10 Mar 2026

Finite Difference Method

Replacing derivatives by difference formulas on a grid for numerical solution of differential equations.

msc semester-i numerical-methods finite-difference

The finite difference method replaces derivatives by differences between function values at nearby grid points. It is widely used for boundary value problems and partial differential equations.

Grid points

Let the grid points be

\[x_i=x_0+ih,\]

where $h$ is the spacing and $y_i=y(x_i)$.

Forward difference

The first derivative may be approximated by

\[\frac{dy}{dx}\bigg|_{x_i} \approx \frac{y_{i+1}-y_i}{h}.\]

This is called the forward difference formula.

Backward difference

Another approximation is

\[\frac{dy}{dx}\bigg|_{x_i} \approx \frac{y_i-y_{i-1}}{h}.\]

This is the backward difference formula.

Central difference

The central difference formula is

\[\frac{dy}{dx}\bigg|_{x_i} \approx \frac{y_{i+1}-y_{i-1}}{2h}.\]

It is usually more accurate than forward or backward difference for smooth functions.

Second derivative

A common formula for the second derivative is

\[\frac{d^2y}{dx^2}\bigg|_{x_i} \approx \frac{y_{i+1}-2y_i+y_{i-1}}{h^2}.\]

Key points

• Derivatives become algebraic expressions involving neighboring grid values.
• Smaller grid spacing usually improves accuracy, but may increase round-off effects.
• Boundary conditions are essential for solving differential equations by finite differences.