Research-Stack/3-Mathematical-Models/AMMR/HEAT2D_ADAPTER.md

Heat-2D Adapter

This repository is the 2D diffusion / dissipative-residue sandbox for the pressure-cavitation-vibration model catalog.

Core role

shock / pressure / vibration / combustion / friction event
    -> deposited scalar residue
    -> 2D diffusion
    -> measurable afterimage field

In the broader stack:

Burgers handles shock smoothing.
heat-2D handles deposited residue.

Native PDE

The upstream README defines the supported PDE family as:

\partial_t u = \nabla\cdot(\alpha\nabla u)+f

where u(x,y,t) is the diffusive variable, alpha(x,y) is a space-dependent diffusivity, and f(x,y,t) is a source term.

For catalog integration, reinterpret these as:

Field	Catalog interpretation
u	residual field: heat, deposited shock energy, vibration residue, or pressure-afterimage proxy
alpha	material / geometry-dependent diffusivity
f	source term from cavitation, combustion, friction, impact, or acoustic deposit
boundary conditions	heat/pressure residue sinks, walls, insulation, or imposed driving surfaces

Adapter definition

X_{heat2D}=(u,\alpha,f,\Omega,\partial\Omega,\Delta x,\Delta y,\Delta t,B)

\alpha_{heat2D}:X_{heat2D}\rightarrow U_T(x,y,t)

The adapter is admissible when:

1. boundary conditions are explicit,
2. source units are documented,
3. geometry masks are kept separate from solver logic,
4. stability requirements are checked for explicit integration,
5. implicit or Crank-Nicolson runs include linear-solver receipt metadata.

Candidate source projections

Cavitation collapse deposit

f(x,y,t)=E_{shock}\exp\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma_s^2}\right)\delta(t-t_0)

Rocket cooling proxy

\partial_t T=\nabla\cdot(\alpha\nabla T)+q''_{combustion}-q''_{coolant}

Passive fluidic residue map

Use alpha(x,y) to encode channels, walls, pockets, and damping zones:

\partial_t u=\nabla\cdot(\alpha(x,y)\nabla u)+f

Burgers dissipative afterimage

f_{diss}(x,y,t)=\lambda\nu(\partial_x U)^2

Minimal tests

Test	Purpose	Expected behavior
point impulse	pure diffusion sanity check	radial spread, decaying max
line source	shock-front residue proxy	front thickens like sqrt(t)
geometry mask	passive fluidic damping test	pockets and channels alter residue storage
cooling channel	rocket wall proxy	stable gradient under balanced source/sink

Failure modes

Failure	Meaning
oscillation in pure explicit heat solve	timestep instability
negative values from nonnegative initial/source field	instability or source bug
energy growth with no source	boundary/source accounting error
claiming CFD from diffusion	category error

Scope warning

This repository should not be treated as a full Navier-Stokes, cavitation, rocket-engine, or acoustic solver.

Its correct role is narrower and more useful:

Given a source event, what dissipative residue field remains?