MTH3007B Weekly Problems 6

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6.1. Explicit Scheme for 1D Heat Equation

Question

Consider a rod of length $L=10\text{cm}$ with heat diffusion coefficient $\alpha =0.735{\text{cm}}^2{\text{s}}^{-1}$. Assume boundary conditions $u(t)=100^{\circ }\text{C}$ and $u(t)=25^{\circ }\text{C}$ for all $t\ge 0$. The initial condition at $t=0$ is $u(0,x)=0^{\circ }\text{C}$ for $0<x<10$, with the boundaries fixed at the given temperatures. Take spatial step $\Delta x=0.2\text{cm}$ and time step $\Delta t=0.01\text{s}$ and solve the one-dimensional heat equation

$$\frac{\partial u}{\partial t}=\alpha \frac{{\partial }^{2}u}{\partial x^2}$$

using the explicit (forward Euler in time, central difference in space) finite-difference scheme.

1. Implement the explicit scheme with the given parameters, enforcing the boundary conditions at each time step and using the recurrence

$$u_j^{n+1}=u_j^n+\alpha \frac{\Delta t}{\Delta x^2}(u_{j-1}^n-2u_j^n+u_{j+1}^n),$$

for interior grid points $j$. 2. Plot $u(t,x)$ as a function of $x$ at $t=0\text{s}$, $t=6\text{s}$, and $t=12\text{s}$ on a single graph, and include this plot together with your code in the extra material. 3. Determine the temperature of the rod at time $t=12\text{s}$ and position $x=4\text{cm}$, i.e. compute $u(12\text{s}\text{cm})$, from your numerical solution.

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