Theory and Modern Applications
Table 10 \(L_{1}\) and \(L_{\infty }\) errors CPU time at some different values of time-step size k for Problem 1 with \(\rho =10^{4}\) at time \(2.5\times 10^{-3}\) with spatial mesh size \(\sigma =2.0 \) and \(h=0.01\) using FTCS with artificial viscosity
Time step (k) | \(L_{1}\) error | \(L_{\infty }\) error | CPU (s) |
|---|---|---|---|
\(T_{\max }/53\) | 2.6947 × 10−1 | 1.5155 | 0.737 |
\(T_{\max }/100\) | 6.2546 × 10−2 | 6.8175 × 10−1 | 0.737 |
\(T_{\max }/200\) | 3.0034 × 10−2 | 3.6771 × 10−1 | 0.742 |
\(T_{\max }/300\) | 1.7645 × 10−2 | 2.1809 × 10−1 | 0.758 |
\(T_{\max }/400\) | 1.1108 × 10−2 | 1.3983 × 10−1 | 0.782 |
\(T_{\max }/500\) | 7.0695 × 10−3 | 8.9529 × 10−2 | 0.800 |
\(T_{\max }/600\) | 4.3263 × 10−3 | 5.5222 × 10−2 | 0.847 |
\(T_{\max }/700\) | 2.3414 × 10−3 | 3.0552 × 10−2 | 0.856 |
\(T_{\max }/800\) | 8.5592 × 10−4 | 1.2040 × 10−2 | 0.907 |
\(T_{\max }/850\) | 3.1048 × 10−4 | 4.4291 × 10−3 | 0.913 |
\(T_{\max }/860\) | 2.2700 × 10−4 | 3.0141 × 10−3 | 0.918 |
\(T_{\max }/870\) | 1.6308 × 10−4 | 1.8608 × 10−3 | 0.923 |
\(T_{\max }/880\) | 1.3833 × 10−4 | 2.3032 × 10−3 | 0.926 |
\(T_{\max }/890\) | 2.3267 × 10−4 | 2.9465 × 10−3 | 0.930 |
\(T_{\max }/900\) | 3.3881 × 10−4 | 3.6042 × 10−3 | 0.951 |
\(T_{\max }/1000\) | 1.2861 × 10−3 | 1.4273 × 10−2 | 0.967 |
\(T_{\max }/1100\) | 2.0648 × 10−3 | 2.3789 × 10−2 | 1.001 |
\(T_{\max }/1200\) | 2.7162 × 10−3 | 3.1756 × 10−2 | 1.042 |
\(T_{\max }/1300\) | 3.2692 × 10−3 | 3.8522 × 10−2 | 1.071 |
\(T_{\max }/1400\) | 3.7445 × 10−3 | 4.4336 × 10−2 | 1.124 |
\(T_{\max }/1500\) | 4.1573 × 10−3 | 4.9385 × 10−3 | 1.142 |