The Finite Difference Method of Discretization in CFD

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Key Takeaways

The numerical method of solving differential equations by approximating them with difference equations is called the finite difference method.
The finite difference method can easily obtain high-order approximations.
The finite difference method requires a structured grid.

Different discretization methods such as the finite element method, finite volume method, and finite difference method are used in CFD modeling

In engineering problems, we often come across linear and non-linear differential equations. These governing equations must be solved, and the analytical method of solving them can not be used due to their complexity. However, it is possible to use computational fluid dynamic techniques to obtain computer-based solutions for these complex equations. Different discretization methods, such as the finite element method, finite volume method, and finite difference method, can be used in the CFD modeling of these engineering problems. In this article, we will explore the finite difference method of discretization in detail.

Solving Complex Equations With CFD

Laws of physics are associated with various physical phenomena like heat transfer and fluid flow. They need to be represented as mathematical equations to simulate these phenomena over various length scales. Take fluid dynamics, for example: governing equations are obtained by applying the fundamental laws of mechanics to the fluid. The conservation of mass equation, the conservation of momentum equation, and the conservation of energy equation form a set of coupled, non-linear partial differential equations–collectively called the Navier-Stokes equations, which describe fluid flow behavior.

The complicated or coupled nature of partial differential equations makes solving them analytically difficult. Numerical techniques have been developed to solve such complex differential equations. In physical problems related to fluid flow, heat transfer, and aerodynamics, CFD is prominent in finding accurate solutions. In CFD problem-solving, a physical phenomenon is mathematically modeled and solved using numerical techniques.

Discretization

Numerical techniques are incorporated in CFD exclusively for solving differential equations. The fundamental principle of including numerical methods for solving partial differential equations is to bring the idea of discretization, one of the key aspects of a numerical solution strategy. Discretization converts the governing equations into a set of simple equations.

There are different methods to discretize the complex governing equations, called discretization methods. A few popular discretization methods utilized in CFD tools are the finite element method, finite volume method, and finite difference method.

We will take a deep dive into the finite difference method in the upcoming section.

The Finite Difference Method

The numerical method of solving differential equations by approximating them with difference equations is called the finite difference method. For the purpose of discretization, the derivatives in the governing equations use the truncated Taylor series expansion in the finite difference method. The implementation of this method is easy and is one of the widely-used approaches to solving partial differential equations numerically.

The finite difference method is based on the secant line approximation of the derivatives. In the place of actual derivatives, the finite difference method uses the difference approximation. Use of the finite difference numerical method results in the generation of a set of algebraic equations that can be solved for the dependent variables. The set of algebraic equations is solved at the discrete grid points in the physical domain under consideration.

To formulate the finite difference scheme, the first step is to discretize the domain into grid points. At each of these grid points, the derivatives are written in the form of differences. The differences can be the central difference, backward difference, or forward difference. The differences relate the values of the variable at each grid point to its neighboring points. After completing this process on all the grid points in the physical domain under consideration, a set of equations are obtained. By numerically solving this set of algebraic equations, the solution for the partial differential equation is determined.

Here is a table that clearly lays out the advantages and disadvantages of the finite difference method.

Advantages	Disadvantages
This method is fairly simple.	If not given special care, the physical principle of conservation is not automatically enforced in the finite difference method.
Can easily obtain high-order approximations.	This method can be implemented only on simple geometries since the method cannot be implemented with convenience on unstructured grids.
Gives higher-order accuracy of the spatial discretization.	The finite difference method requires a structured grid.

Classification of Difference Formulae

The difference formulae can be classified into two based on:

The geometrical relationship of the neighboring grid points. They are central, forward, and backward differences.
The accuracy of the expression. The forward and backward differences are first-order accurate, whereas the central difference is second-order accurate.

The fundamental philosophy of the finite difference method is to replace the derivates in the governing equations with algebraic differences. The complete set of CFD simulation software from Cadence can support you when solving physical phenomena such as heat transfer, fluid flow, and aerodynamics.

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