[iilw1314]

hi guys ,can you help me to understand what is the difference between the high resolution scheme and high order scheme in details.i am confused.in some cases are they the same ?if i use coarse meshes ,what do high order scheme and low order scheme mean to me?and what do high resolution scheme and low resolution scheme mean to me?

[bjohn]

Hi,

Here is my understanding:

1. The term, 'high-resolution', means 2nd-order or higher accuracy and no oscillations (primarily focusing on applications to shock waves, in comparison with monotone but too dissipative 1st-order schemes which is low-resolution).

2. The term, 'high-order', means simply 3rd-order or higher accuracy.

I guess the term 'high-resolution' is obsolete these days since many existing codes
use 'high-resolution' schemes. By the same logic, the term 'high-order' may become
obsolete in future.

bjohn

[ganesh]

Dear Lin,

Resolution refers to error on a given grid and accuracy refers to error decay on a sequence of grids. A high resolution scheme can have a lower order accuracy, but yield much smaller errors on a given mesh, on which a higher order scheme might give larger errors. While it is true that these terms ar eoften used interchangably and loosely in literature, resolution has to do more with ability of the scheme to resolve all wave numbers on a given mesh. You could probably look into equaivalent wave number concepts and finite difference schemes to get a more clearer picture.

Hope this helps.

Ganesh

[jed]

The term "high resolution" in this context was (I think) coined by Ami Harten in 1983.
Broadly speaking, "high resolution" refers to non-oscillatory methods of order 2 or higher (in smooth regions, no method can be more than first order in L^1 for discontinuous solutions).
In short, bjohn's answer is correct.

"High order" typically refers to any method of order 2 or higher (usually 3 or higher for elliptic problems with nice coefficients), including those that are oscillatory. Sometimes "very high order" is used specifically for methods of at least third order.

I do not agree that "high order" will ever become obsolete, and "high resolution" is still very much used, although sometimes people use the more specific TVD (Total Variation Diminishing, almost always refers to a second order method because no TVD method can have order higher than 2) and TVB (Total Variation Bounded, includes higher order methods like (W)ENO that are not TVD, but are still very much usable for shocks).

[granzer]

The term 'High-resolution schemes' could be used to mean -able to 'resolve(lution)'(ie capture or handle) sharp gradients like the shock wake.

[iilw1314]

Originally Posted by granzer

The term 'High-resolution schemes' could be used to mean -able to 'resolve(lution)'(ie capture or handle) sharp gradients like the shock wake.

Thanks, although this question come from 12 years ago when I start my PHD, the answers from all above inspire me so much! Thank you all! Wish you good luck! You know that this website may live longer than any of us...Enjoy your life...

[FMDenaro]

It is interesting to see old posts and the dated answers ...



At present, I would say that any terms related to high resolution is associated to ratio between the Nyquist frequency of the mesh+numerical method and the physical highest frequency of the flow problem.
High order is related only to the accuracy of a scheme. Thus, we can work with a high order accurate scheme but having low resolution.


I don't know if you agree with my point of view.

[iilw1314]

Originally Posted by FMDenaro

It is interesting to see old posts and the dated answers ...



At present, I would say that any terms related to high resolution is associated to ratio between the Nyquist frequency of the mesh+numerical method and the physical highest frequency of the flow problem.
High order is related only to the accuracy of a scheme. Thus, we can work with a high order accurate scheme but having low resolution.


I don't know if you agree with my point of view.

Hi, High resolution scheme is usually used in the community of compresible flow simulation and high resolution method can capture the shock sharper than low resolution method. I agree with bjohn, ganesh, jed and granzer.