HPC Tier Module
2D Heat Equation (Transient)
Solve the 2D transient heat equation in your browser, checked against an exact analytic eigenmode.
This tool runs inside GDBS HPC. What it computes and the authorities it applies are below.
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.
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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.