SOLVER STORIES · ORIENTATIONAlternative ways to carry a fluid
Before choosing an algorithm
How should a fluid remember where it came from?
A crest passes under a buoy. A dye filament stretches. A vortex crosses a mesh edge. Every solver has to decide what information survives that journey—and where forces and constraints are computed afterward.
Begin on the frog’s actual triangle mesh, then unfold the comparison onto a flat periodic chart. The random mode prescribes projected spectral fields; the water mode evolves height and tangent velocity through surface shallow-water eigenmodes.
loading tree-frog mesh…drag to orbit · scroll to zoom
Loading the tree-frog mesh and Laplace–Beltrami basis…
What changes between the two views? Both views begin from a localized height disturbance at rest. Pressure creates velocity; the divergence of mass flux changes height; and the resulting fronts travel across the flat torus or the frog. The previews solve the linearized shallow-water equations without artificial damping. They are propagation references—not yet the nonlinear transported Clebsch solver—and make a trustworthy baseline for the larger student project.
A vocabulary before the names
Ask four questions of every fluid method.
A method becomes much easier to compare when each design decision is placed on the right axis.
01
What is stored?
A grid velocity? Particles with velocity moments? Vorticity? A velocity one-form? Clebsch labels and depth?
02
How is it transported?
Backtrace from grid samples, move particles forward, transport labels, or update conservative cell fluxes.
03
Where do forces act?
On an Eulerian grid, on particles, through an energy, or through a surface differential operator.
04
Which promise is enforced?
Zero divergence, mass conservation, circulation, energy behavior, positivity of depth, or some deliberate compromise.
Three influential stories
They solve different parts of the problem.
Read left to right as a set of possible ingredients, not a tournament with one winner.
A
GRID TRANSPORT · STAM 1999
Backtrace the answer: Stable Fluids.
Stand at each grid sample at the new time, follow its characteristic backward, and interpolate the old field at the departure point. The same semi-Lagrangian idea can carry smoke density, velocity components, or Clebsch labels.
new value samples the old field at the backtraced point
WHY PEOPLE LOVE IT
Large steps remain visually stable and the grid makes forces and a pressure Poisson solve convenient.
WHAT IT SPENDS
Repeated interpolation smooths fine structure. “Stable” does not mean exact conservation, no numerical diffusion, or correct behavior at arbitrarily large steps.
Particles move material through the domain while a background grid handles forces and constraints. APIC augments each particle’s translational velocity with a local affine velocity model, retaining information that ordinary PIC loses during particle–grid transfers.
each particle carries a locally affine velocity
WHY PEOPLE LOVE IT
It keeps PIC’s filtering while greatly reducing transfer dissipation, and preserves angular momentum across the transfer described in the method.
WHAT A SURFACE VERSION NEEDS
Tangent particle states, parallel transport between frames, particle–mesh kernels, boundary handling, and a decision about whether the physics is a thin layer or a free surface in ambient space.
Clebsch variables factor the velocity one-form into scalar data. The labels can be transported semi-Lagrangianly, with particles, or by another characteristic method; afterward the one-form is reconstructed and coupled to the intended constraint or height equation.
exact, label, and harmonic pieces of the velocity one-form
WHY STUDY IT HERE
The representation makes material labels, vorticity, circulation, metric conversion, and topology visible in one pipeline.
WHAT IT DOES NOT CHOOSE FOR YOU
Transport accuracy, gauge, global charts, harmonic periods, pressure or height evolution, boundaries, and the final discretization all remain modeling decisions.
The course’s Clebsch shallow-water experiment already uses semi-Lagrangian backtracing for material labels and a grid flux for height. APIC could become a different transport-and-transfer layer without replacing the Clebsch representation or the surface equations.
1 · REPRESENTh, φ, λ, B, harmonic periods
Choose the state that exposes the invariants and topology you care about.
→2 · TRANSPORTbacktrace, particles, or conservative flux
Choose how each component moves and where interpolation error enters.
→3 · RECONSTRUCTone-form → metric-dual velocity
Use surface operators and restore the harmonic data.
→4 · EVOLVE + AUDITforces, depth, circulation, mass
Apply the governing equations and report what the discrete step preserved.
Other doors worth opening
The landscape is wider than these three.
These methods become relevant when the physical goal changes.
CONSERVATIVE FINITE VOLUMEWhen waves, shocks, and wet/dry fronts matter.
Numerical fluxes make mass conservation and discontinuous solutions central. This is often the more natural baseline for production shallow-water systems.
VORTICITY–STREAMFUNCTIONWhen two-dimensional incompressible rotation is the protagonist.
Transport vorticity and solve a Poisson problem for velocity, while remembering that topology contributes harmonic periods.
VARIATIONAL / GEOMETRICWhen discrete symmetry and long-time structure drive the design.
Derive the step from a discrete action or bracket so Kelvin circulation, momentum maps, or energy behavior shape the method from the start.
Choose your next question
Now return to the water with a sharper comparison.
The random-field lab isolates representation and projection. The shallow-water labs add a governing evolution. The background notebook and reading map hold the research lineage.