ADVECTION-DIFFUSION SOLVER

The picture

A factory fires off a puff of something out of its chimney. The wind catches it and pushes it downwind — that's advection. As it drifts, the puff smears out into a softer, wider cloud — that's diffusion. Almost every "stuff is moving through a medium" problem in engineering is some mix of these two: smokestack plumes, dye in a stream, heat in a metal bar, pollutants in a river.

The curve underneath the scene is the same story stripped to numbers: the concentration U at each position x. The pulse drifts right and flattens at the same time.

The scheme

The solver is an explicit finite-volume scheme on a 1D grid. Upwind flux for the advection term, central difference for diffusion. There's only about fifty lines of math; the rest is bookkeeping.

The CFL condition

Explicit schemes are stable only when the timestep is small relative to how fast information moves across a cell — dt ≤ 1 / (v/dx + 2D/dx²). The Blow-up preset deliberately busts that bound. The plume turns into garbage within a few steps and the canvas freezes on the first non-finite value. That's not a bug in the simulation; it's the simulation telling you it's no longer simulating anything.

The port

The original Fortran 90 code is from 2014 (from a numerical methods course in grad school). To run it in a browser, I had Gemini 3.1 Pro translate the Fortran to C++, then compiled the C++ to WebAssembly with Emscripten. The translation is checked against a frozen snapshot of the original Fortran's output: the test suite runs each preset under WASM and asserts the curves track the Fortran's within a small tolerance. The math is identical to the bit — a tiny residual comes from time += dt accumulating differently across the two compilers' floating-point pipelines, which puts the WASM about one timestep past the Fortran at output time. The original Fortran source is linked below.