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.

Mar 9, 2026

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.

Apr 20, 2026

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.

Apr 22, 2026

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.

Apr 24, 2026

[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.

Apr 25, 2026

[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.

Apr 26, 2026

[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.

May 2, 2026