For leap frog method applied on a first order 1D wave equation, if von neumann analysis is performed, two amplification factors are gained that have the same amplitude but different phase. What can we inferred from having two amplification factor about relative phase error and about the it's overall effect on the solution?
If I understand your question, you are considering the amplification matrix of multi-steps methods. The stability analysis is based on the requirement that the modulus of the eigenvalues of the matrix are lower than one.
The study of the errors in terms of the amplification factors for the leap-frog scheme is illustrated in Sec.7.4.1.4 of the book of Hursch
Filippo Maria Denaro thanks for your answer. I've read this section after your answer. In this book, it is mentioned that leapfrog method has to different relative phase error (for CFL number less than 1) for a given CFL number.But it's not clear for me that with two relative phase error , whether phase error is leading or lagging?
I attached the expression written in the book you recommended.
You see in the book that the phase error is reported to be always lower than 1, therefore is a lagging error.
Filippo Maria Denaro Oh, you're right. Actually my last sentence was incorrect.
In that book it seems that the relative phase error(RPE) with negative sign is omitted,or it may be invalid or sign is not important for RPE,although expression for RPE doesn't give necessarily the absolute value.
Leapfrog is a very interesting method for many reasons. For your specific question I suggest you also look at the book of Durran:
Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, especially the second edition, which is a real masterpiece of presenting how to get accurate solutions to wave & advection problems in terms of phase and amplitude considerations.
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