21.1 INTRODUCTION

Finite difference methods (FDMs) are used for numerical simulation of many important applications in science and engineering. Examples of such applications include

• Air flow in the lungs
• Blood flow in the body
• Air flow over aircraft wings
• Water flow around ship and submarine hulls
• Ocean current flow around the globe
• Propagation of sound or light waves in complex media

FDMs replace the differential equations describing a physical phenomenon with finite difference equations. The solution to the phenomenon under consideration is obtained by evaluating the variable or variables over a grid covering the region of interest. The grid could be one-, two-, or three-dimensional (1-D, 2-D, and 3-D, respectively) depending on the application. An example of 1-D applications is vibration of a beam or string; 2-D applications include deflection of a plate under stress, while 3-D applications include propagation of sound underwater.

There are several types of differential equations that are encountered in physical systems [48, 130, 131]:

Boundary value problem:

(21.1)

where v_x = dv/dx, v_xx = d²v/dx², and f is a given function in three variables and v is unknown and depends on x. The associated boundary conditions are given by

(21.2)

(21.3)

where v₀ is the value of variable v at the boundary x = 0 and v₁ is the value of variable v at the boundary x = 1.

Elliptic partial differential equation (Poisson equation):

(21.4)

(21.5)

These equations describe the electrical potential and heat distribution at steady state. For the 1-D case, when f(x) = 0, the above equation is called the Laplace equation. For the 2-D, Laplace equation results when f(x, y) = 0.

Parabolic partial differential equation (diffusion equation):

(21.6)

where a is a constant and v_xx = ∂²v/∂x² and v_t = ∂v/∂t. This equation describes gas diffusion and heat conduction in solids in 1-D cases such as rods.

Hyperbolic partial differential equation (wave equation):

(21.7)

where v_tt = ∂²v/∂t². This equation describes the propagation of waves in media such as sound, mechanical vibrations, electromagnetic radiation, and transmission of electricity in long transmission lines.

In the following section, we will study the wave equation as an example. The analysis can be easily applied to the other types of differential equations.