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Table 5 The results of \(L_{\infty }\) error norms and the related temporal rate of convergence in Test Problem 5.3 with \(M=10\), \(T_{f}=1\) and diverse Δt

From: Time accurate solution to Benjamin–Bona–Mahony–Burgers equation via Taylor–Boubaker series scheme

Δt

TBM (M = 10)

\(\mathcal{E}_{\infty }\)

\(\operatorname{ROCt}_{\infty }\)

\(\frac{1}{2}\)

2.5261−3

\(\frac{1}{4}\)

7.7570−4

1.7033

\(\frac{1}{8}\)

1.9739−4

1.9745

\(\frac{1}{16}\)

4.9553−5

1.9940

Δt

FDCP [28]

CLM [26]

Lucas [27]

τ

\(L_{\infty } (N=18)\)

\(L_{\infty } (N=36)\)

Order

\(L_{\infty } (N=18)\)

\(\frac{1}{2}\)

\(\frac{2}{10^{3}}\)

5.0010−08

2.048−07

5.0011−08

\(\frac{1}{4}\)

\(\frac{1}{10^{3}}\)

1.2502−08

5.119−08

2.00

1.2503−08

\(\frac{1}{8}\)

\(\frac{2}{10^{4}}\)

5.0011−10

2.047−09

2.00

5.0014−10

\(\frac{1}{16}\)

\(\frac{1}{10^{4}}\)

1.2496−10

5.117−10

2.00

1.2511−10