我有一个耦合偏微分方程组，如1所示。
如果按分量写出方程式，则得到9个方程式。 为简单起见，可以将$ \ rho $和$ h $设置为常量。
我将方程式放入mathematica笔记本并使用'NDSolve'。 我在系统中计算了268个衍生品。但是，我没有找到87个以上的条件。
当我运行笔记本时，我收到消息
NDSolve::pdord: Some of the functions have zero differential order, so the equations will be solved as a system of differential-algebraic equations. >>
LinearSolve::sing: Matrix SparseArray[Automatic,<<3>>] is singular. >>
NDSolve::mconly: For the method IDA, only machine real code is available. Unable to continue with complex values or beyond floating-point exceptions. >>
我理解第一个错误：$ \ phi $中没有时间导数，所以系统严格来说是一个微分代数系统。
大约40分钟后，mathematica说：
No more memory available.
Mathematica kernel has shut down.
Try quitting other applications and then retry.
我可以改变什么来降低内存成本，或者 - 更好 - 我可以改变哪些来解决系统更优雅？
这是mathematica代码：
NDSolve[{
eqn1 == 0,
eqn2 == 0,
eqn3 == 0,
eqn4 == 0,
eqn5 == 0,
eqn6 == 0,
eqn7 == 0,
eqn8 == 0,
eqn9 == 0,
(*Initial Conditions*)
T[0, x, y, z] == 300,
p[0, x, y, z] == 101300,
phi[t, x, y, z] == 75 x /. t -> 0,
u1[0, x, y, z] == 0,
(* ect... *)
(*BOUNDARY CONDITIONS*)
phi[t, x, y, z] == 0 /. x -> 0,
phi[t, x, y, z] == 750 /. x -> 10,
phi[t, x, y, z] == 75 x /. y -> 0,
phi[t, x, y, z] == 75 x /. y -> 10,
phi[t, x, y, z] == 75 x /. z -> 0,
phi[t, x, y, z] == 75 x /. z -> 10,
\!\(
\*SubscriptBox[\(\[PartialD]\), \(y\)]\(phi[t, x, y, z]\)\) == 0 /.
y -> 0,
\!\(
\*SubscriptBox[\(\[PartialD]\), \(y\)]\(phi[t, x, y, z]\)\) == 0 /.
y -> 10,
(* ect..... *)
},
{p, u1, u2, u3, T, phi, a1, a2, a3},
{t, 0, 10}, {x, 0, 10}, {y, 0, 10}, {z, 0, 10}
]
以下是方程式的Latex代码：
\begin{align}
%%% first
\frac{\partial \rho}{\partial t} + \vec{u} \cdot \nabla \rho + \rho \nabla \cdot \vec{u} &=0 \\
%%% second
\rho \frac{\partial \vec{u}}{\partial t}
+ \rho \vec{u} \cdot \nabla \vec{u} \pm \nabla p
+ \nabla \cdot \left[- \mu \left( \nabla \vec{u} + \nabla \vec{u}^t \right) \right] \nonumber
\\ -
\left[ \sigma
\left( - \nabla \phi - \frac{\partial \vec{A}}{\partial t} \right)
+ \vec{u} \times (\nabla \times \vec{A}) \right]
%
\times \left[ \nabla \times \vec{A} \right]
&= \vec{0} \\
%%% third
\rho \frac{\partial h}{\partial t}
+ \rho \vec{u} \cdot \nabla h
+ \nabla \cdot \nabla T
- \frac{\partial p}{\partial t}
- \overset{.}{Q_j} + \overset{.}{Q_r}
&= 0 \\
%%% fourth
\nabla \cdot \left[ \sigma
\left( - \nabla \phi - \frac{\partial \vec{A}}{\partial t} \right)
+ \vec{u} \times (\nabla \times \vec{A}) \right] &= 0 \\
%%% fifth
\frac{\partial \vec{A}}{\partial t} + \nabla \phi - \vec{u} \times \nabla \times \vec{A} - \eta \nabla ^2 \vec{A} &= \vec{0}
\end{align}
我在这里添加了简单的mathematica代码：
Needs["NDSolve`FEM`"]
kB = N[ 8.6*10^-5 ];
AG = N[1.2*10^6 ];
mu0 = 1.257*10^(-6);
sigma = N[20.];
eta = 1/(mu0 * sigma);
rho = N[1.];
h = N[1672000.];
r = 1/ 287.058 ;
mu = N[1.];
kappah = N[5.];
cp = N[2.4];
lbb = NDSolve[{
0 == D[rho, t] + u1[t, x, y, z]*D[rho, x] +
u2[t, x, y, z]*D[rho, y] + u3[t, x, y, z]*D[rho, z] +
rho*(D[u1[t, x, y, z], x] + D[u2[t, x, y, z], y] +
D[u3[t, x, y, z], z]),
0 ==
rho*D[u1[t, x, y, z], t] +
rho*u1[t, x, y, z]*D[u1[t, x, y, z], x] - D[p[t, x, y, z], x] -
mu*D[D[u1[t, x, y, z], x], x] -
mu*D[D[u1[t, x, y, z], y], y] -
mu*D[D[u1[t, x, y, z], z], z] -
(((-sigma)*D[phi[t, x, y, z], y] -
sigma*D[a2[t, x, y, z], t] +
u3[t, x, y,
z]*(D[a3[t, x, y, z], y] - D[a2[t, x, y, z], z]) -
u1[t, x, y,
z]*(D[a2[t, x, y, z], x] - D[a1[t, x, y, z], y]))*(D[
a2[t, x, y, z], x] - D[a1[t, x, y, z], y]) -
((-sigma)*D[phi[t, x, y, z], z] -
sigma*D[a3[t, x, y, z], t] +
u1[t, x, y,
z]*(D[a1[t, x, y, z], z] - D[a3[t, x, y, z], x]) -
u2[t, x, y,
z]*(D[a3[t, x, y, z], y] - D[a2[t, x, y, z], z]))*(D[
a1[t, x, y, z], z] - D[a3[t, x, y, z], x])),
0 ==
rho*D[u2[t, x, y, z], t] +
rho*u2[t, x, y, z]*D[u2[t, x, y, z], y] - D[p[t, x, y, z], y] -
mu*D[D[u2[t, x, y, z], x], x] -
mu*D[D[u2[t, x, y, z], y], y] -
mu*D[D[u2[t, x, y, z], z], z] -
(((-sigma)*D[phi[t, x, y, z], z] -
sigma*D[a3[t, x, y, z], t] +
u1[t, x, y,
z]*(D[a1[t, x, y, z], z] - D[a3[t, x, y, z], x]) -
u2[t, x, y,
z]*(D[a3[t, x, y, z], y] - D[a2[t, x, y, z], z]))*(D[
a3[t, x, y, z], y] - D[a2[t, x, y, z], z]) -
(D[a2[t, x, y, z], x] -
D[a1[t, x, y, z], y])*((-sigma)*D[phi[t, x, y, z], x] -
sigma*D[a1[t, x, y, z], t] +
u2[t, x, y,
z]*(D[a2[t, x, y, z], x] - D[a1[t, x, y, z], y]) -
u3[t, x, y,
z]*(D[a1[t, x, y, z], z] - D[a3[t, x, y, z], x]))),
0 ==
rho*D[u3[t, x, y, z], t] +
rho*u3[t, x, y, z]*D[u3[t, x, y, z], z] - D[p[t, x, y, z], z] -
mu*D[D[u3[t, x, y, z], x], x] -
mu*D[D[u3[t, x, y, z], y], y] -
mu*D[D[u3[t, x, y, z], z], z] -
(((-sigma)*D[phi[t, x, y, z], x] -
sigma*D[a1[t, x, y, z], t] +
u2[t, x, y,
z]*(D[a2[t, x, y, z], x] - D[a1[t, x, y, z], y]) -
u3[t, x, y,
z]*(D[a1[t, x, y, z], z] - D[a3[t, x, y, z], x]))*(D[
a1[t, x, y, z], z] - D[a3[t, x, y, z], x]) -
((-sigma)*D[phi[t, x, y, z], y] -
sigma*D[a2[t, x, y, z], t] +
u3[t, x, y,
z]*(D[a3[t, x, y, z], y] - D[a2[t, x, y, z], z]) -
u1[t, x, y,
z]*(D[a2[t, x, y, z], x] - D[a1[t, x, y, z], y]))*(D[
a3[t, x, y, z], y] - D[a2[t, x, y, z], z])),
0 == rho*D[h, t] +
rho*(u1[t, x, y, z]*D[h, x] + u2[t, x, y, z]*D[h, y] +
u3[t, x, y, z]*D[h, z]) +
kappah*(D[T[t, x, y, z], x, x] + D[D[T[t, x, y, z], y], y] +
D[D[T[t, x, y, z], z], z]) - D[p[t, x, y, z], t],
0 == D[
sigma*(-D[phi[t, x, y, z], x] - D[a1[t, x, y, z], t]) +
u2[t, x, y, z]*(D[a2[t, x, y, z], x] - D[a1[t, x, y, z], y]) -
u3[t, x, y, z]*(D[a1[t, x, y, z], z] - D[a3[t, x, y, z], x]),
x] + D[sigma*(-D[phi[t, x, y, z], y] - D[a2[t, x, y, z], t]) +
u3[t, x, y, z]*(D[a3[t, x, y, z], y] - D[a2[t, x, y, z], z]) -
u1[t, x, y, z]*(D[a2[t, x, y, z], x] - D[a1[t, x, y, z], y]),
y] +
D[sigma*(-D[phi[t, x, y, z], z] - D[a3[t, x, y, z], t]) +
u1[t, x, y, z]*(D[a1[t, x, y, z], z] - D[a3[t, x, y, z], x]) -
u2[t, x, y, z]*(D[a3[t, x, y, z], y] - D[a2[t, x, y, z], z]),
z],
0 ==
D[a1[t, x, y, z], t] + D[phi[t, x, y, z], x] -
u2[t, x, y, z]*(D[a2[t, x, y, z], x] - D[a1[t, x, y, z], y]) -
u3[t, x, y, z]*(D[a1[t, x, y, z], z] - D[a3[t, x, y, z], x]) -
eta*D[D[a1[t, x, y, z], x], x],
0 ==
D[a2[t, x, y, z], t] + D[phi[t, x, y, z], y] -
u3[t, x, y, z]*(D[a3[t, x, y, z], y] - D[a2[t, x, y, z], z]) -
u1[t, x, y, z]*(D[a2[t, x, y, z], x] - D[a1[t, x, y, z], y]) -
eta*D[D[a2[t, x, y, z], y], y],
0 ==
D[a3[t, x, y, z], t] + D[phi[t, x, y, z], z] -
u1[t, x, y, z]*(D[a1[t, x, y, z], z] - D[a3[t, x, y, z], x]) -
u2[t, x, y, z]*(D[a3[t, x, y, z], y] - D[a2[t, x, y, z], z]) -
eta*D[D[a3[t, x, y, z], z], z], T[0, x, y, z] == 300,
p[0, x, y, z] == 101300,
phi[t, x, y, z] == 75*x /. t -> 0, u1[0, x, y, z] == 0,
u2[0, x, y, z] == 0, u3[0, x, y, z] == 0,
a1[0, x, y, z] == 0, a2[0, x, y, z] == 0,
a3[0, x, y, z] == 0, D[T[t, x, y, z], x] == 0 /. x -> 0,
D[T[t, x, y, z], x] == 0 /. x -> 10,
D[T[t, x, y, z], y] == 0 /. y -> 0,
D[T[t, x, y, z], y] == 0 /. y -> 10,
D[T[t, x, y, z], z] == 0 /. z -> 0,
D[T[t, x, y, z], z] == 0 /. z -> 10,
phi[t, x, y, z] == 0 /. x -> 0,
phi[t, x, y, z] == 750 /. x -> 10,
phi[t, x, y, z] == 75*x /. y -> 0,
phi[t, x, y, z] == 75*x /. y -> 10,
phi[t, x, y, z] == 75*x /. z -> 0,
phi[t, x, y, z] == 75*x /. z -> 10,
D[phi[t, 0, y, z], x] == (-sigma^(-1))*0.5*AG*T[t, 0, y, z]^2*
Exp[-4/(kB*T[t, 0, y, z])],
D[phi[t, 10, y, z], x] == (-sigma^(-1))*0.5*AG*T[t, 10, y, z]^2*
Exp[-4/(kB*T[t, 10, y, z])],
D[phi[t, x, y, z], y] == 0 /. y -> 0,
D[phi[t, x, y, z], y] == 0 /. y -> 10,
D[phi[t, x, y, z], z] == 0 /. z -> 0,
D[phi[t, x, y, z], z] == 0 /. z -> 10,
p[t, x, y, z] == 101300 /. x -> 0,
p[t, x, y, z] == 101300 /. x -> 10,
p[t, x, y, z] == 101300 /. y -> 0,
p[t, x, y, z] == 101300 /. y -> 10,
p[t, x, y, z] == 101300 /. z -> 0,
p[t, x, y, z] == 101300 /. z -> 10, u1[t, x, y, z] == 0 /. x -> 0,
u1[t, x, y, z] == 0 /. x -> 10,
D[u1[t, x, y, z], y] == 0 /. y -> 0,
D[u1[t, x, y, z], y] == 0 /. y -> 10,
D[u1[t, x, y, z], z] == 0 /. z -> 0,
D[u1[t, x, y, z], z] == 0 /. z -> 10,
u2[t, x, y, z] == 0 /. x -> 0,
u2[t, x, y, z] == 0 /. x -> 10,
D[u2[t, x, y, z], y] == 0 /. y -> 0,
D[u2[t, x, y, z], y] == 0 /. y -> 10,
D[u2[t, x, y, z], z] == 0 /. z -> 0,
D[u2[t, x, y, z], z] == 0 /. z -> 10,
u3[t, x, y, z] == 0 /. x -> 0, u3[t, x, y, z] == 0 /. x -> 10,
D[u3[t, x, y, z], y] == 0 /. y -> 0,
D[u3[t, x, y, z], y] == 0 /. y -> 10,
D[u3[t, x, y, z], z] == 0 /. z -> 0,
D[u3[t, x, y, z], z] == 0 /. z -> 10,
a1[t, x, y, z] == 0 /. x -> 0,
a1[t, x, y, z] == 0 /. x -> 10,
a1[t, x, y, z] == 0 /. y -> 0,
a1[t, x, y, z] == 0 /. y -> 10,
a1[t, x, y, z] == 0 /. z -> 0,
a1[t, x, y, z] == 0 /. z -> 10,
D[a1[t, x, y, z], x] == 0 /. x -> 0,
D[a1[t, x, y, z], x] == 0 /. x -> 10,
D[a1[t, x, y, z], y] == 0 /. y -> 0,
D[a1[t, x, y, z], y] == 0 /. y -> 10,
D[a1[t, x, y, z], z] == 0 /. z -> 0,
D[a1[t, x, y, z], z] == 0 /. z -> 10,
a2[t, x, y, z] == 0 /. x -> 0,
a2[t, x, y, z] == 0 /. x -> 10,
a2[t, x, y, z] == 0 /. y -> 0,
a2[t, x, y, z] == 0 /. y -> 10,
a2[t, x, y, z] == 0 /. z -> 0,
a2[t, x, y, z] == 0 /. z -> 10,
D[a2[t, x, y, z], x] == 0 /. x -> 0,
D[a2[t, x, y, z], x] == 0 /. x -> 10,
D[a2[t, x, y, z], y] == 0 /. y -> 0,
D[a2[t, x, y, z], y] == 0 /. y -> 10,
D[a2[t, x, y, z], z] == 0 /. z -> 0,
D[a2[t, x, y, z], z] == 0 /. z -> 10,
a3[t, x, y, z] == 0 /. x -> 0,
a3[t, x, y, z] == 0 /. x -> 10,
a3[t, x, y, z] == 0 /. y -> 0,
a3[t, x, y, z] == 0 /. y -> 10,
a3[t, x, y, z] == 0 /. z -> 0,
a3[t, x, y, z] == 0 /. z -> 10,
D[a3[t, x, y, z], x] == 0 /. x -> 0,
D[a3[t, x, y, z], x] == 0 /. x -> 10,
D[a3[t, x, y, z], y] == 0 /. y -> 0,
D[a3[t, x, y, z], y] == 0 /. y -> 10,
D[a3[t, x, y, z], z] == 0 /. z -> 0,
D[a3[t, x, y, z], z] == 0 /. z -> 10
},
{p, u1, u2, u3, T, phi, a1, a2, a3},
{t, 0, 10}, {x, 0, 10}, {y, 0, 10}, {z, 0, 10}]
非常感谢！