ � ✪ ✌ ✧ ✂ � ✂ ✩ ★ ✧ � ✄ ✗ ✖ ✡ ✧ ✩ ✧ ✥ ✪ ✝ ✧ ✩ ★ ✧ ✩ ★ ✬ ☎ � ✄ ✄ ✮ ✭ ✂ ✧ ✩ ★ ✦ ✧ �✔✬ �✔✬ ✧✱✡ �✔✬ �✯✬ ✝✜✛ �✔✬

✂ ✡ ✵ ✵ ✓ ✄ ✥ ✖ ✝ ✧ ☎ ✵ ✧ ★ ✩ ✥ ☎ ✧ ✡ ☞ ✴ ✩ ✡ ✧ ✧ ✒ ☎ ✒ ✍ ✡ ☞ ✡ ☎ ★ ✧ ✩ ✥ ☎ ✩ ✩ ✡ ☞ ✴ ☎ ☎ Box scheme vs standard Crank-Nicholson: Parasitic solution? Crank-Nicholson ✧✳✲ Box scheme const

✗ ✾ ✖ ✡ ✾ ✗ ✛ ✖ ✡ ☞ ✗ ✛ ✖ ✡ ✾ ✿ ✛ ✾ ✛ ✾ ☞ ✛ ☎ ✄ ✍ ✌ ✫ ✝ ✡ ✺ ✡ ✞ ★ ✻ ✼ ✖ ✽ ✫ ✡ Accuracy? One step: . Approximation of ? ✶✸✷✹ 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 Box , Exact and Box , Box , C-N(4,2) , C-N(2,2)

✘ ✖ ✫ ✡ ❀ ❁ ✗ ✽ ❂ ☞ ✛ ❂ ✘ ✾ ✛ Approximation at (4 points per wavelength) Phase speed for 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Box, C-N(4,2), C-N(2,2)

❆ ❆ ✝ ✗ ❁ ❆ ✄ ❅ ❃❄ ✝ ✗ ❁ ✄ ✾ ❉❊ ❅ ☎❈ ✾ ✥ ✝ ✗ ❁ ☎ ✄ ❅ ✂ ☎❈ ✡ ✬ ✡ ❍ ☎ ✤ ☎ ❇ ✤ ✡ ✤ ✩ ★ � ❅ ❇ ✤ ● ✡ ❇ ✝ ✗ ❁ ❆ ✄ ❉❊ ❃❄ ✩ ☞ ✄ ✂ ✝ ✡ ❇ ✥ ✩ ★ ✬ � ❇ ✗ ★ ❆ ❃❄❅ ☞ ✡ ✞ � ☎ ✂ ✁ � ❀ ✾ ☎❈ ❅ ❉❊ ☞ ❇ ✫ ✽ ✡ ❆ ✌ ✫ ✭ ✡ ✾ ☞ ✡ ✝ ✌ ❇ ✂ ✩ ★ ❆ ❇ ✄ ✗ ❆ Note : problem with Unconditional stability if Fourier transform Constant coefficient �❋✬ �✯✬ , periodic solutions . �❋✬ ! �✯✬ �❋✬ �✯✬

� ✝ ✡ ✝ ☞ ✌ ✍ � ✌ ✏ ✄ ✄ ✙ ✡ ✝ ✝ ☞ ✄ ✚ Initial-boundary value problem: ✌✑✏ ✄✆✍ Solve from left to right t x

☎ ✻ ✻ ✤ ✤ ✚ ✤ ✤ ✩ ❑ ❁ ✓ ❑ ✘ ✤ ✩ ✤ ✤ ✤ ✓ ✤ ❏ ✩ ★ ✤ ✂ ✓ ✡ ❑ ✡ ✬ ◆ ❁ ✭ ☎ ✡ ✾ ✛ ✝ ✖ ✌ ✓ ✾ ✄ ▼ ▲ ▲ ✡ ✓ ❑ ✌ ✝ ✖ ✌ ✓ ✾ ✄ ❊ ❉ ❏ ✤ � ☎ ✡ ✻ ✓ ✤ ✤ ✩ ★ ✬ � ✪ ✗ ✭ ✤ ✤ ✤ ✂ ✻ ✓ ✤ ✤ ✩ ★ ✦ ✤ ✤ ☞ ✢ ✤ ✤ ✦ � ✄ ✗ ✭ ☎ ✥ ✻ ✓ ✤ ✤ � ✬ ✫ ✻ ✓ ✗ ✭ ✬ ✤ ☎ ✤ ✤ ✂ ✪ ✤ Stability estimate Energy conservation: �■✬ const �❋✬ �■✬ ✝✜✛ (1)

☎ ✂ ✧ ✬ � ✄ ❙ ✂ ✧ ✩ ★ ☎ ✾ ✖ ✌ ☎ ✾ ✥ ✖ ✥ ✡ ✩ ★ ✧ ✩ ★ ✬ ✝ ❚ ● ✛ ✫ ❁ ✭ ✡ ✾ ✝ ✫ ✄ ✩ ❚ ✌ ✛ ✡ ✛ ✌ ✖ ✂ ✰ ❚ ✌ ✝ ✫ ✄ ✰ ❚ � ☞ ✄ ★ ✂ ✩ ★ ✧ ✬ � ✥ ✩ ★ ✧ ✩ ✬ ✡ � ◗ ✍ ✘ ✍ ✘ P ✍ ✌ ☞ ✍ ★ ✩ ✧ ★ ✝ ✧ ✥ ✧ ✂ ✧ ✩ ★ ✥ ✥ ✩ ✧ ✄ ✧ ✂ ✩ ☎ ✾ ★ Unstable! ✝✣❖ �❘✬ �❘✬ �✯✬ ? �❘✬ �❘✬ �✯✬ �✯✬

✄ ✡ ✌ � ✫ ✍ ❲ ✬ ★ ✩ ✝ ✡ ❯ ✂ ☞ ❑ ✭ ✌ ✏ ✬ ★ ✩ ✭ ❑ ❑ ✭ ✡ ✛ ✡ ✌ ☎ ❑ ❲ ❯ ✁ ✂ ✖ ✗ ✄ ✄ ❑ ✂ ✝ ❱ ❯ ✝ ❱ ✂ ✖ ✗ ✄ ❑ ✂ ☎ ✝ ❯ ✫ The shifted box scheme Add constant to the velocity: True solution obtained by a shift ✌✑✏ ✄✆✍ ✝✜✛ Choose (or more generally integer)

The shifted box scheme t x

❭ ❳ ❩ ❨ ❪ ❭ 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 −0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a) Velocity (b) Initial function ❨❬❩

Characteristics x 0.4 0.6

✖ � ✡ ✾ ✌ ☞ ☞ ✿ ✡ ❫ ✝ ✏ ✌ Original box scheme unstable 2.5 1.6 1.4 2 1.2 1.5 1 0.8 1 0.6 0.5 0.4 0.2 0 0 −0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (c) Original box scheme (d) Shifted box scheme after 18 steps, ✄✆✍

☞ ✄ ✌ ✾ ❑ ✌ ✝ ✍ � ✡ ✡ ✖ ✌ ❫ ✡ ✿ ☞ ✖ 1.6 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (e) t=2 (f) t=3 ✌✑✏

☞ ✌ ☞ ✛ ❂ ✌ ✌ ✝ ✏ � ❑ ✾ ✡ ✗ ✌ ❫ ✡ ✿ ☞ ✡ 1.6 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (g) t=2 (h) t=3 ✄✆✍

✗ ❫ ✻ ✌ ☞ ✘ ✏ ✌ ✡ ✤ ✿ ☞ ☞ ✌ ✾ ✡ ✤ ✝ ✄ � ✤ ✤ ✏ 0.01 0.008 0.006 0.004 0.002 0 0 0.5 1 1.5 2 2.5 3 ✘❵❴

−3 10 −4 10 Error −5 10 Periodic BC Inlet BC −6 10 −4 −3 −2 −1 10 10 10 10 Grid Spacing Second order accuracy

2 space dimensions y u > 0 x v > 0

� ☞ ✌ ✓ ✙ ✡ ✝ ✏ ✌ ✌ ✝ ✍ ✄ � ✌ ✖ ✘ ✎ ✘ ✏ ✌ ✁ ☞ ✝ ✎ ✌ ✍ ✄ ✚ ✡ ✝ ✌ ☞ ✎ ✌ � ✌ ✖ ✘ ✍ ✘ ☞ ✌ ✝ � ✌ ✍ ✘ ☞ ✌ ☞ ✝ ✟ ✖ ✄ ✂ ✞ ✝ � ☎ ✄ ✂ ✎ ❖ ✌ ✌ ✎ ✄ ✩ ✙ ✡ ✝ ✏ ✌ ✎ ☞ ✄ � ✌ ✏ ✘ ☞ ✛ ✄✆✍ ✠☛✡ ✄✆✍ ✌✑✏

✬ ❝ ✂ ✦ � ✝ ✬ ✷ ❞❡ ✂ ✦ ✞ ✪ ✠ ✧ ✷ � ✝ ✬ ★ ✩ ✷ ✧ ✂ ✄ ✟ � ✝ ✧ ✩ ✷ ✧ ✞ ✦ ✠ ✄ ✌ ★ ✩ ✷ ✧ ✥ ☞ ✷ ✝ ★ ✂ ✭ ✡ ✦ ✠ ✪ ✞ ❝ ✄ ☎ � ✝ ✬ The box scheme t �❘✬ y �✯✬ ✗❜❛ x ✄✆✟ ✄✆☎ ✧❵❞

☎ ★ ✂ � ✬ ★ ✩ ✐ ❥ ✧ ✩ ✐ ✝ ✓ ❞ ✫ ✠ ✥ ✾ ✠ ✧ ✩ ❢ ✬ ❝ ✄ � ✬ ✐ ✧ ✂ � ✐ ★ ❥ ✧ ★ ✩ ✝ ✓ ✂ ✄ ✖ ✐ ✢ ✷ ✩ ✷ ❏ ✂ � ✬ ★ ✩ ★ ✬ ✩ ❥ ❏ ✝ ✓ ❞ ✫ ✞ ★ � ✲ ✂ ✩ ✷ ❣ ✰ ❝ ✄ ✷ ❏ � ✄ ✬ ✷ ★ ✩ ❥ ❏ ✝ ✓ ✂ ✰ ✛ ✩ ✞ ✦ ✠ ✪ ✞ ✂ ✟ ✦ ✪ ✭ ✠ ✝ � ✬ ★ ✩ ✤ ✤ ✗ ✤ ✻ ✞ ☞ ✌ ✟ ✢ ☞ ✤ ✤ ✦ ✦ ✤ ✠ ★ ✩ ✤ ✤ ✓ ✻ ✂ ✲ ✓ ✡ ✤ ✟ ✦ ✞ ✪ ✠ ✝ � ✬ ✤ ✤ ✓ ✻ ✥ ✾ ✞ ✖ ❢ ❏ ✂ ✞ ✪ ✓ ✤ ✦ ✞ ✦ ✠ ✤ ✤ ✻ ✂ ✤ ✤ ✭ ✗ ✦ ✠ Unconditional stability: ✧❤❣ �✯✬ �✯✬ �✯✬ ✄✆☎ ✄✆☎ �❘✬