# Computational Analysis of PDE-Based Shape Analysis Models by Exploring the Damped Wave Equation
## 5 Introductory Synthetic Tests
### Experiments
#### 5.2.1 Details for the Damped Wave Equation  
We already noticed in our last work [10] that the second-order methods CN and SOI\({}_{0}\) did not perform favourably compared to the first-order method IE. Using numerical solvers for PDEs will always result in a solution with an approximation error compared to the exact solution. We could, in principle, attempt to conduct an analytical error analysis of the numerical schemes to determine accounts of the wave propagation and damping parts in the leading order errors, but this is actually a quite tedious issue for the SOI\({}_{0}\) scheme, which is of importance in our numerical study. As it turns out, when approaching this task, the resulting terms are not as clear to interpret as desirable. It is even more challenging and a great deal beyond the scope of this study to perform a theoretical error analysis for the setting on a shape. Thus, we opt to present here a more vivid approach that clearly illustrates the underlying mechanisms. Since we have the analytical damped feature descriptor (48), we can adjust the damping parameter \(k\) in a way that it will fit the feature descriptors obtained from one of the numerical solvers. For the evaluation, we chose \(\tau=0.1\) as the time increment and \(h=0.05\) as the lattice parameter. In Figure 5, we present the results of our approach. In this figure, the red circles indicate the analytical feature descriptor obtained from the IE method. The blue triangles belong to the CN scheme, the solid black line is the analytical damped feature descriptor, and the solid red line shows the difference between the second-order methods. In the top row in Figure 5, we changed the damping parameter \(k\) from the analytical damped feature descriptor to fit best with the feature descriptors from the IE (left) or the CN method (right). For the fit to the IE method, we used a damping parameter of \(k=\tau/2\). In the top right plot, we used \(k=\tau/8\) as the damping parameter. This indicates that the numerical solution of a wave equation obtained via the IE method is closer to the analytical solution of the damped wave equation than to the analytical solution of the wave equation. This phenomenon is well known under the topic of modified equations [23]. In the bottom right plot, we showed that it is possible to imitate the analytical feature descriptor from the IE method with the analytical damped feature descriptor from the CN method. The damping parameter was set to \(k=\tau\). In total, this means that the solution of the wave equation using the IE method and the solution of the damped wave equation using the CN method are approximately the same. This leads us to conjecture that it is possible, with an adequate choice of the damping parameter, to describe beneficial properties of the IE method in an analytical way. This may give an indication about a potentially necessary (or beneficial) damping behaviour in continuous-scale PDE models that may be useful in a more general shape matching scenario. In the bottom left picture of Figure 5, we present the difference between the two second-order methods. We observe that the difference becomes slightly larger during time evolution, but for our time interval, they are close to each other. This result leads us to plot only the CN method as representative of the second-order methods. The error between the second-order methods will grow, and at a certain point, the results of both schemes are not equal anymore. To show this, we set up Figure 6. In this plot, we see the cosine initial condition at the top and the second-order schemes at different iterations on the bottom. We have chosen iterations close to the 100 th, 250 th and 500 th iteration to show the differences between both solutions. We chose iterations close to the mentioned ones because we want to show the turning point for the SOI\({}_{0}\) method. In doing so, the comparison to the CN gets easier. Both solutions will simply oscillate up and down without any movement to the sides.