Relation between energy conservation and numerical stability

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SUMMARY

The discussion centers on the relationship between energy conservation and numerical stability in the context of isothermal linear incompressible flow governed by mass and momentum equations. It is established that demonstrating energy conservation does not necessarily imply convergence or stability of the numerical strategy employed. The participants highlight that while physical laws may conserve energy, numerical algorithms can fail to do so, particularly when using straightforward discretizations of wave equations. Additionally, a free energy approach can indicate unconditional stability, relying on a monotonic decrease in energy rather than strict conservation.

PREREQUISITES
  • Understanding of isothermal linear incompressible flow
  • Familiarity with mass and momentum equations
  • Knowledge of numerical methods for solving differential equations
  • Concept of free energy approaches in numerical analysis
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  • Research the implications of energy conservation in numerical simulations
  • Explore stability analysis techniques for numerical schemes
  • Learn about discretization methods for wave equations
  • Investigate free energy approaches and their applications in numerical stability
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Researchers, numerical analysts, and engineers working on fluid dynamics simulations, particularly those focused on energy conservation and stability in computational methods.

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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You can have physical laws which conserve energy, but a numerical algorithm which fails to conserve energy.
 
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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.
 

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