## 3 Results
### Third-Order Linear Dispersive Wave (LdW) Equation  
The third-order LdW equation can be considered as a linearized form of the KdV equation analyzed in the next section. It is defined as:  
\[u_{t}+c^{-2}u_{xxx}=0. \tag{16}\]  
Figure 4: Solution errors of reconstructed derivatives calculated based on the results obtained applying CD schemes with \(M=1\) and various \(N\). (**a**) 1 st derivative, (**b**) 2 nd derivative, (**c**) 3 rd derivative, (**d**) 5 th derivative. For the initial condition \(u(x,0)=\sin(cx)\) and the periodic boundary conditions in the domain \(x\in[0,2\pi]\), the exact solution is \(u(x,t)=\sin(c(x+t))\). The solution to the Equation (16) is a wave that moves to the left along the '\(x\)'-direction. [PERSON] et al. (2018) solved (16) with \(c=1\) and \(c=8\) using 4 th and 6 th-order FD and CD schemes. They showed better performance of the CD schemes than the FD schemes of the same order. Here, we solve (16) with the parameter \(c=1,\,8,\) and 16 on the meshes consisting of \(K=33,49,\,65,\ldots,257\) nodes in the time span \(t\in[0,1]\). Note that increasing the \(c\) parameter increases the frequency of the analytical solution, both in time and space (see Figure 5), and this makes the solution of (16) more difficult compared to what has been done in (2018). The computations are performed using CD schemes with \(M=1-4\) and various \(N\) up to the 32 nd order. Figure 5 shows the numerical solutions obtained on the mesh with \(K=257\) nodes for the CD scheme of the 26 th order with \(M=1\) and \(N=13\) for various \(c\). The grey lines indicate the local solution error. As can be seen, the agreement with the analytical solution is excellent. The calculated errors for the schemes with \(M=1\) and different \(N\), depending on the parameter \(c\), are presented in Figure 6. The error is computed in the same way as in (15) with the derivatives replaced by the function values. It can be seen that for \(c=1\) for all \(M\) and \(N\) combinations, the error is already very small on coarse meshes and the increase of the mesh points does not improve the solution accuracy, i.e., the error stagnates at the level \(\mathcal{O}(10^{-10})\). For \(c=8\) and \(c=16\), the situation is different, i.e., when the number of nodes increases, the errors almost linearly drop from the initial levels. The slopes of the error lines are in good agreement with the expected orders of approximations. Ultimately, for dense meshes, the errors remain on the same level as for \(c=1\), but for \(c=8\) and \(c=16\), this starts from a different number of the mesh point \(K\). Moreover, it can be seen that using a more accurate scheme (larger \(N\)) speeds up the achievement of the minimum error. It is worth noting that its level corresponds well to the minimum error rate calculated for the 3 rd derivative approximation in Section 2.3 (see Figure 4). Here, however, it does not increase with \(K\), which may be due to either the dissipative character of the Runge-Kutta method or a different analytical solution. The error lines for the case with \(c=16\) computed using CD schemes with various \(M\) and \(N\) combinations are shown in Figure 7. It can be seen that the expected error behavior indicated by the straight reference lines is reached in all the cases. Additionally, one can observe that CD schemes of the same order, i.e., the same \((M+N)\), are more accurate when the \(M\) values are larger. For instance, the combinations \(M=1\), \(N=7\) and \(M=2\), \(N=6\), and \(M=3\), \(N=5\) as well as \(M=4\), \(N=4\) all resulted in the discretization scheme of the 14 th order. Table 1 shows the exact error values of particular CD schemes. It can be seen that the schemes with a larger \(M\) are generally more accurate. As mentioned in Section 2.1, this is due to the differences in the leading terms of the truncation errors. Hence, one can conclude that having a given computational stencil \((M+N)\), it is better to choose the scheme with a larger \(M\).