Changing One Relaxation Factor Moved the Converged Solution — Where Rhie–Chow Face Velocities Remember Under-Relaxation
In SIMPLE, cutting α_u from 0.7 to 0.3 changes the converged velocity. Standard Rhie–Chow keeps α_u on the damping term; Majumdar's face store removes it.
I Gave Curved Cells Their True Area and ∂x/∂x Stopped Being 1 — Why the Diagonal Cross-Product Area Fits Green–Gauss
Cell-center Jacobian volumes make ∂x/∂x miss 1 on a wavy FVM grid. The diagonal cross product matches straight faces, so a linear field stays exact.
When Gradients Go Wrong on Skewed Meshes — Green–Gauss vs. Weighted Least-Squares Reconstruction
On a flat cell, Green–Gauss can swing the gradient nearly sideways. Distance-weighted least squares cuts the curved-field error from 0.0293 to 0.0222.
The Fake Extrema a Second-Order Reconstruction Invents on Unstructured Grids — Barth–Jespersen and Venkatakrishnan Limiters
A second-order face value on an unstructured cell can leave the neighbor range. Barth–Jespersen clips that slope; Venkatakrishnan rounds the clip with K.
When the Mesh Skews, the Laplacian Leaks — Non-Orthogonal Diffusion Correction
The face-area vector splits into an implicit piece and a deferred correction. Minimum correction hits ratio 1 at 45 degrees and stops contracting.
Wrapping a Rectangular Grid Around a Wing — Curvilinear Coordinate Transforms and Grid Metrics
On a wavy mesh the Jacobian runs from 0.62 to 1.38 and stays positive. Write the flux as y_η and the freestream residual falls to about 1e-16.
Advecting Without Splitting Directions — The Bell-Colella-Glaz Unsplit Upwind Scheme
Directional splitting twists a diagonal scalar. Bell-Colella-Glaz keeps the rotating disk's peak with a (1-CFL) slope weight plus a transverse correction.
The Euler Eigensystem in an Arbitrary Direction — Three Waves, Two Repeats, One Singularity
At n_x=0.01 the R-1 left-eigenvector row divides by 100, and at zero it diverges. A 30 degree face keeps R-1; a face along y switches to R-2.
Saving Gradients on Unstructured Grids — Moving Least-Squares Approximation
With h=0.07, order 2 and 25 scattered points, the Taylor MLS derivative of sin(2πx) has RMSE 0.21, about 3% of the 2π amplitude.
The Star That Started Moving — Well-Balanced FVM and Angular Momentum in Polar Coordinates
Mis-discretize 2P/r and a quiet star will leak gas out of its center.
Skewed Cells Bend Gradients — Green–Gauss vs Least-Squares Reconstruction
At skew 0.9 the printed Green–Gauss error is 31% and least squares is 9%. Their ratio stays near 3.4 from the symmetric ring onward.
FDM vs FEM vs FVM: The Essential Differences Between Three Discretization Techniques
FDM discretizes the operator, FEM projects onto a basis, and FVM balances face fluxes. That choice fixes conservation, shocks, and unstructured meshes.