Higher order discretization on staggered grid
Hello everybody,
I am delving for how to descretize a term, say DELu/DELx, using higher order (say 4th order) central difference scheme on STAGGERED grid. For 2nd order, this is "[u(i+1/2)  u(i1/2)]/[x(i+1/2)  x(i1/2)]" where 'u' is defined on facecenter of the grid. But I don't know how to extend it for higher order discretization. Along with this, I wanted to know how to use higher order upwinding scheme(say 3rd order) on STAGGERED grid. The aforementioned discretizations are fine when the grid system used is nonstaggerred, but I haven't got any reference to how to get these done on staggered grid. Staggered grid is of particular interest for me since working on it is equivalent to dealiasing procedure in DNS, in which aliasing error happen to be one of the measure causes of getting the solution divergerd. It would be very nice if any of you could please suggest me something in this regard. Thanks a lot in anticipation, Chandra Shekhar 
Re: Higher order discretization on staggered grid
Chandra,
It all starts from a Taylor series expansion to the required order. Suppose you wish to find the first derivative approximation at a given order at at x=a (e.g., x(i) or x(i+1/2) ). Write explicitly the Taylor series to the desired order for its neighbours. You should do so for enough neighbours so that you can then solve the linear algebraic system for u'(a). Usually one uses neighbours at both sides of a, but this is not required. Also, frequently this is done on a uniform grid, but may also be done on a nonuniform grid (with more algebra, though). This treatment is applicable regardless of the grid arrangement (i.e., staggered or not). I hope this helps, Rami 
Re: Higher order discretization on staggered grid
Hi, Chandra I hope this article will be quite appropriate:
F. Bianco, G. Puppo & G. Russo (1998) High order central schemes for hyperblic systems of conservation laws. As they say they increase the order of 2LxF or NT method. Going to read it as well, so your possible questions will be very stimulating. Good luck 
Re: Higher order discretization on staggered grid
You might find the following articles useful:
Santhanam Nagarajan, Sanjiva K. Lele, Joel H. Ferziger, "A robust highorder compact method for large eddy simulation" Journal of Computational Physics 191 (2003) 392–419 Albert E. Honein and Parviz Moin Higher entropy conservation and numerical stability of compressible turbulence simulations Journal of Computational Physics, Volume 201, Issue 2, 10 December 2004, Pages 531545 Dave 
Re: Higher order discretization on staggered grid
Hi,
If your solution diverges with a central scheme your resolution is probably too poor. You can try to solve this with an upwinding scheme but this probably means that you add some numerical diffusion. In fact, you are doing an LES with an illdefined subgrid model. A better solution would be to implement a dynamic LES model for example. I have seen few people performing DNS with an upwinding scheme, so be warned. Tom 
Re: Higher order discretization on staggered grid
Heyy, thanks for your suggestions!! I was also thinking that discretization scheme should not depend on grid system i.e. whether the grid to be implemented is staggered or simple one. However, I got a material on "http://les.colorado.edu/~vasilyev/Publications/JCP_2003c.pdf" where they have proposed a scheme claiming it to be high order fully conservative difference scheme on nonuniform staggerred grid. For this, during discretization, they have introduced a quantity called interpolation weight, which is obtained after solving a system of linear equations (page number15, equation66). however, I couldnt get how have they got that set of linear equations!!!?? Anyway, after substituting these interpolation weights, the finally obtained descritized equations DOES NOT match with those conventional ones on simple grid obtaned from Taylor series expansion!!
Any suggestion?? chandra shekhar 
Re: Higher order discretization on staggered grid
Hi there,
You need to check the following paper: Y. Morinishi, T. S. Lund, O. V. Vasilyev, P. Moin, "Fully conservative higher order finite difference schemes for incompressible flow," J. Comp. Phys 143, 90124, 1998. It is for cartesian staggered mesh schemes and especially geared towards incompressible DNS/LES computations. Hope this helps. Sincerely, Frederic Felten 
Re: Higher order discretization on staggered grid
Oh yes...this is the same paper what I reffered to my last response!!
Thanks a lot to all of u guys!! However, one thing still remained is how to have the 'upwinding scheme' on staggerred grid....i mean, higher order upwinding scheme!! There are ways to get an upwinding scheme on regular nonstaggerred grid by adding a high order derivative term... e.g. (DELu/DELx)_third_order_upwinding = (DELu/DELx)_4th_order_central_difference + k*(DEL4u/DELx4)_central_difference; where k is an arbitrary constant chosen depending on the amount of dissipation to be added. However, I dont think if this idea works on staggerred grid also. I would be very thankful if any of you could suggest something in this regard!! chandra shekhar 
Re: Higher order discretization on staggered grid
Try going to Bengt Fornberg's website and downloading:
Spatial finite difference approximations for wavetype equations (BF and M. Ghrist). SIAM J. Num. Anal. 37 (1999), 105130. 
Re: Higher order discretization on staggered grid

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