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CFD

working pillars 3, 4 · Discretization, solvers

The machinery that turns the governing equations into numbers: how the domain is divided, how fluxes are built, and how the resulting system is advanced.

The pipeline, and where each stage goes wrong

  • Mesh — quality metrics decide accuracy before a single equation is solved.
  • Discretize in space — face fluxes, reconstruction, limiters.
  • Discretize in time — stability limits, or an implicit solve.
  • Solve the linear system — conditioning, driven largely by mesh anisotropy.
  • Verify — grid convergence and manufactured solutions, before believing anything.

Pages in this section

Page Covers
Finite volume Integral form, face reconstruction, limiters, gradients, non-orthogonal correction
Time integration Explicit and implicit schemes, CFL and diffusive limits, stiffness, dual time stepping
Meshing and grid quality Quality metrics, boundary-layer sizing, \(y^+\) targets, AMR, grid convergence

The recurring theme

Three of the most expensive mistakes in CFD are not errors of theory but of attribution — a symptom blamed on the wrong stage:

  • First-order upwind "fixing" an oscillation, when the oscillation was a mesh or boundedness problem and the fix replaced it with numerical diffusion thousands of times the physical viscosity.
  • Multigrid convergence collapsing, blamed on the solver, caused by boundary-layer cell aspect ratio.
  • A grid study on a wall-function mesh, where refinement changes the turbulence modelling rather than the discretization error.

Each is recorded in the failure-modes block of the relevant page.

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