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- Apr 14, 2013

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Hey!!!

I am implementing in a program the finite difference method for the heat equation.

The problem is the following:

$$u_t(x,t)=(g(x,t)u_x(x,t))_x+f(x,t), \forall (x,t) \in [0,1]x[0,1]$$

$$u(0,t)=u(1,t)=0, \forall t \in [0,1]$$

$$u(x,0)=0, \forall x \in [0,1]$$

where $f(x,t)=\pi x \cos(\pi x t) (1-x)-g(x,t)(2 \pi t \cos(\pi x t)-(x-1) \pi^2 t^2 \sin(\pi x t))+g_x(x,t) (\sin(\pi x t)+\pi (x-1) t \cos(\pi x t))$

$u(x,t)=\sin( \pi x t)(1-x)$

$g(x,t)=2+ \sin(xt)$

The method is the following: ($U_j^n$ is the approximation of $u(x_j,t_n), h=\frac{1}{J}, Dt=\frac{1}{N}, x_j=jh, t_n=nDt$)

$$U_j^0=0, j=0,...,J$$

$$n=0,...,N-1:\frac{U_j^{n+1}-U_j^n}{Dt}=\frac{1}{h}[g(x_j+\frac{h}{2},t_{n+1})\frac{U_{j+1}^{n+1}-U_j^{n+1}}{h}-g(x_j-\frac{h}{2},t_{n+1})\frac{U_j^{n+1}-U_{j-1}^{n+1}}{h}]+f(x_j,t_{n+1}), j=1,...,J-1$$

$$U_0^{n+1}=U_J^{n+1}=0$$

I found the following errors:

For $J=N=10: 0.179505$

For $J=N=20: 0.089506$

Could you tell me if these errors are correct?

I am implementing in a program the finite difference method for the heat equation.

The problem is the following:

$$u_t(x,t)=(g(x,t)u_x(x,t))_x+f(x,t), \forall (x,t) \in [0,1]x[0,1]$$

$$u(0,t)=u(1,t)=0, \forall t \in [0,1]$$

$$u(x,0)=0, \forall x \in [0,1]$$

where $f(x,t)=\pi x \cos(\pi x t) (1-x)-g(x,t)(2 \pi t \cos(\pi x t)-(x-1) \pi^2 t^2 \sin(\pi x t))+g_x(x,t) (\sin(\pi x t)+\pi (x-1) t \cos(\pi x t))$

$u(x,t)=\sin( \pi x t)(1-x)$

$g(x,t)=2+ \sin(xt)$

The method is the following: ($U_j^n$ is the approximation of $u(x_j,t_n), h=\frac{1}{J}, Dt=\frac{1}{N}, x_j=jh, t_n=nDt$)

$$U_j^0=0, j=0,...,J$$

$$n=0,...,N-1:\frac{U_j^{n+1}-U_j^n}{Dt}=\frac{1}{h}[g(x_j+\frac{h}{2},t_{n+1})\frac{U_{j+1}^{n+1}-U_j^{n+1}}{h}-g(x_j-\frac{h}{2},t_{n+1})\frac{U_j^{n+1}-U_{j-1}^{n+1}}{h}]+f(x_j,t_{n+1}), j=1,...,J-1$$

$$U_0^{n+1}=U_J^{n+1}=0$$

I found the following errors:

For $J=N=10: 0.179505$

For $J=N=20: 0.089506$

Could you tell me if these errors are correct?

Last edited: