[Matteo991]

Hi all, I'm new on CFD online, and I want to start this experience with a question for you.

The question is which are the differences between Explicit and Implicit method?

I mean I know that in the Explicit method the RHS is expressed as a function of the previous time t_n and I calculates the solution in j-th node at time t_n+1 from previous known values, whereas in the implicit time discretization method I need to solve the sistem iteratively.

Now for what regards the stability, at least linear stability, it can be said that implicit schemes are unconditionally stable and don't need any further condition in terms of CFL, whereas the explicit do.

I have read a thread that someone states: in general for unsteady problems (better transient problems) it should be better employed explicit methods rather than the Implicit ones.
Instead the implicit methods can be used when you dealing with unsteady problems, but you are interesting in the steady state solution, if this last one exists.

So my question is why this statement? Why the explicit method should be used in transient problems and Implicit don't?

I thought that this could be associated with the type of problem that I'm treating, I mean, if I want to solve a viscous N.S. equations I know that they are Parabolic PDEs in time, so it is a time marching problem, and the equations need to depend from what happens before (dependence dominio), so if I use a molecule (stencil) that includes also the nodes at time step t_n+1, in some way do I loose the link with the physics of the problem?

Could be anyone explain better this concepts?
Moreover, in someway there is a link with the consistency concept?

and what about the type of flow (compressible or not, super/sub/transonic) I mean in which way the kind of flow can affects the statement before: I need to use explicit methods in transient simulations?

Are there some cases instead it is better employe the implicit methods? if yes which are?

Every advises, suggestions, references, examples will be helpfull to undestand better.

Thanks in advance who will answer.

[FMDenaro]

Originally Posted by Matteo991

Hi all, I'm new on CFD online, and I want to start this experience with a question for you.

The question is which are the differences between Explicit and Implicit method?

I mean I know that in the Explicit method the RHS is expressed as a function of the previous time t_n and I calculates the solution in j-th node at time t_n+1 from previous known values, whereas in the implicit time discretization method I need to solve the sistem iteratively.

Now for what regards the stability, at least linear stability, it can be said that implicit schemes are unconditionally stable and don't need any further condition in terms of CFL, whereas the explicit do.

I have read a thread that someone states: in general for unsteady problems (better transient problems) it should be better employed explicit methods rather than the Implicit ones.
Instead the implicit methods can be used when you dealing with unsteady problems, but you are interesting in the steady state solution, if this last one exists.

So my question is why this statement? Why the explicit method should be used in transient problems and Implicit don't?

I thought that this could be associated with the type of problem that I'm treating, I mean, if I want to solve a viscous N.S. equations I know that they are Parabolic PDEs in time, so it is a time marching problem, and the equations need to depend from what happens before (dependence dominio), so if I use a molecule (stencil) that includes also the nodes at time step t_n+1, in some way do I loose the link with the physics of the problem?

Could be anyone explain better this concepts?
Moreover, in someway there is a link with the consistency concept?

and what about the type of flow (compressible or not, super/sub/transonic) I mean in which way the kind of flow can affects the statement before: I need to use explicit methods in transient simulations?

Are there some cases instead it is better employe the implicit methods? if yes which are?

Every advises, suggestions, references, examples will be helpfull to undestand better.

Thanks in advance who will answer.

Well, your question has been already posted and you can search for similar discussions.


Anyway, the choice between implicit or explicit time integration depends on a lot of things and, often both methods are used together.
As a general framework, if your problem is unsteady and you want to describe accurately the transient evolution, the time step is constrained by physical requirements and must be small. Thus, using the implicit method would be computationally too expensive. On the other hand, in terms of numerical stability you must be aware that the cfl is strictly depending on the viscous part and you indeed get a region of stability in the (cfl,Reh) plane. For low Reh generally the required dt would become very very small and an implicit method for the diffusive part of the momentum equation is better.
What are the textbooks you are reading?

[Matteo991]

Originally Posted by FMDenaro

Well, your question has been already posted and you can search for similar discussions.


Anyway, the choice between implicit or explicit time integration depends on a lot of things and, often both methods are used together.
As a general framework, if your problem is unsteady and you want to describe accurately the transient evolution, the time step is constrained by physical requirements and must be small. Thus, using the implicit method would be computationally too expensive. On the other hand, in terms of numerical stability you must be aware that the cfl is strictly depending on the viscous part and you indeed get a region of stability in the (cfl,Reh) plane. For low Reh generally the required dt would become very very small and an implicit method for the diffusive part of the momentum equation is better.
What are the textbooks you are reading?

Thanks you for your reply.
I am reading Computational Techniques for Flud Dynamics C.A.J. Fletcher; however can you suggest a book that treat well these concepts?
However, regarding your answer you say before that is too expensive use the implicit methods in the transient problems. Can I say that I have to avoid to use them also for the numerical diffusion? I mean since generally speaking the implicit methods are less accurate than explicit and they could given you a solution but not the exact solution?
Moreover when you say physical requirement do you mean timescale of the flow? Can you explain this clear please?
Furthermore, when I have a small reynolds of cell, yes I need a time step very very small because the stability issues in the case of explicit method, but I can refinement the mesh size or this solution is more expensive than using implicit method? But if I have a time step too small, the given solution using the implicit method is not too diffusive?

Regarding the nature of the equations (elliptic, parabolic and hyperbolic) this does affect the choise of the method, because the consistency?

Thanks again for the aswer.

Cheers,

Matteo

[FMDenaro]

Originally Posted by Matteo991

Thanks you for your reply.
I am reading Computational Techniques for Flud Dynamics C.A.J. Fletcher; however can you suggest a book that treat well these concepts?
However, regarding your answer you say before that is too expensive use the implicit methods in the transient problems. Can I say that I have to avoid to use them also for the numerical diffusion? I mean since generally speaking the implicit methods are less accurate than explicit and they could given you a solution but not the exact solution?
Moreover when you say physical requirement do you mean timescale of the flow? Can you explain this clear please?
Furthermore, when I have a small reynolds of cell, yes I need a time step very very small because the stability issues in the case of explicit method, but I can refinement the mesh size or this solution is more expensive than using implicit method? But if I have a time step too small, the given solution using the implicit method is not too diffusive?

Regarding the nature of the equations (elliptic, parabolic and hyperbolic) this does affect the choise of the method, because the consistency?

Thanks again for the aswer.

Cheers,

Matteo

Not at all. For example, if you use either implicit or explicit Euler method, it is easy to show by the modified equation that you get the same accuracy order (first order in time).
The accuracy order does not depend on the implicit or explicit discretization.


I can suggest many other textbooks, for example Blazek or the last edition of Peric, Ferziger & Street. On a more advanced level the Leveque.
Morton & Mayers is a textbook on general numerical methods for PDE.


PS: note that elliptic equations are generally defined for an equilibrium state, not for evolutionary problems.

[FMDenaro]

However, you answered for a lot of topics, please number your main questions so that is clear the list.

[Matteo991]

Originally Posted by FMDenaro

Not at all. For example, if you use either implicit or explicit Euler method, it is easy to show by the modified equation that you get the same accuracy order (first order in time).
The accuracy order does not depend on the implicit or explicit discretization.


I can suggest many other textbooks, for example Blazek or the last edition of Peric, Ferziger & Street. On a more advanced level the Leveque.
Morton & Mayers is a textbook on general numerical methods for PDE.


PS: note that elliptic equations are generally defined for an equilibrium state, not for evolutionary problems.

Regarding the accurancy, yes indeed I have said "generally speaking", maybe I have used a word not properlyI had to say usually.
So basically can I say that I have to use explicit methods only in order to reducing the computational efforts? both cost and implementation.

Regarding the question about low Reynolds cell, why I have to use implicit method instead of refine the mesh? Is it the last one more expensive in terms of cost?

Thanks for your suggestions, for sure I'lI take a look them.
However what about the physicals requirement that you have mentioned?



PPS: yes I know the elliptic ones are not time marching but I quoted them for example of the equations in order to express better what I have meant. Sorry for the incomprension but the question is the choise of a method rather than the other is affected by the nature of the equations that I have to solve?

[FMDenaro]

- So basically can I say that I have to use explicit methods only in order to reducing the computational efforts? both cost and implementation.


R. Not exactly. The problem is when the physics you want to simulate is transient in nature and has small time scale you have to describe, as usually happens in DNS/LES. Clearly an explicit scheme is computationally more affordable than an implicit.





- Regarding the question about low Reynolds cell, why I have to use implicit method instead of refine the mesh? Is it the last one more expensive in terms of cost?
R. No, you are wrong. If you refine the mesh you reduce the Reh and, consequently, the cfl for the stability is smaller. Often the explicit method is adopted for the convective terms and the implicit method for the diffusive term. This way you avoid that the time step becomes very very small for the diffusive requirement.


- Thanks for your suggestions, for sure I'lI take a look them.
However what about the physicals requirement that you have mentioned?


R. You can read aslo the well known Anderson textbook or Hirsch. The physical requirement is mainly in the nature of the problem. If you want only a steady state, describing accurately the small time scales is not necessary and you can use implicit method and large time step. But if you are interested in resolving the small physical time scale, no matter about implicit or explicit method you have to use a computational time step of the order of the characteristic physical one.

[Matteo991]

Originally Posted by FMDenaro

However, you answered for a lot of topics, please number your main questions so that is clear the list.

Well, yes actually it is becaming confused.
However:

1) Explicit vs Implicit: as well as for reasons due computational cost, there are some other reasons for which explicit method, in transient problem, has to be used? I mean other reasons related to the physics of the under investigation problem?

2) In the implicit method, there are some issues related to nature of the equations?For example if the problem is Hyperbolic, I can use the nodes of t n+1 to compute the solution despite these nodes are out of dependece dominio?

3) Could be happen in someway that using an implicit method following a "fast dynamic" the getting solution are unphysics? if yes why?

4)In the flow with low Reynolds Cell, I have a small time step, cause the instability so you said me that it is better to use an implicit method instead of explicit. But in order to avoid the instability I can refine my mesh for example or this solution is more expensive than using implicit?

5) What about the flow regime? Which are the consideration about the choise depending the regime of the flow (sub/super/transonic)?

I hope now it is clear

Thanks,

Matteo

[FMDenaro]

1) Explicit vs Implicit: as well as for reasons due computational cost, there are some other reasons for which explicit method, in transient problem, has to be used? I mean other reasons related to the physics of the under investigation problem?
R. see the above answer


2) In the implicit method, there are some issues related to nature of the equations?For example if the problem is Hyperbolic, I can use the nodes of t n+1 to compute the solution despite these nodes are out of dependece dominio?
R. Think about the characteristic method for hyperbolic equation. You can move upward or downward along the characteristic line to determine the solution. You are always within the correct domain of dependence. Different is the issue when you consider the spatial stencil, it is this latter that can cause the fail in the location of the domain od dependence.


3) Could be happen in someway that using an implicit method following a "fast dynamic" the getting solution are unphysics? if yes why?
R. Yes if the implicit method is used with a large time step that filters out the high frequency that the flow physically has.



4)In the flow with low Reynolds Cell, I have a small time step, cause the instability so you said me that it is better to use an implicit method instead of explicit. But in order to avoid the instability I can refine my mesh for example or this solution is more expensive than using implicit?
R. No. see the above answer



5) What about the flow regime? Which are the consideration about the choise depending the regime of the flow (sub/super/transonic)?
R. Think mainly in terms of steady or unsteady physical solution. This is the main issue, in any type of regime.



I hope now it is clear

[Matteo991]

1) I have seen the above answer Can you explain this better?

2) Ok it is clear

3) Ok

4) yes I was confused with CFL. However if I decreases delta_x Reh decrease and so the time step also because CFL.

5) Ok

Thanks you for your availability

[FMDenaro]

Originally Posted by Matteo991

1) I have seen the above answer Can you explain this better?

What is not clear?

[Matteo991]

Originally Posted by FMDenaro

What is not clear?

Then I make you an example if I interesting in a steady state solution of a Unsteady problem, but this problem is combustion problem, I have 2 time scale, fluid dynamics and reaction, but I am interesting in time scale in that case can I use implicit again? I suppose no, because stiff, so the steady state depends from how you have followed the dynamics. I mean this issues related to the physics. There is some rules to follow to choose the method?

[FMDenaro]

Originally Posted by Matteo991

Then I make you an example if I interesting in a steady state solution of a Unsteady problem, but this problem is combustion problem, I have 2 time scale, fluid dynamics and reaction, but I am interesting in time scale in that case can I use implicit again? I suppose no, because stiff, so the steady state depends from how you have followed the dynamics. I mean this issues related to the physics. There is some rules to follow to choose the method?

First, if the problem admits a steady solution it is not unsteady. You just have a physical transient starting from the initial condition and ending at the final steady solution. Now if you are interested only in the steady solution you can think of a numerical transient that does not represent the physical one and that means you don't care about the physical characteristic time-scales. This is the case where implicit methods can be efficient as you use a large time step (at least provided the steady solution exists and is unique).
Conversely, if your analysis requires to simulate also the physical transient (even if the final solution is steady), you are in the condition to use a small computational time step comparable to the magnitude of the physical characteristic times. In such a case the explicit integration is superior in terms of computational cost. But be aware about what I already addressed, it could be convenient to use an implicit integration (for example the Crank-Nicolson) only for the diffusive terms in order to alleviate the numerical stability constraint.

[Matteo991]

Originally Posted by FMDenaro

First, if the problem admits a steady solution it is not unsteady. You just have a physical transient starting from the initial condition and ending at the final steady solution. Now if you are interested only in the steady solution you can think of a numerical transient that does not represent the physical one and that means you don't care about the physical characteristic time-scales. This is the case where implicit methods can be efficient as you use a large time step (at least provided the steady solution exists and is unique).
Conversely, if your analysis requires to simulate also the physical transient (even if the final solution is steady), you are in the condition to use a small computational time step comparable to the magnitude of the physical characteristic times. In such a case the explicit integration is superior in terms of computational cost. But be aware about what I already addressed, it could be convenient to use an implicit integration (for example the Crank-Nicolson) only for the diffusive terms in order to alleviate the numerical stability constraint.

Very usefull! Thanks you!
A last step, after that I will stop to ask I promise . However, in the case of steady solution, if I intersting in steady state, without the coinstraints following the several time scales, why I need to use the implicit method and don't solve immediatly the elliptical equations associate? I mean solve direclty the steady equations?
and in that sense this stuff it deal with pseudo-transient techniques? I mena using an implicit method to solve the equation in order to find the steady state? and if yes which are the limititation to use this in that sense? I mean if I have a separation on an airfoil? I mean statistically it could be consider steady but actually it doesn't. Why?2

Thanks you for the support I appreciate it.

Matteo

[FMDenaro]

Yes, you can solve without time integration.

If you are thinking about simulation of turbulence, you should be aware about the fact that turbulence is always 3D and unsteady, by definition. However, you can have or not a statistical steady turbulence problem, that depends on the flow problem. When you have a statistically steady case, that is the classical RANS framework, you solve directly the equation without any time derivative and, consequently, any impliciti or explicit time integration. In this framework, the residuals that you should drive theoretically to vanish could be seen as "pseudo-transient". But no physical meaning for such terms.

[Matteo991]

Originally Posted by FMDenaro

Yes, you can solve without time integration.

If you are thinking about simulation of turbulence, you should be aware about the fact that turbulence is always 3D and unsteady, by definition. However, you can have or not a statistical steady turbulence problem, that depends on the flow problem. When you have a statistically steady case, that is the classical RANS framework, you solve directly the equation without any time derivative and, consequently, any impliciti or explicit time integration. In this framework, the residuals that you should drive theoretically to vanish could be seen as "pseudo-transient". But no physical meaning for such terms.

Thanks a lot for your e explanations and your availability.

Matteo