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for Flux Reconstruction Will Trojak Scope Why Shock Capturing? - - PowerPoint PPT Presentation

Oct 02, 2023 583 likes β€’780 views

Shock Capturing Methods for Flux Reconstruction Will Trojak Scope Why Shock Capturing? Invariance Preserving Methodology Preliminary Results Summary and Future Developments Why Shock Capturing? More Physics Currently

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SLIDE 1

Shock Capturing Methods for Flux Reconstruction

Will Trojak

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SLIDE 2

Scope

  • Why Shock Capturing?
  • Invariance Preserving Methodology
  • Preliminary Results
  • Summary and Future Developments
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SLIDE 3

Why Shock Capturing?

  • More Physics
  • Currently parametric
  • P-adaptive methods present

several issues

π‘ž = 0, DoF = 20𝑙

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SLIDE 4

Current Shock Capturing

Method Parametric

  • Op. Cheap

β€œStable” AV/Per-Olof βœ“ βœ“ βœ—? Filtering βœ“ βœ“ βœ“? Adaptation/ Moving mesh βœ—? βœ— βœ“?

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SLIDE 5

Methodology

𝑗 ℐ(𝑗)

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SLIDE 6

Methodology

πœ–π‘’π’—π‘— = βˆ’ ෍

π‘˜βˆˆβ„(𝑗)

π•˜π‘˜ βˆ™ π’…π‘—π‘˜ So... + ෍

π‘˜βˆˆβ„(𝑗)

π‘’π‘—π‘˜(π’—π‘˜ βˆ’ 𝒗𝑗) Now… ෍

π‘˜βˆˆβ„(𝑗)

π’…π‘—π‘˜ = 0 = ෍

π‘˜βˆˆβ„(𝑗)

π‘’π‘—π‘˜

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SLIDE 7

Methodology

πœ–π‘’π’—π‘— = βˆ’ ෍

π‘˜βˆˆβ„(𝑗)

(π•˜π‘˜+π•˜π‘—) βˆ™ π’…π‘—π‘˜ βˆ’ π‘’π‘—π‘˜(π’—π‘˜ βˆ’ 𝒗𝑗)

π‘’π‘—π‘˜ = max(πœ‡π‘›π‘π‘¦ π’π‘—π‘˜, 𝒗𝑗, π’—π‘˜ π’…π‘—π‘˜ , πœ‡π‘›π‘π‘¦(π’π‘˜π‘—, π’—π‘˜, 𝒗𝑗)|π’…π‘˜π‘—|)

Which we solve by using…

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SLIDE 8

Methodology

πœ–π‘’π’—π‘— = βˆ’π‘‡ ෍

π‘˜βˆˆβ„(𝑗)

(π•˜π‘˜+π•˜π‘—) βˆ™ π’…π‘—π‘˜ βˆ’ π‘’π‘—π‘˜(π’—π‘˜ βˆ’ 𝒗𝑗) 𝑇 = βˆ‡ βˆ™ π•˜ βˆ‡ βˆ™ π•˜ + 𝑩𝑾

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SLIDE 9

Any Quick Questions

  • n Methodology?
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SLIDE 10

Test 1 (Sod)

DoF = 255 DoF = 510 𝒙𝑀 = 𝜍 = 1 𝑣 = 0 π‘ž = 1 , 𝒙𝑆 = 𝜍 = 0.125 𝑣 = 0 π‘ž = 0.1

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SLIDE 11

Test 2 (Shu-Osher)

DoF = 510 𝒙𝑀 = 𝜍 β‰ˆ 3.8 𝑣 β‰ˆ 2.6 π‘ž β‰ˆ 10.3 , 𝒙𝑆 = 𝜍 = 1 + 0.2 sin 5𝑦 𝑣 = 0 π‘ž = 1

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SLIDE 12

How can we improve this?

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SLIDE 13

Sparse Graph-Viscosity

𝑗 ℐ(𝑗) 𝑗 𝒯(𝑗)

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Sparse Graph-Viscosity

πœ–π‘’π’—π‘—

π‘š = βˆ’ ෍ π‘˜βˆˆπ’―(𝑗)

π•˜π‘˜ βˆ™ ො π’…π‘—π‘˜ βˆ’ π‘’π‘—π‘˜(π’—π‘˜ βˆ’ 𝒗𝑗) πœ–π‘’π’—π‘—

β„Ž = βˆ’ ෍ π‘˜βˆˆβ„(𝑗)

π•˜π‘˜ βˆ™ π’…π‘—π‘˜ Low Order… High Order…

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SLIDE 15

Summary and Future

  • Non-parametric shock capturing

method

  • Developing sparse methods for FR
  • Currently working on PyFR

implementation

  • Developing GPU accelerated convex

limiting

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Any Questions?

Thanks to Tarik Dzanic for his work

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References

  • β€œSecond-Order Invariant Domain Preserving Approximation of the

Euler Equations Using Convex Limiting” Jean-Luc Guermond, Murtazo Nazarov, Bojan Popov, and Ignacio Tomas, SIAM Journal on Scientific Computing 2018 40:5, A3211-A3239

  • β€œSparse invariant domain preserving discontinuous Galerkin methods

with subcell convex limiting” Will Pazner, 2020 ArXiV 2004:08503

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