Algorithms and Parallel Computing

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 ...