## 4 Numerical Experiments
The solution of the semi-discretized Equation (16) was also calculated numerically by the ode15s MATLAB routine with a very stringent tolerance. The spatial domain was extended to \(x_{0}=-10\), \(x_{N}~{}=10\), but all other circumstances of Experiment 1, including the analytical solution (20) and the spatial step size \(\Delta x=0.02\), still hold. In Figure 7, we present the errors for six methods when not only the analytical, but the numerical reference was used for the error calculation in Formula (19). One can see that if the numerical reference is used, the residual errors disappear, and for very small time step sizes, the error of the third-order CLL method goes below the errors of the second-order methods. Experiment 3. In this subsection, \(x_{0}=-5\), \(x_{N}~{}=5\), \(t^{0}=0.3\), \(t^{\rm fin}=0.5\). The space step size \(\Delta x\) varied, but all other parameters of Experiment 1 were preserved. For comparison purposes, we used the very common implicit (Euler) method:  
\[u_{i}^{n+1}=u_{i}^{n}+r\Big{(}u_{i-1}^{n+1}-2u_{i}^{n+1}+u_{i+1}^{n+1}\Big{)},\]  
which was implemented by matrix inversion. In our previous publications, we measured the running times and conveyed plenty of examples where the traditional methods were outperformed by the explicit and unconditionally stable ones. Therefore, our main purpose with these standard schemes was not the comparison of the performances, but to compare the behavior of the errors of the 20 conditionally consistent schemes with an unconditionally consistent case, so that we could explain properly the connection between the truncation errors and the observed errors. Figure 6: The numerically calculated temperatures \(u\) for the three-stage ASH method as a function of the space variable \(x\) for different time step sizes in Experiment 1. The red and the black solid lines are for the initial and the analytically calculated temperatures, respectively. First, the space step is considered as a parameter with six different values of \(\Delta x\) (five in the case of the implicit method, since it has much larger running times). The error-curves are presented for the first-order UPFD and implicit methods in Figure 8, while Figure 9 contains the results for the second-order LNe and LH methods. The curves of the UPFD, LNe and LH method cross one another, because if the space step size is smaller, the convergence is slower due to the \(\Delta t^{2}\Delta x^{-4}\) type terms in the truncation error. It is easy to see that for a fixed time step size, decreasing space step sizes does not yield better accuracy. In the case of the implicit method, however, the error decreases if _either_ the time _or_ the space step is decreased. In the case of the UPFD and the LNe method, the horizontal distance between the neighboring curves (belonging to a certain \(\Delta x\) and its half \(\Delta x/2\)) is a factor of 4, while it is only a factor of 2 for the LH method. This can be explained by the truncation error terms of \(\Delta t\Delta x^{-2}\) of the UPFD and \(\Delta t^{2}\Delta x^{-4}\) of LNe schemes, and, on the other hand, the term \(\Delta t^{2}\Delta x^{-2}\) of the LH scheme. If the exponent in the denominator is twice as large as in the numerator, then \(\Delta t\) must be divided by four to counterbalance the division of \(\Delta x\) by two in order to keep the error constant. Figure 7: The errors as a function of the time step size for six methods in Experiment 2 when not only the analytical but a numerical reference solution was used to calculate the errors. Figure 8: Errors as a function of the time step size for the UPFD and the implicit Euler (ImpEu) methods with different space step sizes in Experiment 3. \(\Delta x\) changes from 0.2 to 0.00625 and to 0.0125 for the UPFD and the implicit method, respectively.