# higher order derivative on NON-UNIFORM grids

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 July 4, 2006, 04:41 higher order derivative on NON-UNIFORM grids #1 Chandra Guest   Posts: n/a Hi, I came across a problem in which I need to discretize "d4u/dx4" term on a NON-UNIFORM grid using 2nd order Central Difference. I tried to derive the form using general Taylor series (u[i+2], u[i+1], u[i], u[i-1], & u[i-2]) but I strucked in between. The difficulty is the following: a*(u[i+2]-u[i]) + b*(u[i+1]-u[i]) + c*(u[i-1]-u[i]) + d*(u[i-2]-u[i]) = function(du/dx, d2u/dx2, d3u/dx3, .....) On the right hand side after expansion, coeff of term [d5u/dx5] = 0 coeff of term [d4u/dx4] = 1 coeff of term [d3u/dx3] = 0 coeff of term [d2u/dx2] = 0 Solve these 4 relations to get a,b,c, and d. However, we still have the term du/dx on the right hand side whose co-efficient is NON-ZERO. My questions is that what to do with this term?? Should I incorporate more points in the Taylor-series (e.g. u[i+3] etc.) or just discretize the "du/dx" using CD-2 and use it in the above expression to get d4u/dx4? If any of you knows about this, please let me know. It will be a great help to me. Thank you very much in anticipation, -Chandra

 July 4, 2006, 05:06 Re: higher order derivative on NON-UNIFORM grids #2 Tom Guest   Posts: n/a Write down the discrete formula for q=d^2u/dx^2 on your nonuniform grid then apply the formula to calculate d^2q/dx^2. Elimate q from the discrete system and you have your result.

 July 4, 2006, 05:10 Re: higher order derivative on NON-UNIFORM grids #3 Chandra Guest   Posts: n/a Hey Tom, Thank you very much for your reply. I will try to implement it. -Chandra

 July 4, 2006, 11:50 Re: higher order derivative on NON-UNIFORM grids #4 Runge_Kutta Guest   Posts: n/a OK, here's the stencil: df_{i} = cl*f_{i+CL} + bl*f_{i+BL} + al*f_{i+AL} + A*f_{i} + ar*f_{i+AR} + br*f_{i+BR} + cr*f_{i+CR} + If the stencil had been designed for a uniform grid then CL = -3; BL = -2; AL = -1; AR = +1; BR = +2; CR = +3; If you want d^4/dx^4 on a nonuniform grid, you generally lose one order over the symmetrical stencil. Here's the first order stencil: A = 24/(AL AR BL BR) bl = 24/ ((AL - BL) (AR - BL) BL (BL - BR)) al = 24/ (AL (AL - AR) (AL - BL) (AL - BR)) ar = -24/ ((AL - AR) AR (AR - BL) (AR - BR)) br = -24/ ((AL - BR) BR (-AR + BR) (-BL + BR)) Truncation Error terms 0 *(xi^0) 0 *(xi^1) 0 *(xi^2) 0 *(xi^3) 1 *(xi^4) ((AL + AR + BL + BR)/5) *(xi^5) and here's the 3rd-order stencil: A = (24 (BL BR + BL CL + BR CL + BL CR + BR CR + CL CR + AR (BL + BR + CL + CR) + AL (AR + BL + BR + CL + CR))) ---------------------------------------------------------------------------------------------------------- (AL AR BL BR CL CR), cl = 24 (BL BR + BL CR + BR CR + AR (BL + BR + CR) + AL (AR + BL + BR + CR)) ----------------------------------------------------------------------- (AL - CL) (AR - CL) (BL - CL) (BR - CL) CL (CL - CR) bl = 24 (BR CL + BR CR + CL CR + AR (BR + CL + CR) + AL (AR + BR + CL + CR)) ----------------------------------------------------------------------- (AL - BL) (AR - BL) BL (BL - BR) (BL - CL) (BL - CR) al = 24 (BR CL + BR CR + CL CR + BL (BR + CL + CR) + AR (BL + BR + CL + CR)) ----------------------------------------------------------------------- AL (AL - AR) (AL - BL) (AL - BR) (AL - CL) (AL - CR) ar = -24 (BR CL + BR CR + CL CR + BL (BR + CL + CR) + AL (BL + BR + CL + CR)) ------------------------------------------------------------------------ (AL - AR) AR (AR - BL) (AR - BR) (AR - CL) (AR - CR) br = -24 (BL CL + BL CR + CL CR + AR (BL + CL + CR) + AL (AR + BL + CL + CR)) ------------------------------------------------------------------------ (AL - BR) BR (-AR + BR) (-BL + BR) (BR - CL) (BR - CR) cr = -24 (BL BR + BL CL + BR CL + AR (BL + BR + CL) + AL (AR + BL + BR + CL)) ------------------------------------------------------------------------ (AL - CR) CR (-AR + CR) (-BL + CR) (-BR + CR) (-CL + CR) Truncation Error terms 0 *(xi^0) 0 *(xi^1) 0 *(xi^2) 0 *(xi^3) 1 *(xi^4) 0 *(xi^5) 0 *(xi^6) (BL BR CL + BL BR CR + BL CL CR + BR CL CR + AR (CL CR + BR (CL + CR) + BL (BR + CL + CR)) + AL (BR CL + BR CR + CL CR + BL (BR + CL + CR) + AR (BL + BR + CL + CR)) ) / 210 *(xi^7) So, there's your answer. It was done in Mathematica using Fourier analysis. Good luck! Don't ask me for a tutorial ...

 July 4, 2006, 21:32 Re: higher order derivative on NON-UNIFORM grids #5 Chandra Guest   Posts: n/a Hey Runge_Kutta, thank you very much for your detailed reply. I am going through it. Also, if I need, I will be looking for some relevent book related to the details you mentioned...to avoid a tutorial from you By the way, this is really going to be quite useful, thank you, -Chandra

 July 5, 2006, 07:20 Re: higher order derivative on NON-UNIFORM grids #6 Chandra Guest   Posts: n/a Hello Runge_Kutta, well, i need one more favor from you. Could you please tell me the same thing for d2u/dx2 also. Also, if you have some link to these details, that would also be very useful. I tried to find out in the mathematica's documentation on internet but unfortunately I couldn't. I am doing DNS for turbulent flow and that is why I need all these. Thank you very much in advance, your help will be too helpful to me indeed. -Chandra

 July 7, 2006, 01:18 Re: higher order derivative on NON-UNIFORM grids #7 Runge_Kutta Guest   Posts: n/a Third-order version of d^2/dx^2 =============================== 2 (BL BR + AR (BL + BR) + AL (AR + BL + BR)) {{A -> --------------------------------------------, AL AR BL BR 2 (AR BR + AL (AR + BR)) > bl -> --------------------------------, (AL - BL) (AR - BL) BL (BL - BR) 2 (BL BR + AR (BL + BR)) > al -> --------------------------------, AL (AL - AR) (AL - BL) (AL - BR) 2 (BL BR + AL (BL + BR)) > ar -> ---------------------------------, (AL - AR) AR (-AR + BL) (AR - BR) -2 (AR BL + AL (AR + BL)) > br -> ----------------------------------}} (AL - BR) BR (-AR + BR) (-BL + BR) Truncation Error terms 0 *(xi^0) 0 *(xi^1) 1 *(xi^2) 0 *(xi^3) 0 *(xi^4) [AR BL BR + AL (BL BR + AR (BL + BR)) ]*(xi^5)/60 ============================================ I've never seen a book with this stuff. I hope you can read the Mathematica dump after this software mangles it!

 July 7, 2006, 04:31 Re: higher order derivative on NON-UNIFORM grids #8 Chandra Guest   Posts: n/a Hey, thank you very much indeed. This will be too useful for me. -Chandra

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