2.5.2.1.1. Finite Difference Method

The most common discretization techniques available for the numerical solution of partial differential equations are the finite difference method. The second is finite element method and another one is finite volume method.

The Finite Difference Method (FDM) is one of the primary techniques used to analyze partial differential equations by converting them into algebraic equations. By using FDM, engineers can approximate continuous functions at specific grid points in a flow field. This is essential in computational fluid dynamics because it allows us to solve complex fluid flow problems numerically.

Imagine you are trying to find your way around a city. You could either explore the whole city, which might take a long time, or you could use a map with marked points (grid points) to find your way. The finite difference method is like using the map; it helps you approximate the way through the city by breaking it down into manageable sections.

In finite difference method, the flow field is dissected into a set of grid points. And the continuous functions are approximated by discrete values of these functions calculated at grid points.

The FDM works by taking a continuous flow field and breaking it into discrete sections, identified as grid points. Each grid point represents a specific location where we calculate fluid properties, such as velocity or pressure. This discretization is necessary because it converts complicated equations into simpler forms that can be solved using numerical techniques.

Think of a long road. Instead of viewing the entire road at once, you take several snapshots at regular intervals (grid points). Each snapshot provides information about the condition of the road at that point, rather than trying to observe everything at once, which might be overwhelming.

What we do is calculate the value suppose the velocity and pressure we will calculate at this point using our conversion of partial differential equation to the algebraic equation.

Using the finite difference method, we substitute values at grid points into algebraic equations that approximate the original differential equations. For example, if we want to calculate fluid velocity at a particular grid point, we can derive this from the neighboring grid points using a simple formula. This process helps in understanding the behavior of fluids at different locations in the flow field.

Imagine you're trying to determine the temperature at your home by taking measurements from your kitchen, living room, and bedroom. While you might have a general idea of the temperature in your home based on these readings, observing the exact temperature at every single point can be impractical. The finite difference method allows us to find approximate temperatures based on nearby measurements.

If this is I, j, this point is going to be i, j + 1, this point is going to be i + 1, j + 1.

In the finite difference method, the arrangement of the grid points is critical. The notation (i, j) represents the position of a grid point in a two-dimensional matrix. Using this systematic notation, we can easily identify and locate the neighboring grid points, which are essential for performing calculations based on surrounding values.

Picture a chessboard. Each square can be identified by its row and column (like grid points). When determining the position of a piece, you refer to its specific location and also consider the positions of pieces around it. This helps you make better decisions about your next move, similar to how grid points help in computing fluid behavior based on nearby points.

This process of distribution was developing a set of algebraic equation based on discrete points in the flow domain.

Discretization is crucial in the finite difference method because it transforms the continuous flow equations into numerical ones that can be solved on a computer. By approximating the behavior of fluid dynamics at discrete points, engineers can efficiently conduct simulations and analyze different flow scenarios. This process makes it practical to handle complex fluid scenarios that would be infeasible with direct observational techniques.

Consider a theater stage where a magician performs tricks. Instead of changing the entire stage setup for each magic act, the magician uses specific props strategically placed around the stage (discrete points) to create the illusion. By manipulating these props, the magician can create diverse effects without needing constant major transformations.

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