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Theory and Modern Applications

Table 1 Numerical results for sine-Gordon equation Example 5.1 with \(N{_{e}}=20\) and \(T=1\)

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

 

k = 0.1

k = 0.01

N

\(L_{2}\operatorname{err}\)

\(L_{\infty }\operatorname{err}\)

RMSerr

\({L_{2}}\operatorname{err}\)

\(L_{\infty }\operatorname{err}\)

RMSerr

1

2.4093e−01

1.6930e−01

1.1473e−02

2.4629e−01

1.7297e−01

1.1728e−02

2

1.2518e−01

7.9776e−02

3.0531e−03

1.3130e−01

8.3774e−02

3.2024e−03

3

2.2023e−02

9.9553e−03

3.6104e−04

2.4071e−02

1.0268e−02

3.9460e−04

4

3.9778e−03

1.9421e−03

4.9109e−05

6.4771e−03

3.4862e−03

7.9964e−05

5

3.5862e−03

1.3858e−03

3.5507e−05

1.0154e−03

5.3085e−04

1.0053e−05

6

3.5050e−03

1.3304e−03

2.8967e−05

3.5765e−04

1.5567e−04

2.9558e−06

7

3.7733e−03

1.2734e−03

2.6761e−05

3.5359e−04

1.2447e−04

2.5078e−06