Calculator Inputs
Formula Used
Heat: u(i,n+1) = u(i,n) + r[u(i+1,n) - 2u(i,n) + u(i-1,n)] + dt·S.
Wave: u(i,n+1) = 2u(i,n) - u(i,n-1) + λ²[u(i+1,n) - 2u(i,n) + u(i-1,n)] + dt²·S.
Laplace: u(i,j) is updated from neighboring cells until the maximum change is below tolerance.
The calculator uses finite difference approximations. Smaller grid steps usually improve detail but require more calculation time.
How to Use This Calculator
- Select heat, wave, or Laplace mode.
- Enter domain size, grid nodes, time steps, coefficients, and boundary values.
- Choose an initial profile for heat or wave calculations.
- Press the solve button. Results appear above the form and below the header.
- Review the stability note, chart, and table.
- Use the CSV or PDF buttons to save the current result.
Example Data Table
| Use case | Equation | Suggested grid | Key coefficient | Boundary example |
|---|---|---|---|---|
| Cooling rod | Heat, 1D | 41 nodes, 500 steps | alpha = 0.15 | Left 0, right 0 |
| String vibration | Wave, 1D | 61 nodes, 700 steps | c = 1.2 | Fixed ends |
| Plate potential | Laplace, 2D | 31 by 31 nodes | S = 0 | Top 10, others 0 |
| Poisson field | Poisson, 2D | 41 by 31 nodes | S = 2 | Mixed edge values |
Guide to PDE Solver Calculations
Why PDE Models Matter
Partial differential equations describe change across space and time. They appear in heat flow, waves, fluids, finance, and image processing. A solver turns the continuous problem into a grid. Each grid point stores an estimated value. The calculator then updates those values with finite difference rules.
Heat Solver
The heat option uses the explicit FTCS method. It estimates how temperature spreads from one cell to the next. The stability number r controls the result. Small time steps and larger spatial steps usually improve stability. If r is too high, the model may oscillate or grow without meaning.
Wave Solver
The wave option models motion traveling through a line. It uses a second order time rule. The Courant number explains whether the time step is safe. Values near or below one are often preferred for this simple scheme. Boundary values hold the ends steady.
Laplace Solver
The Laplace option solves a steady surface. It repeatedly averages neighboring cells. This is useful for potential fields, plates, and smooth equilibrium shapes. More iterations usually improve convergence. A tighter tolerance stops the loop when change becomes small.
Practical Workflow
This page is useful for quick experiments. You can change the domain length, total time, grid size, coefficients, and boundary values. You can also select an initial profile. The result table shows final values. The chart helps reveal patterns that are hard to see in raw rows.
Accuracy Notes
Finite difference solvers are approximations. They do not replace specialist simulation software. They are best for teaching, planning, checking assumptions, and building intuition. Always compare important results with trusted references or refined grids.
Best Practice
A good workflow starts with a simple case. Use a smooth initial profile. Run a coarse grid first. Then increase the grid count and compare outputs. If the answer changes greatly, refine the setup. Watch the stability notes beside the result. Export the CSV or PDF when the result is ready for a report. For deeper checks, compare methods against known textbook examples. Try zero boundaries first. Then add unequal boundary values. Observe how the curve bends. Keep units consistent. Do not mix meters with centimeters. Name each run clearly before exporting files. These notes make review easier and reduce mistakes during checks.
FAQs
What does this PDE solver calculate?
It solves simple heat, wave, Laplace, and Poisson style models with finite difference methods. It gives grid values, summaries, and a chart.
Is this calculator suitable for advanced simulations?
It is useful for study, prototyping, and checking ideas. For critical engineering work, compare results with validated software and refined meshes.
What is the heat stability number?
The heat stability number is r = alpha·dt/dx². In this explicit method, values at or below 0.5 are usually safer.
What is the Courant number?
The Courant number is c·dt/dx for the wave solver. Values above one can make the simple central scheme unstable.
How does the Laplace option stop?
It stops when the largest cell change is below the chosen tolerance. It also stops if the maximum iteration limit is reached.
Can I export the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a report-ready summary and computed table.
Why do results sometimes look unstable?
Large time steps, small spatial steps, or high coefficients can break simple explicit schemes. Check the stability note after solving.
Which grid size should I choose?
Start with a moderate grid, such as 31 nodes. Then refine the grid and compare changes before trusting final values.