## 4 Numerical Experiments
\[u^{\text{analytic}}(x,t)=\frac{x}{t^{7/2}}\bigg{(}1-\frac{x^{2}}{3\alpha t}+ \frac{x^{4}}{60\alpha^{2}t^{2}}\bigg{)}e^{-\frac{x^{2}}{64\alpha t}}. \tag{20}\]0 Errors as a function of the time step size for the LNe and the LH methods with different space step sizes in Experiment 3. Experiment 4. We conducted a case study considering \(\Delta t\to 0\) when \(\Delta t/(\Delta x)^{2}=0.2\) was fixed. All parameters were the same as in Experiment 3. The error curves can be seen in Figure 10 for all the 20 methods. Two groups of the methods can be clearly distinguished. The errors of the first group tended to constant values, since their truncation errors contained terms where the exponent of the space step size in the denominator was twice as large as that of the time step size in the numerator. These algorithms were not convergent under these circumstances. On the other hand, the errors of the second group tended to zero, which means they did not have truncation error terms of \(\Delta t^{n}\Delta x^{-2n}\). The results of this numerical investigation are consistent with the analytically calculated truncation error expressions which were detailed in Section 2. The exceptions are the SH and the ASH methods, for which we were not able to calculate the \(\tau\) errors, only the \(\varepsilon\) ones (see Equations (A1) and (A2)), which contain the critical \(\Delta t^{n}\Delta x^{-2n}\) terms. In these cases, the \(\varepsilon\) errors do not properly reflect the observed properties of these methods from this point of view; thus, we think that further investigations would be necessary. Experiment 5. Here, everything was the same as in Experiment 1, with the exception that the diffusion coefficient and the final time were increased to \(\alpha=2\) and \(t^{\text{fin}}=5\), while \(\Delta x=0.01\). The error-curves are presented in Figures 11 and 12 for the slowly and the quickly converging methods. We also examined how the maximum errors developed as time elapsed for a given medium-sized time step size. Figures 13 and 14 show that after a sharp increase, the accumulated errors themselves diffuse away and the errors tend slowly toward zero, after some small fluctuations in case of a few quickly converging methods. \[u^{\text{analytic}}(x,t)=\frac{x}{t^{7/2}}\bigg{(}1-\frac{x^{2}}{3\alpha t}+ \frac{x^{4}}{60\alpha^{2}t^{2}}\bigg{)}e^{-\frac{x^{2}}{64\alpha t}}. \tag{20}\]1 The maximum errors as a function of the time step size in Experiment 4 for a constant \(r=\alpha\ \Delta t/\Delta x^{2}\). This means that the space step size also decreases from the right to the left side of the figure. \[u^{\text{analytic}}(x,t)=\frac{x}{t^{7/2}}\bigg{(}1-\frac{x^{2}}{3\alpha t}+ \frac{x^{4}}{60\alpha^{2}t^{2}}\bigg{)}e^{-\frac{x^{2}}{64\alpha t}}. \tag{20}\]2 The maximum errors as a function of the time step size \(\Delta t\) for the slowly converging methods in the case of Experiment 5. \[u^{\text{analytic}}(x,t)=\frac{x}{t^{7/2}}\bigg{(}1-\frac{x^{2}}{3\alpha t}+ \frac{x^{4}}{60\alpha^{2}t^{2}}\bigg{)}e^{-\frac{x^{2}}{64\alpha t}}. \tag{20}\]3 The errors as a function of the time step size for the quickly converging methods, as well as the CLL method, in the case of Experiment 5. \[u^{\text{analytic}}(x,t)=\frac{x}{t^{7/2}}\bigg{(}1-\frac{x^{2}}{3\alpha t}+ \frac{x^{4}}{60\alpha^{2}t^{2}}\bigg{)}e^{-\frac{x^{2}}{64\alpha t}}. \tag{20}\]4 The evolution of the maximum errors as a function of the physical time \(t\) in the case of fast converging methods in Experiment 5. The used time step size is rather large: \(\Delta t=1.25\cdot 10^{-2}\). \[u^{\text{analytic}}(x,t)=\frac{x}{t^{7/2}}\bigg{(}1-\frac{x^{2}}{3\alpha t}+ \frac{x^{4}}{60\alpha^{2}t^{2}}\bigg{)}e^{-\frac{x^{2}}{64\alpha t}}. \tag{20}\]5 The evolution of the maximum errors as a function of the physical time \(t\) for the slowly converging methods in Experiment 5. The used time step size is \(\Delta t=4.7\cdot 10^{-3}\).