4.4. Exercises for time-stepping schemes

Exercise 1

Implement the Newmark time-stepping for general \(\beta,\gamma\). Choose the second order acoustic wave problem on the unit-square with initial data

\[u_0 = \exp(-30(x-1/2)^2),\quad v_0=0\]

and homogeneous Neumann boundary conditions. Argue that the exact solution for \(t=1\) is identical to the initial values. Compute the error at \(t=1\) in the \(L^2\) norm for \(\beta = \gamma/2\) and varying \(\gamma\). What order of convergence with respect to \(\tau\) do you observe? (Pay particular interest to the case \(\gamma=1/2\).)

Exercise 2

Use the example of the previous exercise to check conservation of energies \(\mathbf E\) and \(E_j^{\tau,\beta,\gamma}\) for the parameters \(\gamma_0=1/2,\beta_0=1/4\), \(\gamma_1=1/2,\beta_1 =0\) and \(\gamma_2 = 1/3, \beta_2 = 1/6\).

Exercise 3

Use the example of the previous exercises to verify the CFL condition: Compute the largest magnitude eigenvalue \(\lambda_{\mathrm {max}}\) of the resulting problem \(\mathbf K \mathbf u = \lambda \mathbf M \mathbf u\). Then plot the energies \(\mathbf E, E^{\tau,\beta,\gamma}\) over time for the explicit method \(\beta=0,\gamma=1/2\) and timesteps \(\tau_{\pm}^2=\frac{4}{\lambda_{\mathrm{max}}}(1\pm\varepsilon)\) and small \(\varepsilon>0\).

Exercise 4

Implement the Crank-Nicholson time-stepping for the \(H^1-L^2\) space pairing (4.11) for a similar problem as in the previous exercises. Fix the orders of spaces and time-step and check the convergence of the error with respect to the mesh-size \(h\). How do the orders of the two spaces have to be related to obtain an efficient method?

Exercise 5

Implement the Leap-Frog time-stepping for the first order equation using an \(L^2-H(\mathrm{div}))\) pairing. Preserving the skew-symmetric structure of the system, what options do you find for imposing homogeneous boundary conditions?

Exercise 6

For Galerkin approximations using a discrete space \(V_h\), the error of the spacial discretization is usually governed by the error of the spacial approximation \(I-\Pi_h\) where the projection \(\Pi_h:L^2\to V_h\) is defined by

\[\int_\Omega \Pi_h f v = \int_\Omega f v\]

for all \(v\in V_h\). Compare the projection errors for \(f = \exp\left(-30((x-1/2)^2+(y-1/2)^2)\right)\) on the unit_square for the spaces H1 and L2 for different mesh-sizes \(h\) with respect to the degrees of freedom (H1.ndof, L2.ndof).

Assume you were able to use the space L2 for a discretization of the wave equation. Compare the computational effort for factorizing the mass matrices for the H1 and L2 spaces and \(200\) applications of the inverse mass matrix with respect to accuracy (measured by the projection error) on different meshes. Use also L2.Mass().Inverse() instead of assembling the matrix.

Consider also a three-dimensional example.