University:
Boston University
Course:
MET CS 201 | Introduction to Programming
Academic year:

2022
Views:

202

Pages:

21
Author:

Deja Hood

Memory Bandwidth on a GTX 280 Global Memory Bandwidth [GB/s] 200 180 160 140 120 100 80 60 40 201 Gather Lift Diff Assy. Peak 2 3 4 5 6 Polynomial Order N 7 8 9 Multiple GPUs via MPI: 16 GPUs vs. 64 CPUs GFlops/s Flop Rates: 16 GPUs vs 64 CPU cores GPU 4000 CPU 3000 2000 1000 00 2 4 6 8 Polynomial Order N 10 GPU-DG in Double Precision GFlops/s GPU-DG: Double vs. Single Precision 400 350 300 250 200 150 100 50 00 Single Double Ratio 1.0 0.8 0.6 0.4 0.2 2 4 6 8 Polynomial Order N 100.0 Outline 1 PyCUDA 2 Automatic GPU Programming 3 GPU-DG: Challenges and Solutions Introduction Challenges Benefits of Metaprogramming GPU-DG: Performance and Generality Viscous Shock Capture Nonlinear conservation laws → shocks? u(x) t =0.00 t =0.67 t =1.33 0 2 4 x 6 8 10 Nonlinear conservation laws → shocks? u(x) 0 t =0.00 t =0.67 t =1.33 u(x) t =0.00 t =0.67 t =1.33 2 0 4 2 6 x 4 8x 6 10 8 10 Nonlinear conservation laws → shocks? 1D advection: t =0.00 t =0.67 t =1.33 ∂t u + ∂x u = 0 u(x) u(x) 1D advection with viscosity: t =0.00 t =0.67 t =1.33 ∂t u + v · ∇x u = ∇x · (ν∇x u). Important: Conservation form. Upwind fluxes for advection, IPDG for secondorder 0 2 0 4 2 6 x 4 8x 6 10 8 10 Nonlinear conservation laws → shocks? 1D advection: ∂t u + ∂x u = 0 t =0.00 t =0.67 t =1.33 u(x) u(x) 1D advection with viscosity: Detector → ν? ∂t u + v · ∇x uGPU-suitability? = ∇x · (ν∇x u). Data locality? Properties? [Build on work by Important: Conservation form. Persson/Peraire ‘06] t =0.00 t =0.67 t =1.33 Time integration Upwind fluxes for advection, IPDG for secondImplicit/explicit? Adaptivity? RKC for order 4 0 2 4bigger ∆t6 with viscosity? 0 2 8x 6 10 8 x Accuracy? Near shocks? Away from them? 10 Results: Euler’s Equations of Gas Dynamics Euler’s equations with viscosity: ∂t ρ + ∇x · (ρu) = ∇x · (ν∇x ρ), ∂t (ρu) + ∇x · (u ⊗ (ρu)) + ∇x p = ∇x · (ν∇x (ρu)), ∂t E + ∇x · (u(E + p)) = ∇x · (ν∇x E ). Again: Single ν, sensed on ρ. → Undue pollution of the other field? [Persson/Peraire ‘06] suggest Navier-Stokes-like viscosity. No good: can’t control jumps in ρ. Rusanov fluxes for Euler, IPDG for viscosity. Results: Euler’s Equations of Gas Dynamics Euler’s equations with viscosity: ∂t ρ + ∇x · (ρu) = ∇x · (ν∇x ρ), Problem =5∇and K =80 ∂t (ρu) + ∇Sod's (ρu)) +with ∇x pN = x · (u ⊗ x · (ν∇ x (ρu)), 1.2 ∂t E + ∇x · (u(E + p))ρ= ∇x · (ν∇x E ). 1.0 p ρ, p (exact, L2pollution proj.) Again: Single ν, sensed on ρ. →ρUndue of 2 proj.) 0.8 p (exact, L the other field? 0.6 [Persson/Peraire ‘06] suggest Navier-Stokes-like viscosity. No0.4good: can’t control jumps in ρ. 0.2 for Euler, IPDG for viscosity. Rusanov fluxes 0.0 0.0 0.2 0.4 x 0.6 0.8 1.0 Results: Euler’s Equations of Gas Dynamics Euler’s equations with viscosity: Shock-Wave Interaction ∂t ρ + ∇x · (ρu) = ∇x · (ν∇Problem x ρ), with N =5 and K =80 Sod's Problem =5∇and· (ν∇ K =80(ρu)), ∂ (ρu) + ∇12 · (u ⊗ (ρu)) +with ∇ pN = t 1.2 x x x x ∂10 t E + ∇x · (u(E + p))ρ = ∇x · (ν∇x E ). 1.0 p ρ p (exact, L2pollution proj.) Again: Single8 ν, sensed on ρ. →ρUndue of 2 proj.) 0.8 p (exact, L the other field? ρ, p ρ, p 0.6 6 [Persson/Peraire ‘06] suggest Navier-Stokes-like viscosity. No0.4good: 4 can’t control jumps in ρ. 0.2 2 for Euler, IPDG for viscosity. Rusanov fluxes 0.0 0.0 06 0.2 4 0.4 x 0.6 2 0 x 0.8 2 1.0 4 6 Results: Euler’s Equations of Gas Dynamics Euler’s equations with viscosity: Shock-Wave Interaction ∂t ρ + ∇x · (ρu) = ∇x · (ν∇Problem x ρ), with N =5 and K =80 Sod's Problem =5∇and· (ν∇ K =80(ρu)), ∂ (ρu) + ∇12 · (u ⊗ (ρu)) +with ∇ pN = t 1.2 x x x x ∂10 t E + ∇x · (u(E + p))ρ = ∇x · (ν∇x E ). 1.0 p ρ p (exact, L2pollution proj.) Again: Single8 ν, sensed on ρ. →ρUndue of 2 proj.) 0.8 p (exact, L the other field? ρ, p ρ, p 0.6 6 [Persson/Peraire ‘06] suggest Navier-Stokes-like viscosity. No0.4good: 4 can’t control jumps in ρ. 0.2 2 for Euler, IPDG for viscosity. Rusanov fluxes 0.0 0.0 06 0.2 4 0.4 x 0.6 2 0 x 0.8 2 1.0 4 6 Results: Euler’s Equations of Gas Dynamics Euler’s equations with viscosity: Shock-Wave Interaction ∂t ρ + ∇x · (ρu) = ∇x · (ν∇Problem x ρ), with N =5 and K =80 Sod's Problem =5∇and· (ν∇ K =80(ρu)), ∂ (ρu) + ∇12 · (u ⊗ (ρu)) +with ∇ pN = t 1.2 x x x x ∂10 t E + ∇x · (u(E + p))ρ = ∇x · (ν∇x E ). 1.0 p ρ p (exact, L2pollution proj.) Again: Single8 ν, sensed on ρ. →ρUndue of 2 proj.) 0.8 p (exact, L the other field? ρ, p ρ, p 0.6 6 [Persson/Peraire ‘06] suggest Navier-Stokes-like viscosity. No0.4good: 4 can’t control jumps in ρ. 0.2 2 for Euler, IPDG for viscosity. Rusanov fluxes 0.0 0.0 06 0.2 4 0.4 x 0.6 2 0 x 0.8 2 1.0 4 6 Results: Euler’s Equations of Gas Dynamics Euler’s equations with viscosity: Shock-Wave Interaction ∂t ρ + ∇x · (ρu) = ∇x · (ν∇Problem x ρ), with N =5 and K =80 Sod's Problem =5∇and· (ν∇ K =80(ρu)), ∂ (ρu) + ∇12 · (u ⊗ (ρu)) +with ∇ pN = t 1.2 x x x x ∂10 t E + ∇x · (u(E + p))ρ = ∇x · (ν∇x E ). 1.0 p ρ p (exact, L2pollution proj.) Again: Single8 ν, sensed on ρ. →ρUndue of 2 proj.) 0.8 p (exact, L the other field? ρ, p ρ, p 0.6 6 [Persson/Peraire ‘06] suggest Navier-Stokes-like viscosity. No0.4good: 4 can’t control jumps in ρ. 0.2 2 for Euler, IPDG for viscosity. Rusanov fluxes 0.0 0.0 06 0.2 4 0.4 x 0.6 2 0 x 0.8 2 1.0 4 6 Results: Euler’s Equations of Gas Dynamics Euler’s equations with viscosity: Shock-Wave Interaction ∂t ρ + ∇x · (ρu) = ∇x · (ν∇Problem x ρ), with N =5 and K =80 Sod's Problem =5∇and· (ν∇ K =80(ρu)), ∂ (ρu) + ∇12 · (u ⊗ (ρu)) +with ∇ pN = t 1.2 x x x x ∂10 t E + ∇x · (u(E + p))ρ = ∇x · (ν∇x E ). 1.0 p ρ p (exact, L2pollution proj.) Again: Single8 ν, sensed on ρ. →ρUndue of 2 proj.) 0.8 p (exact, L the other field? ρ, p ρ, p 0.6 6 [Persson/Peraire ‘06] suggest Navier-Stokes-like viscosity. No0.4good: 4 can’t control jumps in ρ. 0.2 2 for Euler, IPDG for viscosity. Rusanov fluxes 0.0 0.0 06 0.2 4 0.4 x 0.6 2 0 x 0.8 2 1.0 4 6 Results: Euler’s Equations of Gas Dynamics Euler’s equations with viscosity: Shock-Wave Interaction ∂t ρ + ∇x · (ρu) = ∇x · (ν∇Problem x ρ), with N =5 and K =80 Sod's Problem =5∇and· (ν∇ K =80(ρu)), ∂ (ρu) + ∇12 · (u ⊗ (ρu)) +with ∇ pN = t 1.2 x x x x ∂10 t E + ∇x · (u(E + p))ρ = ∇x · (ν∇x E ). 1.0 p ρ p (exact, L2pollution proj.) Again: Single8 ν, sensed on ρ. →ρUndue of 2 proj.) 0.8 p (exact, L the other field? ρ, p ρ, p 0.6 6 [Persson/Peraire ‘06] suggest Navier-Stokes-like viscosity. No0.4good: 4 can’t control jumps in ρ. 0.2 2 for Euler, IPDG for viscosity. Rusanov fluxes 0.0 0.0 06 0.2 4 0.4 x 0.6 2 0 x 0.8 2 1.0 4 6 GPU DG Showcase Eletromagnetism GPU DG Showcase Eletromagnetism Poisson GPU DG Showcase Eletromagnetism Poisson CFD GPU DG Showcase Eletromagnetism Poisson CFD Conclusions GPUs and scripting work surprisingly well together Enable Run-Time Code Generation GPU-DG is significantly faster than CPU-DG Method well-suited a priori Numerous tricks enable good performance Further work in GPU-DG: Curvilinear Elements (T. Warburton) Local Time Stepping Shock Capturing for Nonlinear Equations