Relation between energy conservation and numerical stability

In summary, the conversation discussed the topic of energy conservation in isothermal linear incompressible flow. It was mentioned that the kinetic energy equation can be solved to determine if energy is conserved, and a question was raised about the implications of energy conservation on the convergence and stability of a numerical strategy. The conversation also touched on the use of energy approaches in other cases, such as with closed governing equations. One participant mentioned the possibility of a numerical algorithm failing to conserve energy despite physical laws dictating energy conservation. Finally, the use of a free energy approach to show unconditional stability was brought up, but it was noted that this does not necessarily involve energy conservation.
  • #1
hoomanya
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Hi,
Consider the conservation laws for an isothermal linear incompressible flow governed by the mass and momentum equation. The kinetic energy equation is then solved to see if energy conserved. Can anyone tell me if once it is shown energy is conserved, it implies that convergence is obtained and that the numerical strategy is stable?

Are there any other cases, where some kind of an energy approach is utilised even though the governing equations are closed?

Thanks!
 
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  • #2
You can have physical laws which conserve energy, but a numerical algorithm which fails to conserve energy.
 
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Likes hoomanya
  • #3
Cant you just pick any straight forward discretization of a wave equation to have a non conservative solver for a conservative system?

Some times you can show that your numerical scheme is unconditionally stable by using a free energy approach but as far as I know this uses a monotonic decrease in the specified energy not conservation.
 

Related to Relation between energy conservation and numerical stability

1. What is the relation between energy conservation and numerical stability?

The relation between energy conservation and numerical stability is that numerical methods used to solve physical problems must also ensure that energy is conserved in the system. This means that the total energy of the system before and after the numerical solution should be the same.

2. Why is energy conservation important in numerical stability?

Energy conservation is important in numerical stability because it ensures that the results obtained from numerical methods are accurate and reliable. Without energy conservation, the numerical solution may be prone to errors and may not accurately represent the physical system being studied.

3. How does energy conservation affect the accuracy of numerical methods?

Energy conservation plays a critical role in determining the accuracy of numerical methods. If energy is not conserved in the system, it can lead to errors and discrepancies in the numerical solution. Therefore, maintaining energy conservation is essential for ensuring the accuracy of numerical methods.

4. What are some common techniques used to maintain energy conservation in numerical methods?

Some common techniques used to maintain energy conservation in numerical methods include symplectic integration methods, energy-preserving methods, and multi-symplectic methods. These methods are specifically designed to conserve energy in the system and improve the accuracy of the numerical solution.

5. Can numerical methods still be stable without energy conservation?

Yes, numerical methods can still be stable without energy conservation, but their accuracy may be compromised. While energy conservation is not a requirement for stability, it is an essential factor in ensuring the accuracy and reliability of the numerical solution. Therefore, it is crucial to use numerical methods that also conserve energy in the system to obtain accurate results.

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