PDE Solver
Choose one of three preset equation classes, configure its grid and boundary conditions, and inspect the numerical approximation.
A QUICK WALKTHROUGH
How to use this tool
- Choose heat, wave or Laplace and enter valid parameters.
- Select an initial profile or fixed boundary values, then start or solve.
- Pause or reset; read numerical step size, limits and convergence information.
Exact supported scope
Heat: u_t=αu_xx with FTCS; wave: u_tt=c²u_xx with central time differences and zero initial velocity. The domain is x∈[0,1], with fixed zero, reflecting zero-gradient or periodic boundaries. Periodic grids omit the duplicated endpoint. Laplace: u_xx+u_yy=0 on the unit square, with constant Dirichlet values on four sides and averaged corners. No arbitrary equation, forcing, mesh or expression parser is provided.
Numerics and limits
Evolution grids:20–256 points, ≤10,000 steps, heat α=0.0001–10 with Δt=0.4Δx²/α; wave c=0.1–10 with Δt=0.9Δx/c and a half-acceleration first step. Laplace grids:8–64, Gauss-Seidel or SOR ω=1.5, tolerance0.01/0.001/0.0001 on maximum update, ≤20,000 sweeps or 8 seconds of computation. Maximum discrete PDE residual is reported separately. Iteration limits are not convergence. No continuum error bound or engineering qualification is claimed.
GOOD TO KNOW
Common questions
Are open ends absorbing?
No. The zero-gradient option is a reflecting Neumann boundary, not a non-reflecting radiation boundary. Initial endpoint values are adjusted to fixed zero when that option is chosen.
Is a converged Laplace result exact?
No. Convergence here means the maximum iterative update meets the selected tolerance. It does not bound discretization or continuum error. Compare grids and suitable analytical solutions independently for serious use.