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While solving the heat equation in one spatial variable $$u_t = u_{xx}$$ (x goes from 0 to L) with the initial temperature distribution $$T_0 \frac{x(L-x)}{L^2}$$ , and with neumann boundary conditions $$u_x(0,t) = u_x(L,t) = 0$$, I got some really weird behaviour from NDSolve.

My code looks like this:

h[x_] := x*(30 - x)/900;
pde = D[u[t, x], t] == D[u[t, x], x, x]
begin = 0;
end = 30;
bc = {u[0, x] == 100*h[x], (Derivative[0, 1][u])[t, begin] ==
0, (Derivative[0, 1][u])[t, end] == 0};
finaltime = 100

s = NDSolve[{pde}~Join~bc, u, {t, 0, finaltime}, {x, begin, end}];


Since heat cannot flow out through the ends, continuing this in time should yield a smoothening until it reaches the average everywhere. Instead, I get a very weird time evolution which when plotted seems to be of the form $$u_x(x,t) = u_x(x,0) - kt$$. This is particularily infuriating because the problem seems to be intermittent. Taking the square of h does not cause any trouble.

The problem seems to magically fix itself if I instead feed in a truncated cosine series into the code:

rule = t -> FourierCosSeries[t*(2*Pi - t), t, 35];
f[x_] := (t /. rule) /. (t -> x)
g[x_] := f[2*Pi*x/30]/(4*Pi*Pi)


and insert g instead of h into the code above. Trying a function interpolation gave only errors.

Is there a fix which is more general than this quick hack?

While solving the heat equation in one spatial variable $$u_t = u_{xx}$$ (x goes from 0 to L) with the initial temperature distribution $$T_0 \frac{x(L-x)}{L^2}$$ , and with neumann boundary conditions $$u_x(0,t) = u_x(L,t) = 0$$, I got some really weird behaviour from NDSolve.

My code looks like this:

h[x_] := x*(30 - x)/900;
pde = D[u[t, x], t] == D[u[t, x], x, x]
begin = 0;
end = 30;
bc = {u[0, x] == 100*h[x], (Derivative[0, 1][u])[t, begin] ==
0, (Derivative[0, 1][u])[t, end] == 0};
finaltime = 100

s = NDSolve[{pde}~Join~bc, u, {t, 0, finaltime}, {x, begin, end}];


Since heat cannot flow out through the ends, continuing this in time should yield a smoothening until it reaches the average everywhere. Instead, I get a very weird time evolution which when plotted seems to be of the form $$u_x(x,t) = u_x(x,0) - kt$$.

The problem seems to magically fix itself if I instead feed in a truncated cosine series into the code:

rule = t -> FourierCosSeries[t*(2*Pi - t), t, 35];
f[x_] := (t /. rule) /. (t -> x)
g[x_] := f[2*Pi*x/30]/(4*Pi*Pi)


and insert g instead of h into the code above. Trying a function interpolation gave only errors.

Is there a fix which is more general than this quick hack?

While solving the heat equation in one spatial variable $$u_t = u_{xx}$$ (x goes from 0 to L) with the initial temperature distribution $$T_0 \frac{x(L-x)}{L^2}$$ , and with neumann boundary conditions $$u_x(0,t) = u_x(L,t) = 0$$, I got some really weird behaviour from NDSolve.

My code looks like this:

h[x_] := x*(30 - x)/900;
pde = D[u[t, x], t] == D[u[t, x], x, x]
begin = 0;
end = 30;
bc = {u[0, x] == 100*h[x], (Derivative[0, 1][u])[t, begin] ==
0, (Derivative[0, 1][u])[t, end] == 0};
finaltime = 100

s = NDSolve[{pde}~Join~bc, u, {t, 0, finaltime}, {x, begin, end}];


Since heat cannot flow out through the ends, continuing this in time should yield a smoothening until it reaches the average everywhere. Instead, I get a very weird time evolution which when plotted seems to be of the form $$u_x(x,t) = u_x(x,0) - kt$$. This is particularily infuriating because the problem seems to be intermittent. Taking the square of h does not cause any trouble.

The problem seems to magically fix itself if I instead feed in a truncated cosine series into the code:

rule = t -> FourierCosSeries[t*(2*Pi - t), t, 35];
f[x_] := (t /. rule) /. (t -> x)
g[x_] := f[2*Pi*x/30]/(4*Pi*Pi)


and insert g instead of h into the code above. Trying a function interpolation gave only errors.

Is there a fix which is more general than this quick hack?

1

NDSolve issue with initial and boundary conditions

While solving the heat equation in one spatial variable $$u_t = u_{xx}$$ (x goes from 0 to L) with the initial temperature distribution $$T_0 \frac{x(L-x)}{L^2}$$ , and with neumann boundary conditions $$u_x(0,t) = u_x(L,t) = 0$$, I got some really weird behaviour from NDSolve.

My code looks like this:

h[x_] := x*(30 - x)/900;
pde = D[u[t, x], t] == D[u[t, x], x, x]
begin = 0;
end = 30;
bc = {u[0, x] == 100*h[x], (Derivative[0, 1][u])[t, begin] ==
0, (Derivative[0, 1][u])[t, end] == 0};
finaltime = 100

s = NDSolve[{pde}~Join~bc, u, {t, 0, finaltime}, {x, begin, end}];


Since heat cannot flow out through the ends, continuing this in time should yield a smoothening until it reaches the average everywhere. Instead, I get a very weird time evolution which when plotted seems to be of the form $$u_x(x,t) = u_x(x,0) - kt$$.

The problem seems to magically fix itself if I instead feed in a truncated cosine series into the code:

rule = t -> FourierCosSeries[t*(2*Pi - t), t, 35];
f[x_] := (t /. rule) /. (t -> x)
g[x_] := f[2*Pi*x/30]/(4*Pi*Pi)


and insert g instead of h into the code above. Trying a function interpolation gave only errors.

Is there a fix which is more general than this quick hack?