Table 11 \(L_{1}\) and \(L_{\infty }\) errors and CPU time at some different values of time step size k for Problem 1 at time \(2.5\times 10^{-3}\) with spatial mesh size \(h=0.01\), \(\sigma =2.0\) using NSFD with artificial viscosity method

\(T_{\max }/25\)

1.8982 × 10⁻¹

9.8811 × 10⁻¹

0.891

\(T_{\max }/50\)

1.2102 × 10⁻¹

9.1869 × 10⁻¹

0.893

\(T_{\max }/100\)

6.8010 × 10⁻²

6.9128 × 10⁻¹

0.897

\(T_{\max }/200\)

3.3252 × 10⁻²

3.8653 × 10⁻¹

0.928

\(T_{\max }/300\)

1.9973 × 10⁻²

2.4143 × 10⁻¹

0.998

\(T_{\max }/400\)

1.2961 × 10⁻²

1.5892 × 10⁻¹

0.999

\(T_{\max }/500\)

8.6255 × 10⁻³

1.0838 × 10⁻¹

1.019

\(T_{\max }/600\)

5.6799 × 10⁻³

7.3290 × 10⁻²

1.041

\(T_{\max }/700\)

3.5494 × 10⁻³

4.7746 × 10⁻²

1.089

\(T_{\max }/800\)

1.9544 × 10⁻³

2.8406 × 10⁻²

1.355

\(T_{\max }/900\)

8.4110 × 10⁻⁴

1.3294 × 10⁻²

1.355

\(T_{\max }/950\)

5.2648 × 10⁻⁴

7.1035 × 10⁻³

1.564

\(T_{\max }/1000\)

4.4308 × 10⁻⁴

5.5908 × 10⁻³

1.664

\(T_{\max }/1050\)

7.8758 × 10⁻⁴

8.5810 × 10⁻³

1.683

\(T_{\max }/1100\)

1.1865 × 10⁻³

1.1836 × 10⁻²

1.684

\(T_{\max }/1200\)

1.8866 × 10⁻³

1.9106 × 10⁻²

1.687

\(T_{\max }/1300\)

2.4809 × 10⁻³

2.5706 × 10⁻²

1.688

\(T_{\max }/1400\)

2.9918 × 10⁻³

3.1732 × 10⁻²

1.690

\(T_{\max }/1500\)

3.4356 × 10⁻³

3.6977 × 10⁻²

1.745