Table 7 The \(L_{\infty }\), \(L_{2}\) errors and temporal convergence order with \(h = 1/1000\) for Example 4

0.2

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

3.7493e−008

2.7741e−008

1.6551e−005

1.1499e−005

\(\frac{1}{20}\)

4.3399e−009

3.1109

3.2026e−009

3.1147

5.1209e−006

1.6925

3.5580e−006

1.6925

\(\frac{1}{40}\)

5.2034e−010

3.0601

3.8317e−010

3.0632

1.5620e−006

1.1730

1.0853e−006

1.1730

\(\frac{1}{80}\)

6.2570e−011

3.0559

4.5904e−011

3.0613

4.1734e−007

1.7286

3.2749e−007

1.7285

0.4

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

1.8809e−007

1.3672e−007

6.6853e−005

4.6449e−005

\(\frac{1}{20}\)

2.1674e−008

3.1174

1.5732e−008

3.1195

2.3089e−005

1.5338

1.6043e−005

1.5337

\(\frac{1}{40}\)

2.6010e−009

3.0588

1.8861e−009

3.0602

7.8699e−006

1.5528

5.4682e−006

1.5528

\(\frac{1}{80}\)

3.1627e−010

3.0398

2.2894e−010

3.0424

2.6586e−006

1.5657

1.8472e−006

1.5657

0.6

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

5.6461e−007

4.0656e−007

2.0417e−004

1.4185e−004

\(\frac{1}{20}\)

6.4049e−008

3.1400

4.6091e−008

3.1409

7.9753e−005

1.3562

5.5412e−005

1.3561

\(\frac{1}{40}\)

7.6381e−009

3.0679

5.4945e−009

3.0684

3.0792e−005

1.3730

2.1394e−005

1.3730

\(\frac{1}{80}\)

9.3351e−010

3.0325

6.7150e−010

3.0325

1.1806e−005

1.3830

8.2031e−006

1.3830

0.8

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

1.3917e−006

9.9376e−007

5.5067e−004

3.8256e−004

\(\frac{1}{20}\)

1.5093e−007

3.2049

1.0777e−007

3.2049

2.4518e−004

1.1674

1.7033e−004

1.1673

\(\frac{1}{40}\)

1.7696e−008

3.0924

1.2636e−008

3.0923

1.0803e−004

1.1824

7.5052e−005

1.1824

\(\frac{1}{80}\)

2.1461e−009

3.0436

1.5328e−009

3.0433

4.7334e−005

1.1904

3.2887e−005

1.1904

0.98

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

2.9227e−006

2.0612e−006

–

–

–

–

\(\frac{1}{20}\)

3.0132e−007

3.2779

2.1338e−007

3.2720

–

–

–

–

\(\frac{1}{40}\)

3.4061e−008

3.1451

2.4121e−008

3.1451

–

–

–

–

\(\frac{1}{80}\)

4.0493e−009

3.0724

2.8676e−009

3.0724

–

–

–

–