Numerical Methods

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.

2026.03.09

Discontinuous Galerkin Method (DGM): High-Precision Discretization for Unstructured Grid Compressible Flow Analysis

In 3D the degrees of freedom at K = 1, 2, 3 are 4, 10, and 20. Halving the mesh spacing should cut the error by 4, 8, and 16, not by 2.

2026.04.20

Numerical Dungeon: Learn Numerical Methods Through a Game

Three HP, five stages: the quiz runs from the CFL condition through Godunov limiters and Riemann problems to Abgrall's multiphase pressure fix.

2026.03.08

Mach Architect: Learn Compressible Flow by Designing Nozzles

At γ = 1.4 the choke ratio is 0.5283 and the M = 2 area ratio is about 1.6875. Back pressure between those two parks a normal shock inside the nozzle.

2026.03.11

Choosing a CFD Linear Solver: Comparing Convergence Speeds from Jacobi to BiCGSTAB

On a 60-unknown Poisson system Jacobi is still at residual 2.11e-2 after 2000 sweeps, while CG reaches 1.87e-13 in exactly 60 steps.

2026.04.22

Solving Wall Distance in O(N log N): Eikonal Equation and Fast Marching

Ten million cells and half a million walls mean five trillion distance checks. Fast Marching solves that wall distance as an Eikonal problem in O(N log N).

2026.04.22

When Lax–Wendroff Shook at a Step — Building a 2nd-Order TVD Advection Scheme with Flux Limiters

At CFL 0.5 and 500 steps, Lax–Wendroff overshoots a unit step by 0.181. minmod, van Leer, and MC stay at zero overshoot on the same pulse.

2026.04.24

[Paper Review] When WENO Smears Small Eddies — Fu (2019) Low-Dissipation TENO Finite-Volume Scheme

A 10 percent bump in one smoothness indicator already shifts WENO-JS weights. TENO's 0/1 cutoff does not flip, so the fifth-order linear stencil stays on.

2026.04.25

[Paper Review] Beating the Storm Clock — Casulli–Walters (2000) Semi-Implicit Shallow Water on Unstructured Orthogonal Grids

At 28 m depth the explicit limit is 0.2 s. Keeping only the free-surface gradient implicit is what lets a five-minute step survive the estuary.

2026.04.26

Quadtree AMR — Letting the Mesh Grow Where It Needs To

Level-7 uniform needs 128 cells; one Gaussian keeps 41 leaves, about 32 percent. A quadtree, slope threshold tau, and a 2:1 rule decide which cells split.

2026.04.27

When Characteristics Collide, a Shock Is Born — Burgers and the (x,t) Plane

A sine of amplitude 0.35 and wavenumber 1 breaks at t* ≈ 0.455 when characteristics cross, and the weak solution cuts that fold with one line.

2026.05.01

[Paper Review] Pick the Reconstruction with Smaller Jumps — Deng (2018) MUSCL-THINC-BVD

Deng (2018) builds MUSCL and THINC in every cell, then keeps the one with the smaller jump on the two faces. The rule is called BVD.

2026.05.02

Implicit Diffusion and the Thomas Algorithm — Buying Stability with One Tridiagonal Solve

At r=0.6 explicit diffusion reaches 1e12 in 60 steps. One Thomas sweep keeps the implicit update finite at that same r, and still finite at r=2.

2026.05.04

When FTCS Trembles, Upwind Goes Numb — First-Order Advection, Artificial Viscosity, and the CFL Trade

At c=0.8 the worst FTCS mode grows by about 1.28 each step. Upwind artificial viscosity drops to zero only when c equals 1.

2026.05.08

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.

2026.05.11

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.

2026.05.22

Delaunay Triangulation — the Empty-Circumcircle Rule and Bowyer–Watson Insertion

N=60 and 8 hull points fix the triangle count at 2N-2-h=110. Four cocircular corners make the in-circle determinant zero, so either diagonal is legal.

2026.05.27

How to Make an Outflow Stop Reflecting Sound — Navier–Stokes Characteristic Boundary Conditions

A six-hour LES drifted 0.3 bar because the outlet reflected sound. NSCBC sets the incoming wave with σ near 0.25 so acoustics leave and the mean holds.

2026.05.29

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.

2026.06.01

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.

2026.06.17

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.

2026.07.01

The Man Who Erased Pressure — Chorin's Projection Method and Fractional Step

After the predictor the divergence sits near 1e-2. One FFT projection drops it to 1e-13, and the pressure Poisson still takes most of the step.

2026.07.03

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.

2026.07.06

Two Philosophies for Triangulating a Point Cloud — Bowyer–Watson and Advancing Front

Twelve points become 17 triangles. A sliced airfoil edge is not restored by the empty circumcircle; a pipe or the front has to take that edge back.

2026.07.08

When Pressure Rings in a Checkerboard — SIMPLE and Rhie-Chow Coupling

On a 48-cell periodic line the naive stencil keeps a checkerboard of 0.80, while Rhie-Chow drives the same sawtooth down to 3.1e-16.

2026.07.22

The Mesher That Dies Defending Your Boundary — Constrained Delaunay and Steiner Points

Of 4408 valid models, tetgen dies on about 8.5 percent, and 2 of those failures remain even with infinite-precision arithmetic.

2026.07.29

A 1% Error in Diameter Is a 4% Error in Flow Rate — Half a Cell of Wall and Poiseuille's Constant

Pipe flow rate rides on the fourth power of the diameter. The same exponent explains a mesh with the wall half a cell off and Poiseuille's 1838 tubes.

2026.09.01

202x More Scalar Dissipation, 13% Less Temperature — Where a Laminar Flamelet Quenches

Scalar dissipation rose 202-fold while peak temperature fell only 13%, then the burning solution vanished. Python pins that quench limit.

2026.09.22

Upgrading Only the Viscous Term to Second Order Made the Error 1.82x Worse — Why the Semi-Implicit Gear Scheme Extrapolates Convection

Pairing BDF2 with convection at N^n drops the scheme to first order and inflates the error. The extrapolation 2N^n − N^(n−1) restores second order.

2026.10.12

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.

2026.10.13

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.

2026.10.14