Supported expression syntax
Use variables such as x, y, t, and u. Operators include addition, subtraction, multiplication, division, and powers. Available functions include sine, cosine, tangent, exponential, logarithm, square root, absolute value, minimum, maximum, and power.
Examples include sin(pi*x), exp(-x^2), and x*t+u. Constants pi and e are available. Keep formulas explicit and mathematically finite throughout the chosen domain.
How to use the solver
Select the problem type matching your equation. Enter the domain, grid, coefficients, and conditions. Then calculate and review the stability indicators before interpreting results.
Use finer grids to investigate convergence. Compare numerical values against an exact solution when possible. Export the table after confirming appropriate discretisation and boundary settings.
Example data
| Mode | Example equation or function | Suggested setup | Expected behaviour |
|---|---|---|---|
| Derivative | f(x)=sin(x) | x=1, h=0.01, central | Approaches cos(1) |
| Boundary value | y″=-π²sin(πx) | y(0)=0, y(1)=0 | Approaches sin(πx) |
| Heat | u_t=u_xx | u(x,0)=sin(πx) | Amplitude decays exponentially |
| Wave | u_tt=u_xx | u(x,0)=sin(πx) | Profile oscillates in time |
| Laplace | u_xx+u_yy=0 | Top boundary 100 | Smooth steady interior field |
| Advection–diffusion | u_t+u_x=0.01u_xx | Gaussian initial pulse | Pulse moves and spreads |
Frequently asked questions
What is the finite difference method?
It replaces derivatives with differences between neighbouring grid values. The differential equation becomes algebraic equations. Solving those equations produces an approximate numerical field.
Which stencil should I choose?
Central stencils usually provide better accuracy inside smooth domains. Forward and backward stencils suit one-sided locations. Five-point formulas need more neighbouring values and smoothness.
How many grid points are enough?
No universal grid works for every problem. Repeat calculations using progressively smaller spacing. Stable converging results indicate adequate grid resolution.
What is a stability condition?
A stability condition limits allowable time and space steps. Violating it may amplify numerical errors. Always inspect Courant and Fourier numbers before accepting results.
What boundary conditions are supported?
One-dimensional solvers support Dirichlet, Neumann, and Robin conditions. The two-dimensional solver uses fixed edge values. Mixed boundaries can model many physical constraints.
Why can an iterative solver fail?
The matrix may lack favourable convergence properties. Tolerance can also be unrealistically strict. Try another solver, relaxation factor, or finer modelling setup.
What does truncation error mean?
Truncation error comes from replacing continuous derivatives with finite formulas. Smaller spacing usually reduces this error. Round-off error may eventually limit further improvement.
Can I enter nonlinear equations?
The included modes primarily solve linear discretised equations. Source expressions may depend on current u explicitly. Fully nonlinear systems require Newton or specialised iterative methods.
Are these results suitable for design decisions?
The calculator supports learning and preliminary numerical investigation. Validate models against trusted software and measurements. Qualified review remains essential for safety-critical engineering decisions.