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WENO SCHEME BASICS Help Required!!!

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Old   January 15, 2007, 16:38
Default WENO SCHEME BASICS Help Required!!!
  #1
John
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Hi Guys,

I am trying to implement a basic 1-D hyperbolic equation using WENO scheme in Matlab. I am facing lot of trouble due to my boundary and also . Could you guys have a look at my Matlab code and send a quick comment..!! Itwill be of great help

fweno.m

function [DT]=fweno(u,ng,delx,eps) DTP=zeros(ng,1); DTN=zeros(ng,1); DT=zeros(ng,1);

% %extrapolate the beginning and end points of data %% u is the data with 3 extra points at the beginining and %at the end % ng is the number of spatial divisions

u(3) = 2*u(4)-u(5); u(2) = 2*u(3)-u(4); u(1) = 2*u(2)-u(3); u(ng-2) = 2*u(ng-3)-u(ng-4); u(ng-1) = 2*u(ng-2)-u(ng-3); u(ng) = 2*u(ng-1)-u(ng-2);

D = (u(2:ng)-u(1:ng-1))/delx;

% WENO scheme based on left-based stencil ( k=i-3, ........i+2)

for i=1ng-6)

k = i;

f1 = D(i);

f2 = D(i+1);

f3 = D(i+2);

f4 = D(i+3);

f5 = D(i+4);

SP1=(13/12)*(f1-2*f2+f3).^2+0.25.*(f1-4*f2+3*f3).^2;

SP2=(13/12)*(f2-2*f3+f4).^2+0.25.*(f2-f4).^2;

SP3=(13/12)*(f3-2*f4+f5).^2+0.25.*(3*f3-4*f4+f5).^2;

alphap1= 0.1/((eps+SP1).^2);

alphap2=0.6/((eps+SP2).^2);

alphap3=0.3/((eps+SP3).^2);

WPT=alphap1+alphap2+alphap3; WP1=alphap1./WPT; WP2=alphap2./WPT; WP3=alphap3./WPT;

DP1= f1/3- 7*f2/6 + 11*f3/6; DP2= -f2/6 + 5*f3/6 + f4/3; DP3= f3/3 + 5*f4/6 - f5/6;

DTP(i+3)=WP1.*DP1+WP2.*DP2+WP3.*DP3; DT(i+3)=DTP(i+3);

end

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Old   January 18, 2007, 04:16
Default Re: WENO SCHEME BASICS Help Required!!!
  #2
saygin
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I don't know Matlab coding syntax but I will try to help you.

First of all, what is the meaning of left-based or biased stencil? Do these (k=i-3, ........i+2) indicating nodes or cell centers? Cell centers should be from i-2 to i+2 for the reconstruction to i+1/2(-) and i-1/2(+). (-) (+) indicating the left- and right-side of the cell boundary respectively.

Can you expalin, what do this expression calculates? D = (u(2:ng)-u(1:ng-1))/delx;

i starts to count from 1, so f1=D(1), if it is your 3rd ghost cell it is ok. But the end-side uses ng-2 at most, I think it is not you 3rd ghost cell.

For finding i+1/2(-) and i-1/2(+) you should use different linear weights, and the characteristic decomposition should be done at each boundary of the cell wrt the reconstruction point.

Saygin

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