#### Exercises

Exercise classes will be held in NULLROMMET sentralbygg 2, 3rd floor (unless announced otherwise).

Supervision date Topic description Mfiles
12.01 Boundary value problems Implementation of the numerical solution of simple boundary value problems with various boundary conditions. Numerical solution of a convection diffusion boundary value probelm in 1D (nu*u''-u'=f, u(0)=alpha, u(1)=beta) convdiff.m.Numerical solution of the pendulum probelm ch. 3.3 of the note pendulum.m. You find here an example of how to compute error plots giving numerical evidence that the method is implemented correctly and achieves the expected order.
19.01 Parabolic equations.Implement a finite differences discretization of the heat equation, with various boundary conditions: use the Euler method, the backward-Euler method and the Crank-Nicholson method. Solution of the heat equation with Crank-Nicolson. You find a loglog plot of the error in time for the numerical method and for a fixed space discretization. The solution used for comparison is obtained using ode15s with a very small tolerance. Similarly to compute the error in space, since CN is an unconditionally stable method, you can fix a value for the time step and run the method with different values of h. As a reference solution you will need a numerical solution computed with the same (fixed) time step and with a tiny little space-step. If your numerical method is NOT unconditionally stable then the time step must be choosen according to the stability restriction and the size of h, so for a tiny h you'll have to use an even smaller k. This becomes a challenging numerical experiment, but still doable, unless the exact solution of the problem is available. cnheat.m au.m
26.01 Elliptic problems Implement a discretization of the Laplace equation on the square.
02.02 Advection equation and hyperbolic problems Simple experiments with the linear advection equation and the Burgers equation. Semi-discretization of the KdV equation. Kdv.m Kdvfunc.m Kdvinit.m. The following file is an implementation the methods of ch 7.3 for u_t+au_x=0 owwaveeq.m.
09.02 From Monday 9th of February until 23rd of March supervision of the project every Monday