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how to understand high resolution scheme and high order scheme

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Old   September 27, 2010, 22:52
Default how to understand high resolution scheme and high order scheme
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L.Y LIN
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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?
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Old   September 28, 2010, 21:08
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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I guess the term 'high-resolution' is obsolete these days since many existing codes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
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Old   September 29, 2010, 14:30
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ganesh
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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
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Old   September 30, 2010, 10:43
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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).
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