Dividing the problem into discrete rectangles makes calculating each piece easier.
When researching a problem that involves fluids, whether it be the flow of air past an object or sea waves being cut by a ship, the inherently difficult equations that must be solved are further complicated by the irregular geometry of the real life situation. One common method for dealing with these boundaries is to introduce a Cartesian, usually rectangular, grid. The equations can then be solved within each rectangle, simplifying the overall calculation.
Immersed Boundary Method
One of the most popular methods is the Immersed Boundary Method introduced by Charles Peskin. Originally used to study how blood moves around the heart valves, this method starts by expressing the equations of motion in Langrangian variables but eventually introduces Eulerian variables in such a way that the elasticity equations and fluid dynamic equations have a similar form. This makes calculating the interaction between fluids and objects easier and is the key to this approximation method.
Finite Difference Methods
Finite difference methods replace derivatives with a finite difference scheme, calculated over a grid. For example, instead of solving a differential equation directly, one might set up an equation for the slope of the unknown function at the center of each grid element. Solving the system of equations that this sets up will give an approximation of the function at the center of each grid element. Many finite difference methods exist, with varying accuracy and computational complexity.
Hybrid Methods
While the Cartesian grid simplifies computations for grid elements that do not touch the boundary, the elements that do intersect with the irregular boundary still have a non-uniform shape. To deal with these boundaries, some researchers have implemented hybrid methods that only use the Cartesian grid in the interior, and opt for gridless methods immediately next to the boundary.
Navier Stokes
The difficulty in solving fluid dynamic problems comes from the fact that the equations of motion for a fluid element, known as the Navier-Stokes equations, have not been solved in general. While it can be written as a single vector equation, it actually represents three non-linear, coupled, partial differential equations. Only a few simplified problems in two dimensions have been solved exactly, which is why numerical methods such as the Cartesian grid are so important in fluid dynamics.
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