Theory and Modern Applications
Table 2 Numerical errors and convergence orders in spatial direction with \(\tau =\frac{1}{100}\) and \(\alpha =0.1\) at \(T=1\) for Example 1
h | \(\|u^{n}-u_{h}^{n}\|\) | Rate | \(\|u^{n}-u_{h}^{n}\|_{1}\) | Rate | \(\|I_{h}u^{n}-u_{h}^{n}\|_{1}\) | Rate | \(\|u^{n}-\varPi _{2h}u_{h}^{n}\|_{1}\) | Rate |
|---|---|---|---|---|---|---|---|---|
π/4 | 2.7783e-1 | 1.96 | 6.4095e-1 | 0.99 | 8.8916e-2 | 2.13 | 7.3428e-2 | 1.95 |
π/8 | 7.1625e-2 | 1.99 | 3.2382e-1 | 1.00 | 2.0343e-2 | 2.04 | 1.8993e-2 | 1.96 |
π/16 | 1.8037e-2 | 2.00 | 1.6239e-1 | 1.00 | 4.9636e-3 | 2.01 | 4.8653e-3 | 1.99 |
π/32 | 4.5157e-3 | 2.00 | 8.1256e-2 | 1.00 | 1.2311e-3 | 2.01 | 1.2246e-3 | 2.01 |
π/64 | 1.1277e-3 | ∗ | 4.0636e-2 | ∗ | 3.0493e-4 | ∗ | 3.0451e-4 | ∗ |