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burgersRHS.m
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function [RHS] = burgersRHS(Q)
global turbmodel k Cs Delta tausgs tauhatsgs nx nu dx u convective_flux solution_domain ...
uhat M t x Gf w0hat w1hat w2hat
if (solution_domain == 1)
Q2Umap();
uxx = zeros(size(u));
uxx(2:end-1) = (u(3:end) - 2.*u(2:end-1) + u(1:end-2))./dx^2;
uxx(end) = (u(1) - 2.*u(end) + u(end-1))./dx^2;
uxx(1) = (u(2) - 2.*u(1) + u(end))./dx^2;
if (convective_flux == 1)
%% Upwind Convective Flux
fplus = 0.25*u.*(u + abs(u));
fminus = 0.25*u.*(u - abs(u));
f = zeros(size(u));
f(2:end) = 1./dx*(fplus(2:end) - fplus(1:end-1));
f(1) = 1./dx*(fplus(1) - fplus(end));
f(1:end-1) = f(1:end-1) + 1./dx*(fminus(2:end) - fminus(1:end-1));
f(end) = f(end) + 1./dx*(fminus(1) - fminus(end));
end
if (convective_flux == 2)
f = kurgTadmorFlux(u,nu,dx);
end
if (convective_flux == 3)
f = zeros(size(u));
f(2:end-1) = (u(3:end).^2 - u(1:end-2).^2)/(2.*dx);
f(end) = (u(1).^2 - u(end-1).^2)/(2.*dx);
f(1) = (u(2).^2 - u(end).^2)/(2.*dx);
end
if (turbmodel == 0)
RHS = -f + nu.*uxx;
tausgs = zeros(size(u));
tauhatsgs = zeros(size(u));
end
if (turbmodel == 1)
[nut,tau_eq] = smagorinsky(u,dx);
nut_x = diff1(nut,dx);
u_x = diff1(u,dx);
RHS = -f + nu.*uxx + nut_x.*u_x + nut.*uxx;
tausgs = tau_eq;
tauhatsgs = myfft(tausgs,nx);
end
if (turbmodel == 2)
tscale = -0.1;
u_x = diff1(u,dx);
[nut,tau_eq] = smagorinsky(u,dx);
tau_x = diff1(tausgs,dx);
RHS1 = -f + nu.*uxx - tau_x;
%%% Upwind tau equation
ftplus = 0.5*(u + abs(u)).*tausgs;
ftminus = 0.5*(u - abs(u)).*tausgs;
ft = zeros(size(u));
ft(2:end) = 1./dx*(ftplus(2:end) - ftplus(1:end-1));
ft(1) = 1./dx*(ftplus(1) - ftplus(end));
ft(1:end-1) = ft(1:end-1) + 1./dx*(ftminus(2:end) - ftminus(1:end-1));
ft(end) = ft(end) + 1./dx*(ftminus(1) - ftminus(end));
RHS2 =-ft + 1./tscale * (tausgs - tau_eq);
RHS = zeros(2*nx,1);
RHS(1:2:end) = RHS1;
RHS(2:2:end) = RHS2;
end
if (turbmodel == 3)
tscale = -0.2;
u_x = diff1(u,dx);
[nut,tau_eq] = smagorinsky(u,dx);
tau_x = diff1(tausgs,dx);
RHS1 = -f + nu.*uxx - tau_x;
%%% Upwind tau equation
uplus = 0.5*(u + abs(u));
uminus = 0.5*(u - abs(u));
ft = zeros(size(u));
ft(2:end) = uplus(2:end).*1./dx.*(tausgs(2:end) - tausgs(1:end-1));
ft(1) = uplus(1).*1./dx.*(tausgs(1) - tausgs(end));
ft(1:end-1) = ft(1:end-1) + uminus(1:end-1).*1./dx.*(tausgs(2:end) - tausgs(1:end-1));
ft(end) = ft(end) + uminus(end).*1./dx.*(tausgs(1) - tausgs(end));
RHS2 =-ft + 1./tscale * (tausgs - tau_eq);
RHS = zeros(2*nx,1);
RHS(1:2:end) = RHS1;
RHS(2:2:end) = RHS2;
end
if (turbmodel == 4)
tau_x = diff1(tausgs,dx);
[nut,tau_eq] = smagorinsky(u,dx);
RHS1 = -f + nu.*uxx - tau_x;
%%% Upwind tau equation
ftplus = 0.5*(u + abs(u)).*tausgs;
ftminus = 0.5*(u - abs(u)).*tausgs;
ft = zeros(size(u));
ft(2:end) = 1./dx*(ftplus(2:end) - ftplus(1:end-1));
ft(1) = 1./dx*(ftplus(1) - ftplus(end));
ft(1:end-1) = ft(1:end-1) + 1./dx*(ftminus(2:end) - ftminus(1:end-1));
ft(end) = ft(end) + 1./dx*(ftminus(1) - ftminus(end));
RHS = zeros(2*nx,1);
RHS2 = -2./3.*ft + (nu + 15*nut).*diff2(tausgs,dx) + u.*diff1(tausgs,dx);
RHS(1:2:end) = RHS1;
RHS(2:2:end) = RHS2;
end
if (turbmodel == 5)
%kf = linspace(2*min(k),2*max(k+1)-1,2*nx);
%GfD = heaviside(pi/Delta - abs(kf) )';
%uhat = myifft(Q,nx);
%Lhat = -0.5*cconv([uhat;real(uhat(1)) - 1j*imag(uhat(1))],[uhat;real(uhat(1)) - 1j*imag(uhat(1))],2*nx+1);
%Lhat(nx/2+1:3*nx/2+1) = 0.;
%dxf = 2.*pi/(nx*2);
%xf = linspace(0,2.*pi - dxf,2*nx+1);
%L = interp1(xf,real(myifft(Lhat,2.*nx)),x)';
L = 0.5*smooth(u.*u,'lowess') - 0.5*smooth(u,'lowess').*smooth(u,'lowess');
tausgs = -t.*u.*diff1(L,dx)*0.1;
tauhatsgs = myfft(tausgs,nx);
RHS = -f + nu.*uxx -diff1(tausgs,dx);
end
end
if (solution_domain == 2)
if (turbmodel == 0)
c = myconv(Q,Q,length(Q));
RHS = -0.5.*1j.*k.*c - nu.*k.^2.*Q;
tauhatsgs = 0.5*(Gf.*myconv(Q,Q,nx) - myconv(Gf.*Q,Gf.*Q,nx));
tausgs = myifft(tauhatsgs,nx);
end
if (turbmodel == 1)
utmp = myifft(uhat,nx);
[nut,tau_eq] = smagorinsky(utmp,dx);
nut_x = diff1(nut,dx);
tausgs = tau_eq;
tauhatsgs = myfft(tausgs,nx);
c = myconv(uhat,uhat,length(uhat));
RHS = -0.5.*1j.*k.*c - nu.*k.^2.*uhat - 1j.*k.*tauhatsgs;
end
if (turbmodel == 5)
uhat = Q;
c = myconv(uhat,uhat,length(uhat));
uhat2 = zeros(2*nx,1);
uhat2(nx/2+1:nx/2+nx,1) = uhat;
uhat2(nx/2+nx+1) = real(uhat(1)) - 1j*imag(uhat(1));
tmp = -t.*1j.*M./2.*myconv(uhat2,uhat2,length(uhat2));
tauhatsgs = .35*myconv(uhat,tmp,length(uhat));
RHS = -0.5.*1j.*k.*c - nu.*k.^2.*uhat - 1j.*k.*tauhatsgs;
tausgs = myifft(tauhatsgs,nx);
end
if (turbmodel == 6)
kc = pi/Delta;
c = myconv(uhat,uhat,length(uhat));
[val,index] = min(abs(k - kc));
nut = 0.47.*ones(size(k)).*(uhat(index).*conj(uhat(index))/kc).^0.37;
tauhatsgs = -1j.*k.*nut.*uhat;
%tauhatsgs = myconv(-nut,1j.*k.*uhat);
RHS = -0.5.*1j.*k.*c - nu.*k.^2.*uhat - 1j.*k.*tauhatsgs;
tausgs = myifft(tauhatsgs,nx);
end
if (turbmodel == 7)
uhat = Q;
DeltaD = Delta*2;
GfD = heaviside(pi/DeltaD - abs(k));
uhatFilt = GfD.*uhat;
uReal = myifft(uhat,nx);
uReal_x = diff1(uReal,dx);
uRealFilt = myifft(uhatFilt,nx);
uRealFilt_x =diff1(uRealFilt,dx);
L = 0.5*GfD.*myconv(uhat,uhat,nx) - 0.5*myconv(GfD.*uhat,GfD.*uhat,nx);
val = myfft(Delta^2*abs(uReal_x).*uReal_x,nx);
val2 = myfft(DeltaD^2*abs(uRealFilt_x).*uRealFilt_x,nx);
M = GfD.*val - val2;
Cs2 = L./M;
nut = Cs2.*Delta.^2.*abs(uReal_x);
tausgs = -nut.*uReal_x;
tauhatsgs = myfft(tausgs,nx);
c = myconv(uhat,uhat,length(uhat));
RHS = -0.5.*1j.*k.*c - nu.*k.^2.*uhat - 1j.*k.*tauhatsgs;
end
if (turbmodel == 8)
DeltaD = Delta*2;
GfD = heaviside(pi/DeltaD - abs(k));
L = 0.5*GfD.*myconv(Q,Q,nx) - 0.5*myconv(GfD.*Q,GfD.*Q,nx);
tauhatsgs = 0.25*L;
tausgs = myifft(tauhatsgs,nx);
c = myconv(Q,Q,length(Q));
RHS = -0.5.*1j.*k.*c - nu.*k.^2.*Q - 1j.*k.*tauhatsgs;
end
if (turbmodel == 9)
uhat2 = zeros(2*nx,1);
uhat2(nx/2+1:nx/2+nx,1) = Q;
uhat2(nx/2+nx+1) = real(Q(1)) - 1j*imag(Q(1));
kf = linspace(-nx,nx-1,2*nx)';
p = zeros(2*nx,1);
q = kf - p;
p(nx/2+1:nx/2+nx+1) = kf(nx/2+1:nx/2+nx+1);
res = zeros(2*nx,1); res(nx/2+1:nx/2+nx+1) = 1;
unres = ones(2*nx,1) - res;
nxf = length(uhat2);
PLu = -1j.*kf./2.*myconv(uhat2,uhat2,nxf) - nu.*kf.^2.*uhat2;
PLu_p = res.*PLu;
PLu_q = unres.*PLu;
tmp = -t.*1j.*M./2.*myconv(uhat2,uhat2,length(uhat2));
tauhatsgs = 0.35*myconv(uhat,tmp,length(uhat));
PLPLu = -1j.*kf./2.*myconv(2.*uhat2,-1j.*p./2.*myconv(uhat2,uhat2,nxf) - nu.*p.^2.*uhat2 ,nxf) ...
- p.^2.*nu.*(-1j.*kf./2.*myconv(uhat2,uhat2,nxf) - kf.^2.*nu.*uhat2);
PLPLu_p = res.*PLPLu;
PLPLu_q = unres.*PLPLu;
% tau1 = t*myconv(uhat2,PLu_q,nxf);
% tau2 = -t.^2/2.*(myconv(uhat2,PLPLu_q,nxf) + myconv(PLu_p,PLu_q,nxf));
% tauhatsgs = (tau1 + 1.0*tau2);
%
tmod = t.*2.*(-1j.*kf./2.*myconv(uhat2,PLu_q,nxf) - 0.5.*q.^2.*nu.*PLu_q);
t2mod = -t.^2./2.*2.*(-1j.*kf./2.*myconv(uhat2,PLPLu_q,nxf) -0.5.*q.^2.*nu.*PLPLu_q ...
+ -1j.*kf./2.*myconv(PLu_p,PLu_q,nxf) -0.5.*p.^2.*nu.*PLu_q);
tterm = tmod(nx/2+1:nx/2+nx)+ t2mod(nx/2+1:nx/2+nx);
tauhatsgs = tterm./(-1j.*k + 1e-30);
% t2mod = -1j.*kf.*(myconv(uhat2,PLPLu_q,nxf) + myconv(PLu_p,PLu_q,nxf));
% tauhatsgs = t*myconv(uhat2,PLu_q,nxf) - t.^2/2.*(myconv(uhat2,PLPLu_q,nxf) + myconv(PLu_p,PLu_q,nxf));
% tauhatsgs = tauhatsgs(nx/2+1:nx/2+nx);
tausgs = myifft(tauhatsgs,nx);
c = myconv(uhat,uhat,length(uhat));
% RHS = -0.5.*1j.*k.*c - nu.*k.^2.*Q -1j.*k.*tauhatsgs;
RHS = -0.5.*1j.*k.*c - nu.*k.^2.*Q + tterm;
end
if (turbmodel == 10)
L = computeLeonardStress(Gf,Q);
tauhatsgs = 0.25*L;
tausgs = myifft(tauhatsgs,nx);
c = myconv(Q,Q,length(Q));
RHS = -0.5.*1j.*k.*c - nu.*k.^2.*Q -1j.*k.*tauhatsgs;
end
if (turbmodel == 11)
Q2Umap();
uhat2 = zeros(2*nx,1);
kf = linspace(-nx,nx-1,2*nx)';
uhat2(nx/2+1:nx/2+nx,1) = uhat;
uhat2(nx/2+nx+1) = real(uhat(1)) - 1j*imag(uhat(1));
nxf = length(uhat2);
PLu = -1j.*kf./2.*myconv(uhat2,uhat2,nxf) - nu.*kf.^2.*uhat2;
PLu_q = PLu; PLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLu_p = PLu - PLu_q;
PLQLu = 2.*-1j.*kf./2.*myconv(uhat2,PLu_q,nxf) - kf.^2.*nu.*PLu_q;
PLQLu_q = PLQLu; PLQLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLQLu_p = PLQLu - PLQLu_q;
PLQLQLu = 2.*-1j.*kf./2.*myconv(PLQLu_q,uhat2) + 2.*-1j.*kf./2.*myconv(PLu_q,PLu_p) + ...
2.*-1j.*kf./2.*myconv(PLu_q,PLu_q) - nu.*kf.^2.*PLQLu_q;
RHS1 = PLu(nx/2+1:nx/2+nx) + 0.1*w0hat;
RHS2 = PLQLu(nx/2+1:nx/2+nx) + 0.0*w1hat;
RHS3 = PLQLQLu(nx/2+1:nx/2+nx) - 100.*k.^2*nu.*w1hat;
tauhatsgs = 0.02*w0hat./(-1j.*k + 1e-30);
tausgs = myifft(tauhatsgs,nx);
RHS = zeros(nx*3,1);
RHS(1:3:end) = RHS1;
RHS(2:3:end) = RHS2;
RHS(3:3:end) = RHS3;
end
if (turbmodel == 12)
dt0 = 0.1;
dt1 = 0.01;
Q2Umap();
uhat2 = zeros(2*nx,1);
kf = linspace(-nx,nx-1,2*nx)';
uhat2(nx/2+1:nx/2+nx,1) = uhat;
uhat2(nx/2+nx+1) = real(uhat(1)) - 1j*imag(uhat(1));
nxf = length(uhat2);
PLu = -1j.*kf./2.*myconv(uhat2,uhat2,nxf) - nu.*kf.^2.*uhat2;
PLu_q = PLu; PLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLu_p = PLu - PLu_q;
PLQLu = 2.*-1j.*kf./2.*myconv(uhat2,PLu_q,nxf) - kf.^2.*nu.*PLu_q;
PLQLu_q = PLQLu; PLQLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLQLu_p = PLQLu - PLQLu_q;
PLQLQLu = 2.*-1j.*kf./2.*myconv(PLQLu_q,uhat2) + 2.*-1j.*kf./2.*myconv(PLu_q,PLu_p) + ...
2.*-1j.*kf./2.*myconv(PLu_q,PLu_q) - nu.*kf.^2.*PLQLu_q;
RHS1 = PLu(nx/2+1:nx/2+nx) + w0hat;
RHS2 = -2./dt0.*w0hat + 2.*PLQLu(nx/2+1:nx/2+nx) + w1hat;
RHS3 = -2./dt1.*w1hat + 2.*PLQLQLu(nx/2+1:nx/2+nx);
tauhatsgs = w0hat./(-1j.*k + 1e-30);
tausgs = myifft(tauhatsgs,nx);
RHS = zeros(nx*3,1);
RHS(1:3:end) = RHS1;
RHS(2:3:end) = RHS2;
RHS(3:3:end) = RHS3;
end
if (turbmodel == 13)
Q2Umap();
uhat2 = zeros(2*nx,1);
kf = linspace(-nx,nx-1,2*nx)';
uhat2(nx/2+1:nx/2+nx,1) = uhat;
uhat2(nx/2+nx+1) = real(uhat(1)) - 1j*imag(uhat(1));
nxf = length(uhat2);
PLu = -1j.*kf./2.*myconv(uhat2,uhat2,nxf) - nu.*kf.^2.*uhat2;
PLu_q = PLu; PLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLu_p = PLu - PLu_q;
PLQLu = 2.*-1j.*kf./2.*myconv(uhat2,PLu_q,nxf) - kf.^2.*nu.*PLu_q;
PLQLu_q = PLQLu; PLQLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLQLu_p = PLQLu - PLQLu_q;
PLPLu = 2.*-1j.*kf./2.*myconv(uhat2,PLu_p) - kf.^2.*nu.*PLu_p;
PLPLu_q = PLPLu; PLPLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLPLu_p = PLPLu - PLPLu_q;
PLQLQLu = 2.*-1j.*kf./2.*myconv(PLQLu_q,uhat2) + 2.*-1j.*kf./2.*myconv(PLu_q,PLu_p) + ...
2.*-1j.*kf./2.*myconv(PLu_q,PLu_q) - nu.*kf.^2.*PLQLu_q;
PLQLQLu_q = PLQLQLu; PLQLQLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLQLQLu_p = PLQLQLu - PLQLQLu_q;
PLQLQLQLu = 2.*-1j.*kf./2.*myconv(PLQLQLu_q,uhat2) + 4.*-1j.*kf./2.*myconv(PLu_p,PLQLu_q) + ...
4.*-1j.*kf./2.*myconv(PLQLu_p,PLu_q) + 2.*-1j.*kf./2.*myconv(PLPLu_p,PLu_q) + ...
6.*-1j.*kf./2.*myconv(PLQLu_q,PLu_q) + 2.*-1j.*kf./2.*myconv(PLPLu_q,PLu_q) + ...
- nu.*kf.^2.*PLQLQLu_q;
RHS1 = PLu(nx/2+1:nx/2+nx) + w0hat;
RHS2 = PLQLu(nx/2+1:nx/2+nx) + w1hat;
RHS3 = PLQLQLu(nx/2+1:nx/2+nx) + w2hat;
dt2 = 1;
RHS4 = -2./dt2.*w2hat + 2.*PLQLQLQLu(nx/2+1:nx/2+nx);
tauhatsgs = w0hat./(-1j.*k + 1e-30);
tausgs = myifft(tauhatsgs,nx);
RHS = zeros(nx*4,1);
RHS(1:4:end) = RHS1;
RHS(2:4:end) = RHS2;
RHS(3:4:end) = RHS3;
RHS(4:4:end) = RHS4;
end
if (turbmodel == 14)
dt0 = 0.13;
dt1 = 0.07;
dt2 = 0.05;
Q2Umap();
uhat2 = zeros(2*nx,1);
kf = linspace(-nx,nx-1,2*nx)';
uhat2(nx/2+1:nx/2+nx,1) = uhat;
uhat2(nx/2+nx+1) = real(uhat(1)) - 1j*imag(uhat(1));
nxf = length(uhat2);
PLu = -1j.*kf./2.*myconv(uhat2,uhat2,nxf) - nu.*kf.^2.*uhat2;
PLu_q = PLu; PLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLu_p = PLu - PLu_q;
PLQLu = 2.*-1j.*kf./2.*myconv(uhat2,PLu_q,nxf) - kf.^2.*nu.*PLu_q;
PLQLu_q = PLQLu; PLQLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLQLu_p = PLQLu - PLQLu_q;
PLPLu = 2.*-1j.*kf./2.*myconv(uhat2,PLu_p) - kf.^2.*nu.*PLu_p;
PLPLu_q = PLPLu; PLPLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLPLu_p = PLPLu - PLPLu_q;
PLQLQLu = 2.*-1j.*kf./2.*myconv(PLQLu_q,uhat2) + 2.*-1j.*kf./2.*myconv(PLu_q,PLu_p) + ...
2.*-1j.*kf./2.*myconv(PLu_q,PLu_q) - nu.*kf.^2.*PLQLu_q;
PLQLQLu_q = PLQLQLu; PLQLQLu_q(nx/2+1:nx/2+nx+1) = 0.;
PLQLQLu_p = PLQLQLu - PLQLQLu_q;
PLQLQLQLu = 2.*-1j.*kf./2.*myconv(PLQLQLu_q,uhat2) + 4.*-1j.*kf./2.*myconv(PLu_p,PLQLu_q) + ...
4.*-1j.*kf./2.*myconv(PLQLu_p,PLu_q) + 2.*-1j.*kf./2.*myconv(PLPLu_p,PLu_q) + ...
6.*-1j.*kf./2.*myconv(PLQLu_q,PLu_q) + 2.*-1j.*kf./2.*myconv(PLPLu_q,PLu_q) + ...
- nu.*kf.^2.*PLQLQLu_q;
RHS1 = PLu(nx/2+1:nx/2+nx) + w0hat;
RHS2 = -2./dt0.*w0hat + 2.*PLQLu(nx/2+1:nx/2+nx) + w1hat;
RHS3 = -2./dt1.*w1hat + 2.*PLQLQLu(nx/2+1:nx/2+nx) + w2hat;
RHS4 = -2./dt2.*w2hat + 2.*PLQLQLQLu(nx/2+1:nx/2+nx);
tauhatsgs = w0hat./(-1j.*k + 1e-30);
tausgs = myifft(tauhatsgs,nx);
RHS = zeros(nx*4,1);
RHS(1:4:end) = RHS1;
RHS(2:4:end) = RHS2;
RHS(3:4:end) = RHS3;
RHS(4:4:end) = RHS4;
end
end