# Julia DifferentialEquations for state-to-state binary mixture kinetics

Hello, I’m quite new to Julia and I’m considering the following problem. I’d like to solve the (possible stiff) ODE system which describes the relaxation of a flow behind a shock wave according to a state-to-state approach which means that each vibrational level of the molecular species is considered as a pseudo-species with its continuity equation. Here, I consider a binary mixture of N2/N (actually the concentration of N=0).

I have splitted the julia code in several .jl files. In the main, I call the ODE solver as follows:

``````prob = ODEProblem(rpart!,Y0_bar,xspan, 1.)
sol  = DifferentialEquations.solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8, save_everystep=true, progress=true)
``````

where Y0_bar and xspan have been defined earlier and in the rpart.jl file I define the system:

``````function rpart!(du,u,p,t)

ni_b = zeros(l);

ni_b[1:l] = u[1:l]; print("ni_b = ", ni_b, "\n")
na_b = u[l+1];      print("na_b = ", na_b, "\n")
v_b  = u[l+2];      print("v_b = ",  v_b,  "\n")
T_b  = u[l+3];      print("T_b = ",  T_b,  "\n")

nm_b = sum(ni_b);   #print("nm_b = ", nm_b, "\n")
Lmax = l-1;         #println("Lmax = ", Lmax, "\n")
temp = T_b*T0;      #print("T = ", temp, "\n")

ef_b = 0.5*D/T0;    #println("ef_b = ", ef_b, "\n")

ei_b = e_i./(k*T0); #println("ei_b = ", ei_b, "\n")
e0_b = e_0/(k*T0);  #println("e0_b = ", e0_b, "\n")

sigma   = 2.;                     #println("sigma = ", sigma, "\n")
Theta_r = Be*h*c/k;               #println("Theta_r = ", Theta_r, "\n")
Z_rot   = temp./(sigma.*Theta_r); #println("Z_rot = ", Z_rot, "\n")

M  = sum(m); #println("M = ", M, "\n")
mb = m/M;    #println("mb = ", mb, "\n")

A = zeros(l+3,l+3)

for i = 1:l
A[i,i]   = v_b
A[i,l+2] = ni_b[i]
end

A[l+1,l+1] = v_b
A[l+1,l+2] = na_b

for i = 1:l+1
A[l+2,i] = T_b
end
A[l+2,l+2] = M*v0^2/k/T0*(mb[1]*nm_b+mb[2]*na_b)*v_b
A[l+2,l+3] = nm_b+na_b

for i = 1:l
A[l+3,i] = 2.5*T_b+ei_b[i]+e0_b
end
A[l+3,l+1] = 1.5*T_b+ef_b
A[l+3,l+2] = 1/v_b*(3.5*nm_b*T_b+2.5*na_b*T_b+sum((ei_b.+e0_b).*ni_b)+ef_b*na_b)
A[l+3,l+3] = 2.5*nm_b+1.5*na_b

AA = inv(A); println("AA = ", AA, "\n", size(AA), "\n")

# Equilibrium constant for DR processes
Kdr = (m[1]*h^2/(m[2]*m[2]*2*pi*k*temp))^(3/2)*Z_rot*exp.(-e_i/(k*temp))*exp(D/temp); println("Kdr = ", Kdr, "\n")

# Dissociation processes
kd = zeros(2,l)
kd = kdis(temp) * Delta*n0/v0;
println("kd = ", kd, "\n", size(kd), "\n")

# Recombination processes
kr = zeros(2,l)
for iM = 1:2
kr[iM,:] = kd[iM,:] .* Kdr * n0
end
println("kr = ", kr, "\n", size(kr), "\n")

RD  = zeros(l)

for i1 = 1:l

RD[i1] = nm_b*(na_b*na_b*kr[1,i1]-ni_b[i1]*kd[1,i1]) + na_b*(na_b*na_b*kr[2,i1]-ni_b[i1]*kd[2,i1])

end

println("RD = ",  RD,  "\n", size(RD))

B      = zeros(l+3)
for i  = 1:l
B[i] = RD[i]
end
B[l+1] = - 2*sum(RD)

du     = AA*B

end
``````

All constant parameters are correctly set. The problem is that when I run the simulation and plot the solution it looks like nothing happened and all profiles are equal and flat. In fact, the solution at each time-step is equal to itself. So, I think I make some mistake in the update of u and du but I cannot fix it. In the Matlab version I obtain a correct evolution.

Kind regards,
Lorenzo

`du .= AA*B`