10 Week 10 Participation
Name: [Type Your Name Here]
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10.1 Part 1
For this class participation, please complete the following parts
Work on your Midterm 2 Project during one of our class meetings and provide a brief (~5-7 sentences) summary on your progress, any questions that have come up, and what you will do moving forward.
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10.2 Part 2
Watch the following videos on
- Heat Equation Intuition (~ 8 min)
- But what is a PDE? (~17 min)
- Solving the Heat Equation (~14 min)
Provide a 250 - 500 word summary (total) on the three videos.
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10.3 Part 3
- Read and review Sections 6.1 - 6.4 from the textbook.
- Watch Week 10 - Discretizing the Heat Equation
and answer the following
Consider the time dependent heat equation in one dimension
\[ u_t = D \; u_{xx} \]
in 1 spatial dimension. Note that \(D\) is the diffusivity (the rate of diffusion) so in terms of physical problems, if \(D\) is small then the diffusion occurs slowly and if \(D\) is large then the diffusion occurs quickly.
We want to build a numerical approximation to the function \(u(t,x)\) for a given collection of initial and boundary conditions.
First we need to introduce some notation for the numerical solution. As you’ll see in a moment, there is a lot to keep track of in numerical PDEs so careful index and well-chosen notation is essential. Let \(U_i^n\) be the approximation of the solution to \(u(t,x)\) at the point \(t=t_n\) and \(x=x_i\) (since we have two variables we need to two indices). For example, \(U_4^1\) is the value of the approximation at time \(t_1\) and at the spatial point \(x_4\).
Next, we need to approximate both derivatives \(u_t\) and \(u_{xx}\) in the PDE using methods that we have used before. Now, would be a good time to go back to review your notes for how we build approximations of derivatives.
- Give an approximation of \(u_t\), similar to Euler’s method
\[ u_t \approx \frac{??? - ???}{???}. \]
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- Give an approximation of \(u_{xx}\) using the approximation for the second derivative from HW 03
\[ u_{xx} \approx \frac{??? - ??? + ???}{???}. \]
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- Put your answers from parts (a) and (b) together using the 1D heat equation
\[ \frac{??? - ???}{\Delta t} = D \left( \frac{??? - ??? + ???}{\Delta x^2} \right). \]
Be sure that your indexing is correct: the superscript \(n\) is the index for time and the subscript \(i\) is the index for space.
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- Rearrange your result from part (c) to solve for \(U_i^{n+1}\)
\[ U_i^{n+1} = ??? + \frac{D \Delta t}{\Delta x^2} \left( ??? - ??? + ??? \right). \]
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- Read the first 3 pages of these lecture notes and compare them with what you derived above.
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