[JasmineFeng]

Deferred correction is often used for higher-order schemes like QUICK, to make the coefficient diagonally dominant. With deferred correction, the solution can converge to the exact solution `x` of the algebraic equations `Ax=b` using the Gauss-Seidel method. Without it, the iterative error may diverge to infinity.

However, the exact solution of `Ax=b` is not necessarily physically realistic. For example, the coefficient matrix A is given by the central differencing scheme when Peclect number > 2. Consequently, what a higher-order scheme using deferred correction may still lead to unphysical solutions. Things are similar when it comes to the QUICK scheme.

Hence, I wonder why we need deferred correction. I think convergence to an unphysical solution is no better than divergence of the iterative error to infinity.

[FMDenaro]

Originally Posted by JasmineFeng

Deferred correction is often used for higher-order schemes like QUICK, to make the coefficient diagonally dominant. With deferred correction, the solution can converge to the exact solution `x` of the algebraic equations `Ax=b` using the Gauss-Seidel method. Without it, the iterative error may diverge to infinity.

However, the exact solution of `Ax=b` is not necessarily physically realistic. For example, the coefficient matrix A is given by the central differencing scheme when Peclect number > 2. Consequently, what a higher-order scheme using deferred correction may still lead to unphysical solutions. Things are similar when it comes to the QUICK scheme.

Hence, I wonder why we need deferred correction. I think convergence to an unphysical solution is no better than divergence of the iterative error to infinity.

I think you can find the answer of why using deferred correction in many textbooks, for example Ferziger, Peric and Street.
On the other hand, numerical oscillations can appear also without using the DC, see for example the convection-diffusion equation with second order central formula at cell Peclet number >2. Your solution is exact in sense of satisfying the discrete equation Ax=b but if you look at the modified differential equation you see that it is a solution of a differential problem having a dispersive term.

The fact the a Gauss-Seidel iterative method diverges or not is only a topic to be studied in terms of numerical analysis not in term of physical solution.

[sbaffini]

Following your reasoning, every implicit method would be invalid or useless when used in conditions not valid for an equivalent explicit scheme, which is not the case.

Deferred correction is just a way to rearrange terms between implicit and explicit discretizations that, in the end, results in better stability than just using only explicit discretizations. It also has a variety of applications as, in practice, for unstructured grids, it is used for every term that needs higher than 1st order accuracy, including linear problems like just conduction in solids.

Note that stretching the time discretization with time steps well beyond the stability of equivalent explicit schemes obviously incurs in time dicretization errors that still make the scheme mostly useless, but for steady state this has no consequences.

All in all, the method is also proven with linear problems, where you can verify that indeed you reach the correct solution so, roughly speaking, what you say simply isn't true, at least for linear problems.

[JasmineFeng]

Thanks for your replies, FMDenaro and sbaffini. I just found some useful statements in Hrvoje Jasak's PhD thesis to complement my point:

This, however, does not affect the boundedness in spite of the fact that the matrix is now diagonally equal, as the "troublesome" part of the discretisation still exists in the source term.
...
Unfortunately, the sufficient boundedness criterion cannot be used to establish the boundedness of the discretisation, as it does not take into account the influence of the source term.

In this way, what DC does, is just transfer the troublesome coefficients from the matrix into the source term, after which the iteration can converge, but the boundedness of the solution won't change. However, what I've seen in some textbooks, like Versteeg's, is contradictory to my point:

The advantage of this approach (DC) is that the main coefficients are positive and satisfy the requirements for conservativeness, boundedness and transportiveness.

This is why I am confused about DC's contribution to boundedness. Versteeg's statements seem kind of misleading to me.

But after reading your replies and Jasak's thesis, I'm much clearer now. Now I think DC is just a technique to help converge in the iterative process. Also, I must admit its significance in reducing the size of the computational molecule and changing a pentadiagonal matrix into a tridiagonal one. It's also useful in unstructured grids as sbaffini pointed out.

[arjun]

Originally Posted by JasmineFeng

I think convergence to an unphysical solution is no better than divergence of the iterative error to infinity.

Your base statement is not correct. For example differed correction is very much used in older commercial codes like Fluent, StarCD. People got physically good results in them all the time.


In modern commerical codes I do not think much of differed correction is used because now the codes are much more stable.

[sbaffini]

Originally Posted by arjun

Your base statement is not correct. For example differed correction is very much used in older commercial codes like Fluent, StarCD. People got physically good results in them all the time.


In modern commerical codes I do not think much of differed correction is used because now the codes are much more stable.

I don't think this is correct. Whenever a higher order discretization is in place on an unstructured grid, I'm pretty much sure that only the first order part gets the implicit coefficients in pratically any CFD solver, which is, I think, the definition of deferred correction.

If you instead refer to using a first order central vs a first order upwind for the implicit convective part, I am less sure, yet I have doubts on any advantage.

[FMDenaro]

Originally Posted by JasmineFeng

Thanks for your replies, FMDenaro and sbaffini. I just found some useful statements in Hrvoje Jasak's PhD thesis to complement my point:


In this way, what DC does, is just transfer the troublesome coefficients from the matrix into the source term, after which the iteration can converge, but the boundedness of the solution won't change. However, what I've seen in some textbooks, like Versteeg's, is contradictory to my point:


This is why I am confused about DC's contribution to boundedness. Versteeg's statements seem kind of misleading to me.

But after reading your replies and Jasak's thesis, I'm much clearer now. Now I think DC is just a technique to help converge in the iterative process. Also, I must admit its significance in reducing the size of the computational molecule and changing a pentadiagonal matrix into a tridiagonal one. It's also useful in unstructured grids as sbaffini pointed out.

In case of source terms, it is theoretically not correct assuming monotone behavior. The monotonicity is a problem strictly due to the discretization of the convective flux.

[arjun]

Originally Posted by sbaffini

I don't think this is correct. Whenever a higher order discretization is in place on an unstructured grid, I'm pretty much sure that only the first order part gets the implicit coefficients in pratically any CFD solver, which is, I think, the definition of deferred correction.

If you instead refer to using a first order central vs a first order upwind for the implicit convective part, I am less sure, yet I have doubts on any advantage.

I am addressing the part where one says that differed correction produces unphysical results. It can but it does not mean it will always.

Your definition about implicit part is correct. This is how it is used in Fluent and starcd.


Now a days we just put the higher order values directly to the source, while the matrix (or implicit part) see the contribution from first order scheme. Since we solve in delta form the converged results are not first order results.

[JasmineFeng]

Originally Posted by FMDenaro

In case of source terms, it is theoretically not correct assuming monotone behavior. The monotonicity is a problem strictly due to the discretization of the convective flux.

Sorry, I can't understand what "monotonicity" means as to the source terms. Would you please explain it in detail, or just give me some references? Thanks again.

[JasmineFeng]

Originally Posted by arjun

Your base statement is not correct. For example differed correction is very much used in older commercial codes like Fluent, StarCD. People got physically good results in them all the time.


In modern commerical codes I do not think much of differed correction is used because now the codes are much more stable.

I think the reason why DC could still get good results is that, the critical grid Peclect numbers Pe (2 for CD, 8/3 for QUICK and 4 for Fromm) given in many textbooks are too strict. They only apply to the situation which is (1) one-dimensional (2) a linear problem (ρ,u,Γ are constants) (3) without source terms (4) using Dirichlet boundary conditions (5) using a uniform grid. Any violation of the above conditions can lead to an increase of the critical Pe, in which case a bounded solution will be obtained.

[sbaffini]

Originally Posted by arjun

Since we solve in delta form the converged results are not first order results.

This is one thing that I think is worth mentioning. Solving in delta form, which existed well before deferred correction, also uses the same trick, stable first order coefficients and full residuals.

It is just the non-delta form that makes things less trivial, but the matter is pretty much the same

[JasmineFeng]

I introduced an under-relaxation factor into the interpolation formula of the face:


I set different under-relaxation factors: alpha=0.2, 0.5, 0.7 and 1. And test the Central Differencing Scheme with Deferred Correction (CDdc) using one-dimensional cases with different grid Peclect numbers: -0.2, +1, -3, and +10. The RMSE of all cases are listed below. (FUD:First-order Upwind Differencing, PLS:Power Law Scheme)


Pe=-0.2 Pe=1 Pe=-3 Pe=10
__________ ________ _______ __________

CD 0.0029693 0.29762 10.612 433.05
FUD 0.0046994 0.35694 0.79106 0.51616
PLS 0.0010356 0.084203 0.62771 0.51154
CDdc-0 0.0046995 0.35695 0.79106 0.51616
CDdc-0.2 0.0023083 0.18267 0.23052 0.00090799
CDdc-0.5 0.00052839 0.030895 0.12888 3.9064
CDdc-0.7 0.00064569 0.03678 1.3295 22.615
CDdc-1 0.0029692 0.29762 10.612 433.05

Obviously, when alpha=1 (0), CDdc-1 (CDdc-0) is reduced to CD (FUD). As you can see, the RMSE of CDdc-1 (CDdc-0) is exactly the same as CD (FUD) regardless of some round-off errors. This also demonstrates that deferred correction cannot help CD obtain bounded solutions at least in these one-dimensional cases. However, when alpha is adjusted to 0.2, 0.5 or 0.7, they could obtain fairly accurate results, and sometimes perform even better than PLS (CDdc-0.2, Pe=10). So I think the DC technique should be used with the relaxation factor to show its significance.

(Well, this is called higher order term relaxation in Ansys Fluent Theory Guide.)

[sbaffini]

I think there might be something wrong here. DC is not supposed to give an answer when an answer doesn't exist. Nor an answer different from the one expected without DC. It is the same equation with the same discretization, just the terms are rearranged to give certain properties to the matrix.

In your case of linear convection diffusion with central differencing there is an exact numerical solution that should be reached independently from the "time" discretization.

Here I highlight time, even if the equation is steady, because there is an aspect that you maybe are not considering. When you solve the linear system without DC you do it in one shot with everything fixed. When you introduce DC you are actually changing the RHS with iterations, so you really solve a different system each time, which is kind of like if you were solving a non linear system in pseudo time.

[JasmineFeng]

Originally Posted by sbaffini

I think there might be something wrong here. DC is not supposed to give an answer when an answer doesn't exist. Nor an answer different from the one expected without DC. It is the same equation with the same discretization, just the terms are rearranged to give certain properties to the matrix.

In your case of linear convection diffusion with central differencing there is an exact numerical solution that should be reached independently from the "time" discretization.

Here I highlight time, even if the equation is steady, because there is an aspect that you maybe are not considering. When you solve the linear system without DC you do it in one shot with everything fixed. When you introduce DC you are actually changing the RHS with iterations, so you really solve a different system each time, which is kind of like if you were solving a non linear system in pseudo time.

I was indeed dealing with a steady problem. When Pe=10, the answer does not exist for central differencing (alpha=1), but it does exist for the upwind scheme (alpha=0). So CDdc-0 could yield a bounded solution. So could CDdc-0.2. But when alpha increased, CDdc would be like CD more and more instead of FUD. So CDdc-0.5 and CDdc-0.7 could not yield a valid result just like CD.

[FMDenaro]

Originally Posted by JasmineFeng

Sorry, I can't understand what "monotonicity" means as to the source terms. Would you please explain it in detail, or just give me some references? Thanks again.

One of the fundamental property of a convection-diffusion PDE is that it does not create new extrema in the solution. Therefore, it is clear that in a numerical solution the presence of new extrema is not physically meaningful. Monotone schemes have a property to guarantee that.

But, if a source term is added in the PDE, the action of such a term can actually produce new physical extrema.

[FMDenaro]

Originally Posted by JasmineFeng

I introduced an under-relaxation factor into the interpolation formula of the face:


I set different under-relaxation factors: alpha=0.2, 0.5, 0.7 and 1. And test the Central Differencing Scheme with Deferred Correction (CDdc) using one-dimensional cases with different grid Peclect numbers: -0.2, +1, -3, and +10. The RMSE of all cases are listed below. (FUD:First-order Upwind Differencing, PLS:Power Law Scheme)


Pe=-0.2 Pe=1 Pe=-3 Pe=10
__________ ________ _______ __________

CD 0.0029693 0.29762 10.612 433.05
FUD 0.0046994 0.35694 0.79106 0.51616
PLS 0.0010356 0.084203 0.62771 0.51154
CDdc-0 0.0046995 0.35695 0.79106 0.51616
CDdc-0.2 0.0023083 0.18267 0.23052 0.00090799
CDdc-0.5 0.00052839 0.030895 0.12888 3.9064
CDdc-0.7 0.00064569 0.03678 1.3295 22.615
CDdc-1 0.0029692 0.29762 10.612 433.05

Obviously, when alpha=1 (0), CDdc-1 (CDdc-0) is reduced to CD (FUD). As you can see, the RMSE of CDdc-1 (CDdc-0) is exactly the same as CD (FUD) regardless of some round-off errors. This also demonstrates that deferred correction cannot help CD obtain bounded solutions at least in these one-dimensional cases. However, when alpha is adjusted to 0.2, 0.5 or 0.7, they could obtain fairly accurate results, and sometimes perform even better than PLS (CDdc-0.2, Pe=10). So I think the DC technique should be used with the relaxation factor to show its significance.

(Well, this is called higher order term relaxation in Ansys Fluent Theory Guide.)

I don't understand what you are doing. Negative Peclet means an anti-diffusive problem.

Are you just trying to replicate the 1D stready convection-diffusion exercise shown in the textbook of Ferzigr, Peric and Street?

[sbaffini]

Originally Posted by JasmineFeng

I was indeed dealing with a steady problem. When Pe=10, the answer does not exist for central differencing (alpha=1), but it does exist for the upwind scheme (alpha=0). So CDdc-0 could yield a bounded solution. So could CDdc-0.2. But when alpha increased, CDdc would be like CD more and more instead of FUD. So CDdc-0.5 and CDdc-0.7 could not yield a valid result just like CD.

I might have misunderstood what you were doing. In practice, if now I understand correctly, those urf are used like blending factors of the DC source term? Then those differences are expected, I suppose. But then you are not getting solutions to the same problem, but different ones

[arjun]

Originally Posted by JasmineFeng

I introduced an under-relaxation factor into the interpolation formula of the face:


I set different under-relaxation factors: alpha=0.2, 0.5, 0.7 and 1. And test the Central Differencing Scheme with Deferred Correction (CDdc) using one-dimensional cases with different grid Peclect numbers: -0.2, +1, -3, and +10. The RMSE of all cases are listed below. (FUD:First-order Upwind Differencing, PLS:Power Law Scheme)


Pe=-0.2 Pe=1 Pe=-3 Pe=10
__________ ________ _______ __________

CD 0.0029693 0.29762 10.612 433.05
FUD 0.0046994 0.35694 0.79106 0.51616
PLS 0.0010356 0.084203 0.62771 0.51154
CDdc-0 0.0046995 0.35695 0.79106 0.51616
CDdc-0.2 0.0023083 0.18267 0.23052 0.00090799
CDdc-0.5 0.00052839 0.030895 0.12888 3.9064
CDdc-0.7 0.00064569 0.03678 1.3295 22.615
CDdc-1 0.0029692 0.29762 10.612 433.05

Obviously, when alpha=1 (0), CDdc-1 (CDdc-0) is reduced to CD (FUD). As you can see, the RMSE of CDdc-1 (CDdc-0) is exactly the same as CD (FUD) regardless of some round-off errors. This also demonstrates that deferred correction cannot help CD obtain bounded solutions at least in these one-dimensional cases. However, when alpha is adjusted to 0.2, 0.5 or 0.7, they could obtain fairly accurate results, and sometimes perform even better than PLS (CDdc-0.2, Pe=10). So I think the DC technique should be used with the relaxation factor to show its significance.

(Well, this is called higher order term relaxation in Ansys Fluent Theory Guide.)

Try the same problem with Second order upwind, and third order muscl scheme. The problem you have is that you are trying CD. One can apply differed correction for these schemes too like other higher order schemes. What you are finding out is that CD has issues that you think differed correction is introducing them.


The CD has peclet problem which is well noted in literature.


PS: In other words, you asked for CD solution differed correction provided you CD solution. If you ask for other bounded schemes it will provide you that solution. For example you can use WENO schemes in differed correction way and you will not see problems.

[JasmineFeng]

Originally Posted by sbaffini

I might have misunderstood what you were doing. In practice, if now I understand correctly, those urf are used like blending factors of the DC source term? Then those differences are expected, I suppose. But then you are not getting solutions to the same problem, but different ones

Yeah, exactly. Alpha is a blending factor of FUD and CD, ranging from zero to one.

[JasmineFeng]

Originally Posted by arjun

Try the same problem with Second order upwind, and third order muscl scheme. The problem you have is that you are trying CD. One can apply differed correction for these schemes too like other higher order schemes. What you are finding out is that CD has issues that you think differed correction is introducing them.


The CD has peclet problem which is well noted in literature.


PS: In other words, you asked for CD solution differed correction provided you CD solution. If you ask for other bounded schemes it will provide you that solution. For example you can use WENO schemes in differed correction way and you will not see problems.

I got it. Because schemes like SOU and MUSCL are unconditionally stable, so they will always produce bounded results without wiggles.