I think the accuracy of approximation or the speed of convergence is higher in Taylor's one.

This is perhaps too general a question and it certainly doesn't have enough information to give a comprehensive answer. You haven't mentioned if you are solving an ODE or a PDE, if it steady or unsteady in time, what the boundary conditions, Cartesian or polar coordinates, and so on. So I'll make some observations in teh light of some of the thigs that I have been looking at recently.

I am not a all sure that one can formally compare rates of convergence because one isn't comparing like with like. A second order accurate finite difference method has an error which is proportional to h^2, i.e. quadratic convergence. The speed of convergence of a Fourier series depends on the degree of continuity of the function being approximated, and therefore one looks at the power of n in the denominator. Perhaps it would be better to consider how quickly one can evaluate an approximation to one's target function to, say, a relative accuracy of 10^{-6}.

If one is trying to compare the speeds of computation of the steady-state solution of a nonlinear PDE for a given accuracy, then there are many issues to consider. It is much quicker to write a code to employ finite differences to a nonlinear system than it is to write what is in effect a spectral solver. Part of this is due to the multiple interaction of the sinusoids leading, occasionally to triple or even quadruple summations. Such summations require extensive checking and careful coding. Perhaps all of this work may be justified if the resulting code is going to be used very many times. On the other hand, if one requires only a few solutions, then I would use a simple finite difference scheme. It's a bit like the vexed old question of, which is better, a mountain bike or a road bike. The answer depends on what you want to achieve.

As a generalisation, I would imagine that Fourier series solutions will be very quick compared with finite difference solutions, whereas they are likely to be much slower for nonlinear equations.

I guess that the best comparison between the methods themselves but with no account being taken of code creation time will be by means of an operation count for the various methods. Then, the number of iterations required to obtain solutions with a maximum error of 10^{-3], 10^{-4}, 10^{-5} and so on may be used show the variation of the time taken against the error.

The strength of the Fourier series is largely due to its use of orthogonal functions (sine, cosine, etc.) in lieu of Taylor series (simple algebraic functions). For solving differential equations, the analogy is to migrate from the use finite differences to something like a spectral or Galerkin-type method (there are many) that exploits orthogonal functions. More work is required to use these methods, but one can gain improved convergence.

Jordan Michelet

D. Andrew S. Rees Thanks a lot for your helpful explanation.

sorry for my very very holistic question and lack of details.

My issue is accuracy especially when we are facing with complex physics like turbulence.

I think it could be a relatively simple way to predict flow behavior in turbulent regimes instead of using FDM by Taylor's series.

Douglas Vinson Nance Thanks.

would you mind introducing me any references for Galerkin-type method?

There are many such methods, and the method of choice is a matter of personal preference. My favorite is the Discontinuous Galerkin Method (DGM). You may or may not like it, but some references are:

(1) C.A.J. Fletcher, "Computational Galerkin Methods", Springer-Verlag, New York, 1984.

(2) B. Cockburn, G.E. Karniadakis and C.-W. Shu, Eds., "Discontinuous Galerkin Methods, Theory, Computation and Applications", Lecture Notes in Computational Science and Engineering 11, Springer, New York, 2000.

In this reference, you can read the introductory article by the editors, page 3. I also recommend reading the article starting on page 53 by H.L. Atkins. There is also an article on compressible turbulence simulation starting on page 77.

(3) H.L. Atkins and C.-W. Shu, "Quadrature-Free Implementation of Discontinuous Galerkin Method for Hyperbolic Equations", AIAA Journal, Vol. 36. No. 5, 1998.

You will find that the discretization involved in this method is substantially more difficult than FDM. The math is more difficult, and the volumes and surfaces for the problem are all discretized. This property of DGM is a real strength for the method. The article above is where I started.

You might also search out Spectral Methods for physics problems. A reference you can look at is:

C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang Jr., "Spectral Methods in Fluid Dynamics", Springer, New York, 1988.

These areas in Computational Physics cover a VAST amount of territory. In one such area is a career specialty in itself. You find that these references lead you in a different direction for your problem, but these are both powerful methods. Good Luck.