5.5. Exercises for explicit methods

Exercise 1

Estimate the CFL condition of the Leap-Frog timestepping for the DG system. To this end estimate the largest eigenvalue of \(\mathbf M^{-1}\mathbf K\) by \(\mu_N\) for large enough \(N\) given by a power iteration

\[\mathbf x_{n+1}=\frac{1}{\|\mathbf M^{-1}\mathbf K\mathbf x_n\|}\mathbf M^{-1}\mathbf K\mathbf x_n,\quad \mu_n:=\mathbf x_n^\top\mathbf M^{-1}\mathbf K\mathbf x_n\]

with a random starting vector \(\mathbf x_0\). The matrices \(\mathbf M\) and \(\mathbf K\) are the matrices of the equivalent verlet time stepping, i.e. if \(\mathbf M_p\), \(\mathbf M_v\) are the mass matrices of the scalar and vectorial space and \(\mathbf B\) is the discrete gradient we set \(\mathbf M = \mathbf M_p\) and \(\mathbf K=\mathbf B^\top\mathbf M_v^{-1}\mathbf B\).

Exercise 2

Implement the DG-Method for the acoustic wave equation on the unit_square using suitable initial and boundary conditions and the Leap-Frog time-stepping. Choose the time-step as large as possible to satisfy the CFL-condition.

Exercise 3

Implement the first order mass-lumping method by supplying the according IntegrationRule to the differential symbol dx. Explore the sparsity pattern of the resulting mass matrix. Experiment with the argument diagonal = True for the mass BilinearForm. Replace the H1 space by H1LumpingFESpace (only for orders \(1,2\)). The IntegrationRules can by obtained by H1LumpingFESpace.GetIntegrationRules().

Exercise 4

Use a suitable example to compare the efficiency of the explicit methods (Mass-Lumping, DG) to the efficiency of the conforming method (second order system, implicit time-stepping). Plot the respective errors against the computation times.