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

A derivative is defined as the limiting value of a difference quotient. The finite difference method keeps a small but finite spacing and uses nearby function values to approximate derivatives. In this way a differential equation is converted into algebraic equations on a grid.

This method is widely used for boundary value problems and partial differential equations because values at neighboring grid points can be related directly.

Grid points

Let the grid points be

\[x_i=x_0+ih,\]

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

The choice of $h$ controls the resolution of the grid. A smaller $h$ usually gives a better approximation, but it also increases the number of unknowns.

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

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

Derivatives from nearby values

Let

\[y=x^2.\]

Estimate the first and second derivatives at $x=1$ using $h=0.1$. The required values are

\[y(0.9)=0.81,\qquad y(1.0)=1.00,\qquad y(1.1)=1.21.\]

The central difference estimate for the first derivative is

\[\frac{dy}{dx}\bigg|_{x=1} \approx \frac{1.21-0.81}{2(0.1)}=2.0.\]

The second derivative estimate is

\[\frac{d^2y}{dx^2}\bigg|_{x=1} \approx \frac{1.21-2(1.00)+0.81}{(0.1)^2}=2.0.\]

These agree with the exact derivatives of $x^2$ at $x=1$.

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.

Practice questions

1. Write the forward, backward, and central difference formulas for the first derivative.
2. Derive the central difference formula for the second derivative.
3. Estimate $dy/dx$ for $y=x^2$ at $x=1$ using $h=0.1$.
4. Explain why the central difference formula is usually more accurate than the forward difference formula.
5. Why are boundary conditions necessary in finite difference problems?