Model nonlinear diffusion, convection, and reaction across domains. Inspect stability, errors, boundary effects, and trends. Download results, tables, and charts for deeper review later.
This page keeps a single-column structure overall. The input grid becomes three columns on large screens, two on smaller screens, and one on mobile.
This calculator approximates a one-dimensional nonlinear PDE with diffusion, convection, logistic reaction, and an optional power source term.
ut = ∂/∂x [ D(u) ux ] - c ux + r u (1 - u/K) + q sign(u)|u|pThe nonlinear diffusion coefficient is:
D(u) = d0 [ 1 + α |u|n ]The explicit finite difference update at grid point i is:
uik+1 = uik + Δt ( Diffusion + Convection + Reaction + Source )The diffusion term uses midpoint fluxes to handle the nonlinear coefficient more smoothly:
Diffusion ≈ [ Di+1/2(ui+1-ui) - Di-1/2(ui-ui-1) ] / Δx2The displayed CFL indicators help judge whether the chosen step sizes are conservative enough for a stable explicit run.
Enter the spatial domain length, final time, number of spatial nodes, and number of time steps. Higher resolution usually improves accuracy but increases computation.
Choose the base diffusion, nonlinear diffusion strength, diffusion exponent, convection speed, reaction rate, carrying capacity, source coefficient, and source power.
Pick a Gaussian, sine, step, or pulse profile. Then define amplitude, baseline, center location, and width or spread.
Use Neumann for zero-gradient edges, Dirichlet for fixed endpoint values, or periodic if the left and right edges should wrap together.
After submission, the result appears above the form. Review the graph, summary metrics, computed table, and stability indicators before exporting CSV or PDF.
This illustrative table comes from the built-in sample scenario and shows how the initial profile evolves after the selected final time.
| x | Initial u(x,0) | Final u(x,T) | Change |
|---|---|---|---|
| 0.0000 | 0.030750 | 0.060282 | 0.029532 |
| 0.2000 | 0.249352 | 0.282507 | 0.033155 |
| 0.4000 | 0.856997 | 0.804324 | -0.052673 |
| 0.6000 | 0.856997 | 0.827769 | -0.029228 |
| 0.8000 | 0.249352 | 0.313441 | 0.064089 |
| 1.0000 | 0.030750 | 0.066822 | 0.036072 |
Sample settings: Gaussian pulse, nonlinear diffusion, mild convection, logistic reaction, and Neumann boundaries.
It solves a one-dimensional nonlinear evolution equation that combines diffusion, convection, logistic reaction, and an optional power-law source term using a finite difference approximation.
No. It is a numerical solver. It estimates values on a grid over time rather than producing a closed-form analytical expression.
The scheme is explicit, so overly large time steps can cause oscillation or blow-up. The CFL indicators offer a quick practical check for stability risk.
Use Neumann boundaries when you want zero-gradient edges, which is common for insulated, no-flux, or symmetry-based boundary assumptions.
It controls how strongly the diffusion coefficient changes with the current solution magnitude. Larger values make the diffusion response more sensitive to amplitude.
Mass changes when convection, reactions, source forcing, or fixed-value boundaries add or remove content. Pure diffusion with conservative boundaries behaves differently.
Reduce the time step, increase spatial resolution, compare repeated runs, and verify that the displayed stability indicators remain in a conservative range.
Yes, but sharp fronts often need finer grids and smaller time steps. Otherwise, numerical diffusion or instability may distort the solution.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.