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Table 1 Maximum Errors and Rates of Convergence for the Bakhvalov-Shishkin Mesh on.

Theoretical analysis is supported by two numerical examples, and the maximum errors and the convergence order are checked.

The maximum errors and convergence orders in spatial directions with (tau=10^{-3}) and (tin[0,0.05]) are listed in Table 5.

In Tables 1 and 2, the maximum errors and the order of convergence of the approximate solution for different step sizes h are given, which corresponds to order of accuracy (O(h^{6}vert ln hvert )).

In Figures 1-2, the approximating errors and states between the exact solutions and numerical solutions are shown, whereas in Tables 1 and 2, the maximum errors and convergence orders in spatial directions with (tin[0,0.1]) are listed.

In Table 6, the maximum errors and convergence orders in spatial directions with (tau=10^{-4}) and (tin[0,0.01]) are listed, whereas in Figures 5-6, comparison between the exact and numerical solutions is shown under certain conditions.

The maximum error and the coefficient of variation were found to be 4.3% and 4.6%, respectively.

It can reproduce ignition delays with 3.1% maximum error and reproduces pressure rise precisely (error∼10−3%).

When the modulus of the wave is less than 0.95, the maximum error and absolute average error are acceptable for engineering.

The average error, the maximum error and the RMSE in each phase were shown in Table 2 and Table 3 respectively.

However, the two receivers never differed by more than 0.5 Hz (maximum error) and the standard deviation of their difference was 0.045 Hz.