# dissipative and dispersive error of finite difference methods

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 June 21, 2013, 04:29 dissipative and dispersive error of finite difference methods #1 New Member   Join Date: Apr 2013 Posts: 15 Rep Power: 6 Sponsored Links Hi all, We know that for the spatial derivative using a finite difference method, if the leading error term is odd, then the results contain dispersive error, while if the leading error term is even, then it has dissipative error. I want to compare the degree of such errors. Say, the leading error term is 7th order (FD7), compared with the leading error term is 5th order (FD5), which one has more severe dispersive error? For sure, FD7 is more accurate, what's its performance with regard to dispersive error? Thanks. Shu

June 21, 2013, 06:29
#2
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Filippo Maria Denaro
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Quote:
 Originally Posted by shubiaohewan Hi all, We know that for the spatial derivative using a finite difference method, if the leading error term is odd, then the results contain dispersive error, while if the leading error term is even, then it has dissipative error. I want to compare the degree of such errors. Say, the leading error term is 7th order (FD7), compared with the leading error term is 5th order (FD5), which one has more severe dispersive error? For sure, FD7 is more accurate, what's its performance with regard to dispersive error? Thanks. Shu
This is a classical numerical analysis task, you can analyze the local truncation error of the discretization formula, it gives much information about the character of the error.
Furthermore, I suggest using the spectral analysis too, that is the modified wavenumber that each formula produces.
It is important to remark that not necessarily a higher order accurate formula performs better than a lower one on a given grid. Accuracy order is only an asymptotic estimation that does not tell you how the formula behaves for the chosen grid. Thus, the modified wavenumber analysis can help to understand the error distribution for the grid you want to use.

 June 21, 2013, 06:31 #3 Senior Member   cfdnewbie Join Date: Mar 2010 Posts: 557 Rep Power: 13 There is a standard procedure of analysing dispersive and dissipative behavior of FD schemes. Discretize a linear advection equation, plug in a wave with frequency k and amplitude a and check the resulting frequency and amplitude response - that gives you the dissipation and dispersion error of your scheme. FMDenaro and Aeronautics El. K. like this.

June 21, 2013, 06:46
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Filippo Maria Denaro
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Quote:
 Originally Posted by cfdnewbie There is a standard procedure of analysing dispersive and dissipative behavior of FD schemes. Discretize a linear advection equation, plug in a wave with frequency k and amplitude a and check the resulting frequency and amplitude response - that gives you the dissipation and dispersion error of your scheme.
Such analysis can be extended, in a suitable way, also for the non linear equation (e.g., Burgers)

 June 21, 2013, 07:01 #5 Senior Member   cfdnewbie Join Date: Mar 2010 Posts: 557 Rep Power: 13 Hello Prof. Denaro, I have only seen this done for linear equations. I assume that Burger's would lead to a non-linear system to solve for? Do you have any good reference on the details of this? that would be a very interesting read! thanks in advance!

 June 21, 2013, 07:15 #6 Senior Member     Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 742 Blog Entries: 17 Rep Power: 21 I don't know if Prof. Denaro is referencing what i have in mind; however, a possible practical (i.e., non analytical) approach is this: http://www.sciencedirect.com/science...21999111001148 cfdnewbie, FMDenaro and Aeronautics El. K. like this.

 June 21, 2013, 07:20 #7 Senior Member   cfdnewbie Join Date: Mar 2010 Posts: 557 Rep Power: 13 Thank you Paolo, that looks very interesting!

June 21, 2013, 09:12
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Filippo Maria Denaro
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Quote:
 Originally Posted by sbaffini I don't know if Prof. Denaro is referencing what i have in mind; however, a possible practical (i.e., non analytical) approach is this: http://www.sciencedirect.com/science...21999111001148

yes, there are some papers as this one that in these years analysed the non-linear equations.
We also worked on the wavenumber-based analysis for the one and multi-dimensional non-linear equations, using an analytical/numerical approach

http://onlinelibrary.wiley.com/doi/1...d.179/abstract
http://www.sciencedirect.com/science...21999111000933

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