Difference between Crank-Nicolson and Peaceman-Rachford Schemes with ADI

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In summary: This will result in a more accurate approximation of the 2D heat equation, but at the cost of more complex implementation.
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tlonster
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Hi,

I am in a numerical methods for PDE's class and have just been introduced to the ADI method. My teacher has given a general 2D heat equation to numerically compute using ADI with Crank-Nicolson and ADI with Peaceman-Rachford schemes. I understand what ADI is doing, I just don't know what the difference is between these two schemes when ADI is implemented is. I'm missing something here...

I have of course scoured the internet for papers/textbooks that will make me understand how they are different, but all I am getting out of my research is that the CN scheme allowed for ADI and thus, the PR scheme to develop. Basically I am understanding it as if you implement ADI with CN you have the PR scheme. Is that wrong? I then just keep seeing the steps of ADI (i.e. take half points, keep x fixed, keep y fixed etc.). What is different? Do you think the wording of the problem statement is incorrect (like, it should just be use 2D CN, and then use ADI with PR) or is there a distinct difference? Please, if you are familiar with this stuff I would appreciate a response!
 
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The main difference between the Crank-Nicolson and the Peaceman-Rachford schemes is that the Crank-Nicolson scheme uses a linear approximation to the time derivatives, while the Peaceman-Rachford scheme uses a nonlinear approximation. This means that the Peaceman-Rachford scheme is more accurate than the Crank-Nicolson scheme, but it also means that it is more difficult to implement. When you are using the ADI method to solve the 2D heat equation, the Crank-Nicolson scheme is used to approximate the time derivatives in the x-direction, while the Peaceman-Rachford scheme is used to approximate the time derivatives in the y-direction. So when you are asked to use the ADI method with both the Crank-Nicolson and Peaceman-Rachford schemes, what is being asked of you is to use the Crank-Nicolson scheme in the x-direction and the Peaceman-Rachford scheme in the y-direction.
 

1. What is the Crank-Nicolson scheme?

The Crank-Nicolson scheme is a finite difference method used to solve partial differential equations. It is a combination of the explicit and implicit methods, and is known for its second-order accuracy in time and space. It is widely used for solving parabolic equations, such as the heat equation.

2. What is the Peaceman-Rachford scheme?

The Peaceman-Rachford scheme is another finite difference method used to solve parabolic partial differential equations. It is an iterative method that was developed specifically for solving the diffusion equation. It is also a combination of explicit and implicit methods, but has a first-order accuracy in time and second-order accuracy in space.

3. What is the difference between Crank-Nicolson and Peaceman-Rachford schemes?

The main difference between these two schemes is their accuracy. The Crank-Nicolson scheme has a higher accuracy (second-order) in both time and space, while the Peaceman-Rachford scheme has a lower accuracy (first-order) in time and higher accuracy (second-order) in space. Additionally, the Peaceman-Rachford scheme is specifically designed for solving the diffusion equation, while the Crank-Nicolson scheme can be used for a wider range of parabolic equations.

4. What is the ADI method?

The Alternating Direction Implicit (ADI) method is a numerical technique used to solve partial differential equations. It is a type of finite difference method that allows for the efficient solution of equations with high spatial dimensions. The ADI method solves the equations in one direction at a time, alternating between the x and y directions.

5. How do the Crank-Nicolson and Peaceman-Rachford schemes with ADI differ?

The main difference between these two schemes is their order of accuracy. The Crank-Nicolson scheme with ADI has a second-order accuracy in both time and space, while the Peaceman-Rachford scheme with ADI has a first-order accuracy in time and second-order accuracy in space. Additionally, the Peaceman-Rachford scheme with ADI is specifically designed for solving the diffusion equation, while the Crank-Nicolson scheme with ADI can be used for a wider range of parabolic equations.

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