计算科学 - 非线性二维热导方程（数值解） - 吾爱随笔录

我必须为二维方程写一个求解器：我尝试使用显式方法：和和相同，点。我也使用：之前的系数和相同对于点。我在 Mathematica 中编写了以下代码：

\partial_{t} u = u^{2} (\partial_{x}^{2} u + \partial_{y}^{2} u)

$\partial_t u = u^2(\partial_x ^2 u + \partial_y ^2 u)$

\partial_{t} u = \frac{u_{i, j}^{k + 1} - u_{i, j}^{k}}{τ}

$\partial_t u = \frac{u_{i,j}^{k+1} - u_{i,j}^k}{\tau}$

\partial_{x}^{2} u = \frac{u_{i - 1, j}^{k} - 2 u_{i, j}^{k} + u_{i + 1, j}^{k}}{h^{2}}

$\partial_x ^2 u = \frac{u_{i-1,j}^k - 2u_{i,j}^k + u_{i+1,j}^k}{h^2}$

\partial_{y}^{2} u

$\partial_y ^2 u$

j - 1, j, j + 1

$j-1, j, j+1$

u_{i, j}^{k} = \frac{u_{i + 1, j}^{k} + u_{i, j}^{k}}{2}

$u_{i,j}^k = \frac{u_{i+1,j}^k + u_{i,j}^k}{2}$

\partial_{x}^{2}

$\partial_x ^2$

y

$y$

j + 1, j

$j+1, j$

nn = 6;
mm = 200;
h = 1./nn;
tau = 1./mm;
v0[x_, y_] := 1.;


Table[{x[i] = i*h, y[j] = j*h}, {i, 0, nn}, {j, 0, nn}];
Table[t[i] = i*t, {i, 0, mm}];
Table[u[i, j, 0] = v0[x[i], y[j]], {i, 0, nn}, {j, 0, nn}];
Table[u[0, j, k] = 0, {j, 0, nn - 1}, {k, 0, mm}];
Table[u[nn, j, k] = 0, {j, 0, nn - 1}, {k, 0, mm}];
Table[u[i, 0, k] = 0, {i, 0, nn - 1}, {k, 0, mm}];
Table[u[i, nn, k] = 0, {i, 0, nn - 1}, {k, 0, mm}];

Do[ 
  u[i, j, k + 1] = 
   u[i, j, k] + 
    tau*(((u[i - 1, j, k] - 2*u[i, j, k] + u[i + 1, j, k])/
          h^2)*(( u[i, j, k] + u[i - 1, j, k])/
           2)^2 + ((u[i, j - 1, k] - 2*u[i, j, k] + u[i, j + 1, k])/
          h^2)*((u[i, j, k] + u[i, j - 1, k])/2)^2), {k, 0, mm}, {i, 
   1, nn - 1}, {j, 1, nn - 1}];



g[k_] := Table[u[i, j, k], {i, 0, nn}, {j, 0, nn}];
p = Table[
  ListPlot3D[g[k], PlotRange -> {{0, nn}, {0, nn}, {0, 6}}], {k, 0, 
   mm, 20}]

我使用最简单的条件：和 $u(0,0,t) = u(10,10,t) = 0$ $u(x,y,0) = 1$