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PANSOPHIA numerics teaching figures

Fifteen teaching figures on schemes, stability and accuracy, generated from the PANSOPHIA numerics stack

14 images · 1 min read

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PANSOPHIA numerics teaching figures

datasheet // cfd-numerics-teaching

images
14
videos
0
formats
png, webp
text
75 words · 1 min read
studio
/atlas?ch=S34&ex=1
topics
  • scheme comparison ×4
  • stability analysis ×3
  • numerics verification ×1
  • mesh resolution ×1
  • dispersion dissipation ×2
  • nonlinear and space time ×3

Datasheet

  • Source: scripts/run_phase1_educational.py writes 15 figures to figures/edu/ (1849-2984 px wide).
  • Topics covered by the script: scheme comparison, von Neumann stability, stability regions, MMS convergence, CFL sweep, residual history, mesh zones, numerical diffusion, dispersion/dissipation, phase error, solution snapshots, Burgers shock formation, modified wavenumber, space-time heatmaps.
  • Convection schemes in CFDCore.jl/src/schemes/convection: UDS, CDS, LUDS, QUICK, TVD, VOF advection. Diffusion: central difference.
  • Time integrators in CFDCore.jl/src/time: Forward Euler, Backward Euler, Crank-Nicolson, RK2, RK4, AB2, BDF2, IMEX, Strang splitting.
1D advection of a Gaussian and a step profile with UDS, CDS, TVD-minmod and LUDS (CFL 0.45, t=1) Explain CFD and numerical methods
Von Neumann amplification factor |G(theta)| for six scheme/integrator pairings at CFL 0.4 and 0.8 Explain CFD and numerical methods
Spatial scheme comparison (top) and absolute stability regions of seven time integrators (FE, RK2, RK4, AB2, BE, CN, BDF2) Explain CFD and numerical methods
UDS+FE CFL sweep: stable points up to CFL = 1, blow-up above; profiles at CFL 0.5, 0.95 and 1.1 Explain CFD and numerical methods
Diffusion equation: step-change history and final profiles for Forward Euler, Crank-Nicolson and Backward Euler against the exact solution Explain CFD and numerical methods
Uniform and stretched 1D meshes (n = 20, ratio 1.12): cell centres, faces, cell-size distribution and expansion ratio Explain CFD and numerical methods
Numerical diffusion on a step profile with UDS, CDS, TVD minmod and TVD superbee (CFL 0.45, n = 200) Explain CFD and numerical methods
Dissipation |G| and phase-error proxy versus normalised wavenumber for five scheme pairings (CFL 0.5) Explain CFD and numerical methods
Phase error: a sine wave advected to t = 1 with UDS, CDS and TVD minmod (CFL 0.4) Explain CFD and numerical methods
UDS+FE Gaussian advection snapshots from t = 0.17 to t = 1.0 (CFL 0.4, n = 100) Explain CFD and numerical methods
Viscous Burgers equation: a sine wave steepening for viscosity 0.001, 0.01 and 0.05 Explain CFD and numerical methods
Modified wavenumber analysis: phase and dissipation of UDS, CDS and LUDS Explain CFD and numerical methods
Space-time heatmap and maximum value over time for UDS+FE (CFL 0.4) Explain CFD and numerical methods
Space-time heatmap and maximum value over time for TVD-minmod+FE (CFL 0.4) Explain CFD and numerical methods

How to recreate this plot

Watch

A short screen capture of the Math Visualization Studio. Each step is captioned in the clip.

Steps

  1. Open the controls panel
  2. Pick example 2: Implicit Euler is unconditionally stable
  3. Raise the step h to 1.5
  4. Switch the method to explicit Euler
  5. Switch back to the implicit method
Open the scene in the StudioRead the explanation

Code and PDF notes

Free of cost. I approve each request and reply by email.

To request the files, email me at and mention the item id. project-cfd-numerics-teaching