Solving Diffusion Equation u_t=div(A\nabla u) Numerically

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In summary, the diffusion equation, also known as the heat equation, is used to model the flow of heat or other quantities that exhibit diffusion behavior. It can be solved numerically using methods such as finite difference, finite element, or spectral methods. The A variable represents the diffusivity or diffusion coefficient, while the u variable represents the concentration or value of the quantity at a given point in space and time. Boundary conditions play a crucial role in the numerical solution and can affect stability and accuracy. The diffusion equation has various applications in fields such as physics, chemistry, biology, and engineering.
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shayj
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Can someone offer some help on this?

I have this diffusion equation:
u_t = div(A\nabla u)
where A is a 2 by 2 matrix, i.e. it's anisotropic.

I am not sure how I should properly discretize this and solve it numerically. I used simply central difference but it does seem work nicely.

Any help will be greatly appreciated.
 
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Try finite elements instead. You will only need first derivatives, and the discretization works better
 

1. What is the diffusion equation used for?

The diffusion equation, also known as the heat equation, is used to model the flow of heat or other quantities that exhibit diffusion behavior, such as the spread of chemicals or particles through a medium.

2. How is the diffusion equation solved numerically?

The diffusion equation can be solved numerically using methods such as finite difference, finite element, or spectral methods. These methods discretize the equation and solve it for a finite number of points in space and time.

3. What is the purpose of the A and u variables in the diffusion equation?

The A variable, also known as the diffusivity or diffusion coefficient, represents the rate at which the quantity being diffused moves through the medium. The u variable represents the concentration or value of the quantity at a given point in space and time.

4. How do boundary conditions affect the numerical solution of the diffusion equation?

Boundary conditions, which specify the behavior of the solution at the boundaries of the domain, play a crucial role in the numerical solution of the diffusion equation. They can affect the stability and accuracy of the solution, and different types of boundary conditions (e.g. Dirichlet, Neumann, Robin) may require different numerical approaches.

5. What are some applications of solving the diffusion equation numerically?

The diffusion equation and its numerical solution have a wide range of applications in various fields such as physics, chemistry, biology, and engineering. Some examples include modeling heat transfer in buildings, predicting the spread of air pollutants, and simulating drug diffusion in the human body.

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