Important 1

This notebook is designed to be run in Julia.

Important 2

Before handling it please rename this file accordingly as: group_name_assignment_5.ipynb

or

your_last_name_first_name_lab_5.ipynb

All generated code and your presentation must be included in this notebook.

Objectives

  • Become familiar with solving parabolic equations.

  • Test some of the analytic estimates covered in class.

  • Go through the process of implementing a new code.

Problem 1:

a. Take the code given in the class notebook to solve the one-dimensional heat equation and modify it so that it has non-trivial Neumann boundary conditions.

b. Check that with homogeneous Neumann boundary conditions the sum of the vector components sum(sol(t))is constant to a good precision.

c. Check that the energy decays.

d. Check that with homogeneous Dirichlet or Neumann boundary the \(l^2\) norm of the finite difference approximation of a first order derivative satisfies the decay rate found in class. To fit data you can use the library LsqFit.jl. For most data the dacay will be much faster. For which initial data you will approach the decay estimate?

Problem 2:

Solve a 2D problem using the code given in the class notebook. Take a anulus between radious \(R0 = 1\) and \(R2 = 2\). Use \((r, \theta)\) coordinates. In this coordinates the expression for the laplacian is:

\[ \Delta u = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} \]

So, for a function \(u(r, \theta)\):

\[ \boxed{ \Delta u = u_{rr} + \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta\theta} } \]

a. Solve this problem with interior Dirichelet boundary condtion given by \(g_l(\theta) = sin(\theta)\) and Neumann exterior boundary condition. Notice that the \(\theta\) coordinate is periodic, so for it use the periodic finite difference operators. Evolve up to \(T=1.0\) Plot the initial data and the solution at that time. b. Compute the analytic solution and compare. Study convergence. c. If possible plot the results in cartesian coordinates. –-

Problem 3:

a. Solve Burger's equation with dissipation:

\[ \partial_t u - u \partial_x u - ε \partial_x^2 u = 0 \]

In a domain \([-\pi, \pi]\) with Dirichlet boundary conditions on both extremes (using penalties), \(u(-\pi,t) = -1\), \(u(\pi, t) = 1\). Use as a initial data,

\[ f(x) = sin(x/2) \]

b. Plot the results at \(T=1.2\) for different \(ε\)'s. Study what happens when \(\varepsilon \to 0\).