# Stability condition for solving convection equation by FDM

• nazmulislam
In summary, the stability condition for solving the Convection-Diffusion equation using Finite Difference explicit/implicit technique is dependent on the FD approximation of the laplacian. The condition is \Delta t<=(\Delta x)^2/(2*D) for one-dimensional problems and \Delta t<=((\Delta x)^2+(\Delta y)^2)/(8*D) for two-dimensional problems, where D is the diffusion coefficient. However, there may not be a specific stability condition for only the convection equation due to the use of a Fourier series in the calculation, unless the velocity is constant. Advecting a field can also present numerical challenges, especially in the long time limit. The article mentioned in the conversation discusses passive advection and
nazmulislam
Hi,

I know, there is a stability condition for solving the Convection-Diffusion equation by Finite Difference explicit/implicit technique, which is \Delta t<=(\Delta x)^2/(2*D) for one-dimensional or \Delta t<=((\Delta x)^2+(\Delta y)^2)/(8*D) for two-dimensional problem, where D is the diffusion coefficient.
.
Is there any such condition for only the convection equation?

Thanks

The stability criterion is dependent on your FD approximation of the laplacian. Have you tried calculating the stability conditions? I am not sure why the same method wouldn't work for the convection term. I have not done it for the convection though.

Thanks for your response. I am not sure how to calculate the stability condition. I have used the the formula \Delta t<=((\Delta x)^2+(\Delta y)^2)/(8*D) to make my program stable. But if there is no diffusion term,only convection term, how will I calculate the stability condition?

Thanks

After some thought I am not even sure you can do the calculation with the convection term, since the stability of the diffusion equation is calculated using a Fourier series. Unless the velocity is constant i suppose things will not work that well.
Further more as far as I remember just advecting a field is not so easy numerically due to various problems(at least not in the long time limit). What are you advecting by the way? An interface? This article discusses a bit about passiv advection

I think it is nice enough.(and eq 2 might be what you are looking for).

Thanks.

## 1. What is the stability condition for solving the convection equation using FDM?

The stability condition for solving the convection equation using FDM is that the time step must be smaller than the spatial step. This means that the time interval between each calculation must be small enough to accurately capture the changes in the system.

## 2. Why is the stability condition important in solving the convection equation using FDM?

The stability condition is important because it ensures that the numerical solution is accurate and does not produce oscillations or incorrect results. If the stability condition is not met, the solution may become unstable and the results can be significantly different from the actual solution.

## 3. How does the stability condition affect the accuracy of the numerical solution?

If the stability condition is not met, the numerical solution may become unstable and produce inaccurate results. If the condition is met, the solution will be accurate and represent the behavior of the system accurately.

## 4. Can the stability condition be relaxed in order to reduce computation time?

No, the stability condition cannot be relaxed as it is a fundamental requirement for obtaining accurate results. Attempting to relax the condition may result in inaccurate or unstable solutions.

## 5. How can one ensure that the stability condition is met when solving the convection equation using FDM?

The stability condition can be ensured by choosing an appropriate time step and spatial step size. It is also important to regularly check the stability of the solution during the computation process and adjust the steps accordingly.

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