| Case | L | Nodes | ε | Source model | Boundary values | Typical midpoint V |
|---|---|---|---|---|---|---|
| Uniform source benchmark | 1.0 | 41 | 1.0 | Constant, ρ0 = 1 | V(0)=0, V(L)=0 | About 0.125 |
| Linear source ramp | 1.0 | 61 | 2.0 | Linear, ρ0 = 0.5, slope = 2 | V(0)=0, V(L)=1 | Depends on slope and boundary bias |
| Localized charge packet | 2.0 | 81 | 1.5 | Gaussian, amplitude = 3 | V(0)=0, V(L)=0 | Peak shifts near the source center |
This calculator solves the one dimensional Poisson equation:
d²V/dx² = -ρ(x)/ε
The domain is divided into equal spacing
h = L / (N - 1).
Interior nodes use the second order central difference form:
(Vi-1 - 2Vi + Vi+1) / h² = -ρi / ε
Rearranged discrete equation:
Vi-1 - 2Vi + Vi+1 = -(h²/ε) ρi
The electric field is estimated from the solved potential:
E = -dV/dx
Direct mode solves the tridiagonal matrix exactly. Gauss-Seidel mode iterates until the update falls below tolerance.
- Enter the domain length and total grid nodes.
- Set the left and right boundary potentials.
- Choose the source profile that matches your physics problem.
- Enter source parameters such as amplitude, slope, mode, or Gaussian width.
- Select either the direct solver or the Gauss-Seidel iterative solver.
- Press Solve Equation to show the graph and numerical table above the form.
- Review the residual values to check numerical quality.
- Export the solved data as CSV or PDF for reporting.
1) What does this solver calculate?
It computes a one dimensional potential field that satisfies the Poisson equation under fixed boundary values. It also estimates electric field and residuals from the discrete solution.
2) When should I increase the node count?
Increase nodes when the source varies rapidly, the Gaussian is narrow, or you want lower discretization error. Finer grids usually improve accuracy but increase runtime and table size.
3) What boundary conditions are used here?
This tool uses Dirichlet boundary conditions. You specify the potential at the left end and the right end, and the interior values are solved from those fixed limits.
4) What does permittivity change in the result?
Larger permittivity reduces curvature caused by the same source distribution. Smaller permittivity produces stronger bending in the potential profile for identical charge density inputs.
5) Why would I use Gauss-Seidel instead of the direct method?
Gauss-Seidel is useful for learning convergence behavior and iterative solvers. The direct tridiagonal method is usually faster and more precise for this one dimensional problem.
6) What does the residual value mean?
Residual measures how closely the discrete solution satisfies the finite difference equation. Smaller residuals indicate better numerical consistency between the solved profile and the source term.
7) Can I model zero source regions?
Yes. Select the zero source option to solve Laplace type behavior over the same domain. That is useful for checking boundary driven potential shapes without internal forcing.
8) What is exported in the CSV and PDF files?
Both exports summarize the solver settings and computed metrics. The CSV also includes node by node values for position, source, potential, electric field, and residual.