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

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 parametric • P-adaptive methods present several issues 𝑞 = 0, DoF = 20𝑙

Current Shock Capturing Method Parametric Op. Cheap “Stable” ✓ ✓ ✗ ? AV/Per-Olof ✓ ✓ ? ✓ Filtering ✗ ? ✓ ? ✗ Adaptation/ Moving mesh

Methodology 𝑗 ℐ(𝑗)

Methodology So... 𝜖 𝑢 𝒗 𝑗 = − ෍ 𝕘 𝑘 ∙ 𝒅 𝑗𝑘 𝑘∈ℐ(𝑗) + ෍ 𝑒 𝑗𝑘 (𝒗 𝑘 − 𝒗 𝑗 ) 𝑘∈ℐ(𝑗) Now… ෍ 𝒅 𝑗𝑘 = 0 = ෍ 𝑒 𝑗𝑘 𝑘∈ℐ(𝑗) 𝑘∈ℐ(𝑗)

Methodology 𝜖 𝑢 𝒗 𝑗 = − ෍ (𝕘 𝑘 +𝕘 𝑗 ) ∙ 𝒅 𝑗𝑘 − 𝑒 𝑗𝑘 (𝒗 𝑘 − 𝒗 𝑗 ) 𝑘∈ℐ(𝑗) Which we solve by using… 𝑒 𝑗𝑘 = max(𝜇 𝑛𝑏𝑦 𝒐 𝑗𝑘 , 𝒗 𝑗 , 𝒗 𝑘 𝒅 𝑗𝑘 , 𝜇 𝑛𝑏𝑦 (𝒐 𝑘𝑗 , 𝒗 𝑘 , 𝒗 𝑗 )|𝒅 𝑘𝑗 |)

Methodology 𝜖 𝑢 𝒗 𝑗 = −𝑇 ෍ (𝕘 𝑘 +𝕘 𝑗 ) ∙ 𝒅 𝑗𝑘 − 𝑒 𝑗𝑘 (𝒗 𝑘 − 𝒗 𝑗 ) 𝑘∈ℐ(𝑗) ∇ ∙ 𝕘 𝑇 = ∇ ∙ 𝕘 + 𝑩𝑾

Any Quick Questions on Methodology?

Test 1 (Sod) DoF = 255 DoF = 510 𝜍 = 1 𝜍 = 0.125 𝒙 𝑀 = 𝑣 = 0 , 𝒙 𝑆 = 𝑣 = 0 𝑞 = 1 𝑞 = 0.1

Test 2 (Shu-Osher) DoF = 510 𝜍 = 1 + 0.2 sin 5𝑦 𝜍 ≈ 3.8 𝒙 𝑆 = 𝑣 = 0 𝑣 ≈ 2.6 𝒙 𝑀 = , 𝑞 = 1 𝑞 ≈ 10.3

How can we improve this?

Sparse Graph-Viscosity 𝒯(𝑗) 𝑗 𝑗 ℐ(𝑗)

Sparse Graph-Viscosity Low Order… 𝑚 = − ෍ 𝕘 𝑘 ∙ ො 𝜖 𝑢 𝒗 𝑗 𝒅 𝑗𝑘 − 𝑒 𝑗𝑘 (𝒗 𝑘 − 𝒗 𝑗 ) 𝑘∈𝒯(𝑗) High Order… ℎ = − ෍ 𝜖 𝑢 𝒗 𝑗 𝕘 𝑘 ∙ 𝒅 𝑗𝑘 𝑘∈ℐ(𝑗)

Summary and Future • Non-parametric shock capturing method • Developing sparse methods for FR • Currently working on PyFR implementation • Developing GPU accelerated convex limiting

Thanks to Tarik Dzanic for his work Any Questions?

All fields All fields All fields All fields References • “ Second-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 “Sparse invariant domain preserving discontinuous Galerkin methods • with subcell convex limiting” Will Pazner, 2020 ArXiV 2004:08503