ENO scheme (ENORoe) (Shu and Osher).
HI all,
I have a question regarding the ENO scheme. I have been trying to implement an ENO's scheme based on Shu and Osher's paper "ENO and WENO shock capturing schemes II" , Journal of Computational Physics, 83, 3278 (1989). Now, it seems that this method is used when operating on cell centre values. I want to use cell averages. When I look at Laney's book, "Computational GasDynamics", on page 524 he states that "...in one interpretation, f(u(subscript i, superscript n)) is a cellintegral average of f(x)", where f(x) is the f^hat numerical flux. Laney's scheme in his book is Shu and Osher's scheme. This suggests that cell averages can be used. Can someone explain the apparent paradox between the use of cell averages and cell centred values? I am missing something here!! I hope! Regards, J. 
Re: ENO scheme (ENORoe) (Shu and Osher).
I'm not sure I understand your question. The ShuOsher ENO method is a finitedifference method, i.e., it approximates cellcentered values of the solution. The derivation given in my book, which involves cellaverages of the flux function, doesn't imply that the ShuOsher ENO method is finitevolume. If you want a finitevolume method, you'll have to use the original ENO method.
In one dimension, the cellcentered value of the solution is a secondorder approximation to the cellintegral average. Thus, if you can be content with second order accuracy, you can consider any finitedifference method to be finitevolume if you like (in one dimension). Hope this helps. Bert Laney 
Re: ENO scheme (ENORoe) (Shu and Osher).
Hi Bert.
Thanks for your reply. I am aware that a cellcentered value of a solution is a 2nd order approximation to a cellintegral (=finite volume?) average. However, I was hoping to achieve 3rd order accuracy. I would also like to extend the method to 3D. I was under the impression that a cellaverage was the same as finite volume, I take it I was wrong?! Finite volume, cellintegral and cellaverage all seem pretty similar to me? Why are there problems when extending to more than 1dimension? Will the original ENO scheme will work well in 3 dimensions? Thanks again, Regards. J. 
Re: ENO scheme (ENORoe) (Shu and Osher).
Dear J.K:
There is a paper of ChiWang Shu, "Essentially NonOscillatory and Weighted Essentially NonOscillatory Schemes for Hyperbolic Conservation Laws", NASA CR97206253 ICASE Report No. 9765. It can tell you how to implement three or more oder of accuracy of ENO or WENO in FV for a 3d problem. You can download it from http://www.icase.edu/library/reports/rdp/1997.html. Good luck to you! Zhao 
Re: ENO scheme (ENORoe) (Shu and Osher).
I have finished the ENO scheme FV in two dimensions.very difficult. if you want to get higher order than 2nd,you must do some effort to understand the reconstruction for two varriables.not simple dimension splitting technology,you must consider the crossterm.welcome to disscuss with me.

Eno
I am trying to write the matlab code for eno but I am unable to do
for i=2:nx2 k=i1; p1(i)=D1(k); if abs(D2(k+1))<= abs(D2(k+2)) p2(i)=(2*(ki)+1)*D2(k+1); else p2(i)=(2*(ki)+1)*D2(k+2); end Q=p1+p2; end for i=2:nx2 l=i; p11(i)=D1(l); if abs(D2(l+1))<= abs(D2(l+2)) p22(i)=(2*(li)+1)*D2(l+1); else p22(i)=(2*(li)+1)*D2(l+2); end P=p11+p22; end for i=1:nx2 V1(i+1)=Q(i); end for n=1:nt for i=2:nx2 V(n+1,j)=V(n,j)Dt*V1(n,j) i dont know what is wrong here I tried can u please suggest me something 
I am curious as to the difficulties that you talk of "autofly" when using the ENO scheme. It is a straightforward scheme.
I would certainly opt for WENO schemes if you can deal with the increase in solver time. Regarding the original post, the question entirely depends on your interpretation of the terms. The ENO scheme simply selects the smoothest polynomial to calculate the intercell flux values when it comes to FV methods. 
Eno
for example I am solving advection equation u_t+ au_x=0
and with initial data u(x,0)= 0.5x if x<= 0.5 = 0 else okay so I am using ENO via primitive function so I have calulated the average 1) for j=1:nx+3 x(j)=1dx+(j1)*dx; u0(j)=(u_0(x(j)+(dx/2))+u_0(x(j)) + u_0(x(j)(dx/2)))/3; end% 2) the scheme becomes u_{j}^{n+1} = u_{j}^{n} (f_{j+1/2} f_{j1/2}). 3) f_{j+1/2} i sthe numercial flux and now I am calculating numerical flux by Lax friendrich (as u can see in the article of shu page number 355) f_{j+1/2}= h(u_{j+1/2}, u+_{j1/2}). h(a,b)= (1/2)(f(a)+f(b)alpha(ba)). [since this is for u_t+ (f(u(x,t)))_x=0 ] so my f(u)= au. so my f_{j+1/2}=a u_j+1/2. so for calculating f_{j1/2 } i put j=j1 so f_{j+1/2}= au_j1/2. and as in the shu article u_j+1/2= p_(i)(x_{i+1/2}) so for calculating u_j1/2= p_(i1)(x_{i1/2}) I mean polynomail changes, and I am using the code for calculating this polynomail function y = back(u,nx,x,dx) y(1)=0; y(nx+3)=0; for i=1:nx+3 a(i,1)=u(i) end for i=2:nx+2 a(i,2)=(a(i,1)a(i1,1))/(x(i)x(i1)) end for i=2:nx+2 if abs(a(i,2)) < abs(a(i+1,2)) a1(i)=a(i,2)/2 else a1(i)=a(i+1,2)/2 end y(i)=u(i)+a1(i)*(xx(i)); end end so in above function when i put x=x(i)+(dx/2) for claulating the polynomial p(i1) I ahev tried many things but my scheme is not correct I can see very bad behaviour near the kink. I am really tired with this. Please suggets me something where I am wrong Quote:

Please soemone try to answer me and if possible please answer em soon
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