Implicit Methods & Linear Solvers

What Jacobi Cannot Kill in 10,000 Sweeps, a V-Cycle Kills in 10 — The Real Secret of Multigrid

On a 64-cell Poisson problem, twenty V-cycles drive the residual below 1e-10 while Jacobi, at four sweeps a step, stalls near 1e-2.

2026.05.06

When Explicit Blows Up, Solve Implicit Without Inverting the Matrix — LU-SGS

Explicit CFL at 0.3 stalls for days. Backward Euler and two LU-SGS sweeps carry a Burgers shock through CFL 5 in 49 steps, with no full-matrix inverse.

2026.06.22

Solving Without Inverting the Matrix — GMRES and the Arnoldi Subspace

On a 40 by 40 Poisson problem, 1600 unknowns reach a residual of 8.74e-11 in 78 matrix-vector products, and the true residual is 9.02e-11.

2026.06.24

CFL = 1 takes ten thousand steps, CFL = 100 explodes — Switched Evolution Relaxation

Fixed CFL of 3 needs 588 iterations on steady Burgers. SER with p=1 finishes in 34, then the step is Newton only while a finite CFL cap is still in place.

2026.06.29

[Paper Review] Tired of Differentiating Jacobians by Hand? — Operator-Overloading Automatic Differentiation (ADOO)

Fraysse (2019) seeds dual numbers through an unchanged flux routine and reads a machine-precision Jacobian, with no finite-difference step h to tune.

2026.07.05

Letting Sound and Matter Flow Separately — Acoustic-Convective Splitting (Lagrange-Projection)

At M=0.02 the acoustic band stays 2c wide while convection crawls. Feeding R back into the upwind step holds the momentum error at 2.2e-16.

2026.07.11

The Mach Number Goes to Zero, the Time Step Doesn't Budge — Reproducing an IMEX TVD Scheme

At Mach 10^{-4} the time step stays on the convective CFL. Pure second-order IMEX overshoots by about 0.4 unless theta is held to sqrt(2)-1 or below.

2026.07.26

Where Jacobi Dies on a Boundary-Layer Mesh — the Linelet Line-Implicit Preconditioner

On a 64 by 48 mesh, Jacobi climbs from 18 sweeps at aspect ratio 1 to 5471 at 1000, while a wall-normal linelet finishes in 2.

2026.07.31

Shuffling Cell Numbers Made LU-SGS 2.8× Slower — RCM Reordering and the Start Cell That Sets Sweep Direction

Shuffling cell numbers on a 576-cell grid pushed LU-SGS from 9 iterations to 25. RCM cuts bandwidth to 24, but the start corner still splits 9 from 21.

2026.10.16

Halving the Grid Spacing Tripled the Error — What an Increment-Based Stopping Rule Hides in Jacobi Iteration

Stop Jacobi when the update drops below a tolerance and the error left behind grows as 1/(1-ρ) on finer grids. A 2D heat-conduction plate compares Jacobi, SOR and optimal ω.

2026.10.21

A Ball Thrown at 5 m/s Passed Through a 20 mm Wall — Penalty Contact vs. the IPC Log Barrier

In the same implicit Euler step, penalty contact skips over a thin wall while the IPC log barrier with a CCD-filtered line search stops micrometers short. A 1D experiment shows why.

2026.10.29

A Steady-State Preconditioner Inside a Dual-Time Loop Took 19× More Inner Iterations — A Low-Mach Acoustic Pulse Test

Put a Weiss–Smith preconditioner into physical time and the acoustic pulse freezes in place. Put it inside dual time and the answer returns, but inner iterations jump from 16 to 305.

2026.10.30