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Sods problem: Oscillations with WENO schemes

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Old   May 31, 2015, 17:57
Default Sods problem: Oscillations with WENO schemes
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Hey everyone!

When applying my WENO code for the Sods Shock Tube problem, I get some minor oscillations in the solutions:

I am using a central compact WENO scheme with SSP time stepping, which works well in the scalar case (Transport, Burgers, etc.). Initial data and numerical constants are taken from the paper

Thus the mistake must be in the numerical process of cell updating/ calculation the right hand side. I am just copying the necessary code fragments to maintain clarity.

- U is a matrix evaluated rho, m and E at every cell midpoint.
- The function WENO gives back the reconstructed value at the left side and at the right side for each cell for u1, u2, and u3.
- To calculate solutions at cell boundaries I am shifting the solution to the right, calculate the numerical flux and then shift that to the left due to conservation.
- F and the Jacobian matrix are written down explicitly and checked twice.

U = [rho; m; E]; 

%%%%% Calculating righthandside for time stepping %%%%%

function du = rhs(u)
vL2,vR2 = WENO(U)
z = f_num(shmatrixperiodl(vR2),vL2);
y = shmatrixperiodr(z);
du = -(y-z);

%%%%% Shift function routines %%%%%

function v = shmatrixperiodl(U)
v = [U(:,1) U(:,1:(end-1))];

function v = shmatrixperiodr(U)
v = [U(:,2:end) U(:,end)];

%%%%% Calculating numerical flux %%%%%

function Z = f_num(A,B)
% Numerical flux function
% Here: local lambdax-Friedrichs flux
for i=1:n
C(:,i) = max(abs(eig(Jacobmatrix(A(:,i)))), abs(eig(Jacobmatrix(B(:,i)))) );
Z = (F(A)+F(B)-C.*(B-A))./2;

%%%%% Flux Function %%%%%

function Y = F(U)
global gamma
Y = [U(2,:); 

%%%%% Jacobian %%%%%%

function Y = Jacobmatrix(U)
global gamma
Y = [ 0                                      1                                    0 
    -0.5*(3-gamma)*U(2).^2./(U(1).^2)      (3-gamma)*U(2)./U(1)                  (gamma-1);
    -gamma*U(3).*U(2)./(U(1).^2)+(gamma-1)*U(2).^3./(U(1).^3)  gamma*U(3)./U(1)+1.5*(1-gamma)*U(2).^2/(U(1).^2)     gamma*U(2)./U(1)];
I would be glad for some helpful advices!
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