HPC Tier Module
2D Heat Equation (Transient)
Solve the 2D transient heat equation in your browser, checked against an exact analytic eigenmode.
See it run - a worked example, 100% in this browser tab
The problem
Engineers prototyping transient conduction often reach for a heavy desktop solver or an unverified spreadsheet, with no built-in proof that the time-stepping is stable or that the answer matches a known closed form.
The local-first solution
This plugin marches the 2D heat equation with a cited 5-point Laplacian using FTCS explicit or unconditionally stable ADI, computes the von Neumann stability number, and reports the L-infinity / L2 error against an exact analytic eigenmode - all deterministic f64 math in your browser with nothing uploaded.
What it does
FTCS forward-time explicit march with a computed mesh Fourier number and a hard stability flag
Backward-Euler ADI (Peaceman-Rachford) implicit step via the exact Thomas tridiagonal solve
Dirichlet, Neumann (zero-flux insulated), and Robin convective boundary conditions per edge
Validation against the exact separable eigenmode and its analytic decay rate
Steady-state Laplace residual reported as the long-time check
GeoNum conditioning probe of the time-stepping recurrence
Honest scope
EXACT: the 5-point stencil, FTCS update, Thomas solve, stability number, and the analytic eigenmode it is checked against are exact f64 arithmetic over published formulas; diffusivity and geometry are user inputs echoed back. NOT modeled: nonlinear or temperature-dependent diffusivity, sources, anisotropic conductivity, phase change, radiation boundaries, 3D, or curved geometry. This is an engineering/teaching solver, not a certified safety analysis.
Authorities cited
- LeVeque, R. J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM. - 5-point Laplacian (sec 3.2), FTCS heat update (eqn 4.20), ghost-node Neumann/Robin BCs (sec 2.12), ADI (sec 11.6.1). DOI 10.1137/1.9780898717839.
- Strikwerda, J. C. (1989/2004). Finite Difference Schemes and Partial Differential Equations, 2nd ed. SIAM. - von Neumann stability of FTCS for the heat equation; the 2D limit alpha*dt*(1/hx^2+1/hy^2) <= 1/2 (Thm 6.3.1). DOI 10.1137/1.9780898717938.
- Peaceman, D. W., & Rachford, H. H. (1955). The Numerical Solution of Parabolic and Elliptic Differential Equations. J. SIAM 3(1), 28-41. - the alternating-direction implicit (ADI) method. DOI 10.1137/0103003.
- Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids, 2nd ed. Oxford. - separable eigenfunction (sine) solutions and their exponential decay rates for the diffusion equation.
- Crank, J. (1975). The Mathematics of Diffusion, 2nd ed. Oxford. - separable sine-series solutions T = sum sin(n pi x) exp(-D n^2 pi^2 t) on a finite domain.
- Incropera, F. P., & DeWitt, D. P. (2007). Fundamentals of Heat and Mass Transfer, 6th ed. Wiley. - convective (Robin) boundary condition -k dT/dn = h(T - T_inf).
- Press, W. H., et al. (2007). Numerical Recipes, 3rd ed. CUP. sec 2.4 - the Thomas algorithm for tridiagonal systems.
Prototype conduction with a proof
Run the solve in your browser with nothing uploaded, then save the field and error report to a Sandbox workspace or attach it to a Worklog case for review.