Explicit Finite Differences for the Diffusion Equation and Stability
Summary
The document derives an explicit finite-difference scheme for the one-dimensional heat equation. It approximates the time derivative with a forward difference and the spatial second derivative with a centered difference, then advances the solution using values from the current time level. A small discrete initial profile illustrates the time-marching process.
The example shows that choosing the time step equal to the squared spatial spacing produces oscillatory values that do not resemble physical diffusion. The article gives the stability condition that the time-step-to-spacing-squared ratio must be no greater than one half, and demonstrates a stable case at the limiting ratio, where the heat profile spreads smoothly without negative values. It cautions that this restriction forces very small time steps on refined spatial grids. The stability result is stated rather than proved, and the discussion concerns a simple diffusion equation rather than a financial pricing application; an implicit method is identified as a subsequent remedy.
Key ideas
- The explicit scheme combines a forward time difference with a centered spatial second difference.
- Each time step is computed from known values at the preceding time level.
- Stability requires the time step divided by the squared spatial spacing to be at most one half.
- A larger time step can produce oscillatory values that fail to represent diffusion.
- The explicit stability limit can make fine-grid computations expensive in time steps.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.