# Application Examples

One of the most common categories of application that frequently uses parallel computing is the numerical solution of partial differential equations. A partial differential equation, or *PDE*, is an equation containing derivatives of a function of two or more variables. This is in contrast to ordinary differential equations, which are equations containing derivatives of a function of one variable.

Some examples of phenomena that are modeled by PDEs include:

- Air flow over an aircraft wing
- Blood circulation in the human body
- Water circulation in an ocean
- Bridge deformations as its carries traffic
- Evolution of a thunderstorm
- Oscillations of a skyscraper hit by an earthquake
- Strength of a toy

Partial differential equations are solved numerically by several approaches. Two particularly common ones are the *finite element method* (FEM) and the *finite difference method*. A simple conceptual explanation of the difference between these two methods is that finite elements approximates the solution and attempts to fit it to the conditions, whereas the finite-difference method approximates the equations and solves a discrete version of the original system.

We will focus on examples that use finite differencing, since this method is widely used in some areas and is a typical application for point-to-point messaging because it generally involves solving the equations on some form of grid, as we have been studying.

## Linear Second-Order PDEs

Many physical phenomena can be modeled by linear second-order PDEs. Consider such an equation in the two variables $x$ and $y$. The general form is

$$A \dfrac{\partial^{2}{u}}{\partial{x}^{2}} + 2B \dfrac{\partial^{2}{u}}{\partial{x}\partial{y}}+C\dfrac{\partial^{2}{u}}{\partial{y}^{2}} + D \dfrac{\partial{u}}{\partial{x}}+E\dfrac{\partial{u}}{\partial{y}}+Fu=G$$

where A, B, C, D E, F, and G are functions of $x$ and $y$ only. The behavior, and appropriate numerical methods, depends only on $B^2-AC$. There are three possibilities:

### Elliptic Equations

In this case, $B^2-AC \lt 0$. This type of equation is often solved by some iterative method. Elliptic equations occur widely in physical models.

### Parabolic Equations

Here $B^2-AC = 0$. The most widely known example of a parabolic equation is the *diffusion equation* that describes the diffusion of heat or fluids across a gradient.

### Hyperbolic Equations

For this category, $B^2-AC \gt 0$. Hyperbolic equations generally represent wave phenomena.