I need to solve a system of DE's in 2 dimensions over time, The problem is that mathematica complains when I have a boundary condition of the same order as the differential order of the DE's, so I tried using the finiteelement method since it worked for other people who had similar problems, but it just returns itself as if nothing ever happened
P[x_, y_, t_] = e[x, y, t]/(γ - 1) ;
e[x_, y_, t_] = (γ - 1) ρ[x, y, t]/(μ mu ) kb T[x, y, t];
cp = 5/2 kb/(μ mu);
Rgas = 8.3144598;
cv = 5/2 kb/(μ mu) - Rgas;
γ = cp/cv;
g = 28.02*9.81;
μ = 0.6163328197226503`;
mu = 1.66053904*10^-27;
kb = 1.38064852*10^-23;
sol1 = NDSolve[{
D[ρ[x, y, t]*u[x, y, t],
t] == -D[ρ[x, y, t]*u[x, y, t]*u[x, y, t] + P[x, y, t], x] -
D[ρ[x, y, t]*u[x, y, t]*
v[x, y, t],
y],
D[ρ[x, y, t]*v[x, y, t],
t] == -D[ρ[x, y, t]*v[x, y, t]*
u[x, y, t],
y] - D[ρ[x, y, t]*v[x, y, t]*v[x, y, t] + P[x, y, t], y] +
g ρ[x, y, t],
D[ρ[x, y, t], t] == -D[ρ[x, y, t]*u[x, y, t], x] -
D[ρ[x, y, t]*v[x, y, t], y],
D[e[x, y, t], t] == -D[u[x, y, t]*e[x, y, t], x] -
D[v[x, y, t]*e[x, y, t], y] -
P[x, y, t]*(D[u[x, y, t], x] - D[v[x, y, t], y]),
v[0, y, t] == v[12*10^6, y, t],
u[0, y, t] == u[12*10^6, y, t],
T[0, y, t] == T[12*10^6, y, t],
ρ[0, y, t] == ρ[12*10^6, y, t],
(D[u[x, y, t], x] /. x -> 0) == (D[u[x, y, t], x] /. x -> 12000000),
(D[v[x, y, t], x] /. x -> 0) == (D[v[x, y, t], x] /. x -> 12000000),
D[u[0, y, t], y] == D[u[12000000, y, t], y],
D[v[0, y, t], y] == D[v[12000000, y, t], y],
D[u[0, y, t], t] == D[u[12000000, y, t], t],
D[v[0, y, t], t] == D[v[12000000, y, t], t],
(D[T[x, y, t], x] /. x -> 0) == (D[T[x, y, t], x] /. x -> 12000000),
(D[ρ[x, y, t], x] /. x -> 0) == (D[ρ[x, y, t], x] /.
x -> 12000000),
D[T[0, y, t], y] == D[T[12000000, y, t], y],
D[ρ[0, y, t], y] == D[ρ[12000000, y, t], y],
D[T[0, y, t], t] == D[T[12000000, y, t], t],
D[ρ[0, y, t], t] == D[ρ[12000000, y, t], t],
(D[e[x, y, t], x] /. x -> 0) == (D[e[x, y, t], x] /. x -> 12000000),
D[e[0, y, t], y] == D[e[12000000, y, t], y],
D[e[0, y, t], t] == D[e[12000000, y, t], t],
e[x, 0, t] == 3.83767261162,
v[x, 4000000, t] == 0,
v[x, 0, t] == 0,
(D[u[x, y, t], y] /. y -> 0) == 0,
(D[u[x, y, t], y] /. y -> 4000000) == 0,
v[x, y, 0] == 0,
u[x, y, 0] == 0,
T[x, y, 0] == 5770 + 0.00835414960707927 y,
ρ[x, y, 0] ==
1.42*10^-7*1.408*10^3 + 7.3561137493644*10^-10 y
},
{u, v, T, ρ}, {x, 0, 12000000}, {y, 0, 4000000}, {t, 0, 100},
Method -> "FiniteElement" ]
NDSolve
runs fine, and its results are not used later. So, I recommend that you delete it from your question to simplify it a bit. Readers tend to avoid questions in which the length of the code obscures the fundamental issues to be resolved. Your second instance ofNDSolve
fails becauseT
does not appear in any of its PDEs. Perhaps, you miscopied your code. $\endgroup$T
only appears in the boundary conditions. $\endgroup$