# Stability Optimization of Explicit Runge-Kutta Methods with Higher-Order Derivatives
## 5 Numerical Examples
### The Problem of the Linear Advection Equation  
The linear advection equation is written as  
\[\frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x}=0,\quad x\in(0,1 ),\quad t>0, \tag{16}\]  
where \(c>0\) is a constant advection velocity. According to the first derivatives on \(t\) and \(x\), the initial condition and single boundary condition for (16) can be stated as \(u(0,x)=\varphi(x)\), \(x\geq 0\) and \(u(t,0)=0\), \(t\geq 0\). The space grid with constant step with \(N+1\) nodes, \(x_{i}=i\cdot h\), \(i=0,\dots,N\), \(h=\frac{1}{N}\), where \(x_{0}=0\), \(x_{N}=1\), is constructed. After the discretization of space variables, the following system of linear ODEs is obtained:  
\[\frac{d\mathbf{u}}{dt}=\mathbf{A}\mathbf{u}, \tag{17}\]  
Figure 6: Stability domains of fourth-order methods. where \(\text{dim}(\mathbf{u})=N\), \(\text{dim}(\mathbf{A})=N\times N\), \(u_{i}(t)\approx u(t,x_{i})\), \(i=1,\ldots,N\). Matrix \(\mathbf{A}\) is defined by the discretization of the space variable. The matrix is represented as \(-\frac{\epsilon}{h}\mathbf{B}\). The first derivative in the second node is approximated by the simple first-order finite difference, with boundary condition taken into account:  
\[\frac{\partial u}{\partial x}(t,x_{1})\approx\frac{u_{1}-u_{0}}{h}=\frac{u_{1 }}{h}.\]  
In other grid nodes, the same approximation is used:  
\[\frac{\partial u}{\partial x}(t,x_{i})\approx\frac{u_{i}-u_{i-1}}{h}.\]  
Matrix \(\mathbf{B}\) has the following structure: \(B_{11}=-1\), \(B_{ii-1}=1\), \(B_{ii}=-1\), \(i=2,\ldots,N\), and all other elements are zeroes. As can be easily obtained, all eigenvalues of \(\mathbf{B}\) are equal to \(-1\). After the discretization on a time grid, constructed with step \(\Delta t\), and the application of explicit RKM, the following system of difference equations is obtained:  
\[\mathbf{u}^{n+1}=\mathbf{C}\mathbf{u}^{n}, \tag{18}\]  
where \(\mathbf{C}\) is a matrix, represented by the stability polynomial of \(\mathbf{B}\) and hyperbolic CFL number \(\gamma=c\frac{\Delta t}{h}\): \(\mathbf{C}=\mathcal{R}(\gamma\mathbf{B})\). According to the explicit nature of considered RKMs and finite-difference Equation (18), the stability criterion is represented as \(\gamma\leq\hat{\gamma}\). The value of \(\hat{\gamma}\) is obtained empirically as the approximate maximal value of \(\gamma\), which guarantees that the spectral radius of \(\mathbf{C}\) is less than or equal to unity. According to the real nature of the spectrum, we consider only optimal values, obtained by the maximization of \(\mathcal{L}\). For the computational experiment, the case of \(N=1000\) is considered. The results are presented in Table 6. The case of \(c_{3}=2/3\) for TDRKM at \(s=3\) is not considered because the value of \(\hat{\gamma}\) is very close to the case of \(c_{3L}\). \begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline \multicolumn{5}{|c|}{GJKKM} \\ \hline \(c_{2,1L}\) & \(c_{2,1}=1\) & \(c_{2,1}=\frac{1}{2}\) & \(c_{2,1}=\frac{1}{3}\) \\ \hline
6.061 & 1.548 & 3.566 & 2.641 \\ \hline \multicolumn{5}{|c|}{TDRKM, \(s=3\)} \\ \hline \(c_{3L}\) & \(c_{3}=\frac{3}{4}\) & \(c_{3}=\frac{5+\sqrt{5}}{100}\) & \(c_{3}=1\) & \(c_{3}=\frac{4}{5}\) \\ \hline
6.243 & 3.736 & 2.901 & 3.306 & 3.556 \\ \hline \multicolumn{5}{|c|}{TDRKM, \(s=4\)} \\ \hline \(c_{2L},c_{3L},c_{4L}\) & \multicolumn{2}{c|}{\(c_{2}=\frac{1}{4},c_{3}=\frac{2}{3},c_{4}=1\)} \\ \hline
8.138 & \multicolumn{2}{c|}{4.064} \\ \hline \multicolumn{5}{|c|}{ThDRKM, \(s=2\)} \\ \hline \(c_{2L}\) & \multicolumn{2}{c|}{\(c_{2}=2/5\)} \\ \hline
4.344 & \multicolumn{2}{c|}{3.993} \\ \hline \multicolumn{5}{|c|}{ThDRKM, \(s=3\)} \\ \hline \(c_{2L},\widehat{\widehat{a}}_{3,2L}\) & \(c_{2}=\frac{3-\sqrt{2}}{7},\widehat{\widehat{a}}_{3,2}=\frac{122+71\sqrt{2}}{7 03}\) \\ \hline
7.247 & \multicolumn{2}{c|}{5.221} \\ \hline \multicolumn{5}{|c|}{Mead and Renaut methods} \\ \hline RKM & \multicolumn{2}{c|}{RKC} \\ \hline
3.231 & \multicolumn{2}{c|}{3.524} \\ \hline \end{tabular}
\end{table}