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Table 3 Comparison of \(L_{2 }\operatorname{err}\) and \(L_{\infty }\operatorname{err}\) of Example 5.2 with \(N = 7\), \(N_{e}=30\), and \(k=0.001\) at different time levels

From: Legendre spectral element method for solving sine-Gordon equation

Time (s) LSEM Dehghan and Shokri [18] Shukla and Tamsir [30] Mittal and Bhatia [31]
\(L_{2}\operatorname{err}\) \(L_{\infty }\operatorname{err}\) \(L_{2 }\operatorname{err}\) \(L_{\infty }\operatorname{err}\) \(L_{2 }\operatorname{err}\) \(L_{\infty }\operatorname{err}\) \(L_{2}\operatorname{err}\) \(L_{\infty }\operatorname{err}\)
0.25 2.39e−06 4.05e−06 3.91e−05 5.89e−06 2.43e−06 5.56e−06 1.18e−05 2.32e−05
0.5 6.85e−06 7.02e−06 1.30e−04 2.01e−05 5.54e−06 7.39e−06 4.19e−05 4.11e−05
0.75 6.97e−06 7.36e−06 2.35e−04 3.63e−05 6.45e−06 7.40e−06 7.78e−05 1.02e−04
1 1.07e−05 2.23e−05 3.27e−04 5.07e−05 7.84e−06 8.75e−06 1.30e−04 1.64e−04