$ N $ | $ 2^5 $ | $ 2^6 $ | $ 2^7 $ | $ 2^8 $ | $ 2^9 $ | $ 2^{10} $ |
FD (CPU) | 235.9671 | 460.4681 | 922.5431 | 1821.1869 | 3661.3747 | 7446.4113 |
RFD (CPU) | 0.0624 | 0.1092 | 0.2496 | 0.9360 | 3.2448 | 13.5721 |
In this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence order on uniform meshes. Therefore, in order to improve the convergence order, we discrete the Caputo time fractional derivative by a new $ L1-2 $ format on graded meshes, while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. We analyze the approximation about the time fractional derivative, and obtain the time truncation error, but the stability analysis remains an open problem. On the other hand, considering that the computational cost is extremely large, we present a reduced-order finite difference extrapolation algorithm for the time-fraction diffusion equation by means of proper orthogonal decomposition (POD) technique, which effectively reduces the computational cost. Finally, several numerical examples are given to verify the convergence of the scheme and the effectiveness of the reduced order extrapolation algorithm.
Citation: |
Table 1.
The CPU time consumed by FD and RFD with different mesh sizes with
$ N $ | $ 2^5 $ | $ 2^6 $ | $ 2^7 $ | $ 2^8 $ | $ 2^9 $ | $ 2^{10} $ |
FD (CPU) | 235.9671 | 460.4681 | 922.5431 | 1821.1869 | 3661.3747 | 7446.4113 |
RFD (CPU) | 0.0624 | 0.1092 | 0.2496 | 0.9360 | 3.2448 | 13.5721 |
Table 2.
$ N $ | $ \alpha=0.4 $ | $ \alpha=0.6 $ | $ \alpha=0.8 $ | |||||
$ Max\_err $ | rate | $ Max\_err $ | rate | $ Max\_err $ | rate | |||
$ 2^{5} $ | 4.4813e-2 | 2.1049e-2 | 6.0226e-3 | |||||
$ 2^{6} $ | 3.5203e-2 | 0.3482 | 1.5005e-2 | 0.4883 | 4.2132e-3 | 0.5155 | ||
$ 2^{7} $ | 2.7395e-2 | 0.3618 | 1.0381e-2 | 0.5316 | 2.6654e-3 | 0.6606 | ||
$ 2^{8} $ | 2.1178e-2 | 0.3713 | 7.0548e-3 | 0.5572 | 1.6099e-3 | 0.7273 | ||
$ 2^{9} $ | 1.6293e-2 | 0.3783 | 4.7427e-3 | 0.5729 | 9.5001e-4 | 0.7610 | ||
$ 2^{10} $ | 1.2489e-2 | 0.3835 | 3.1668e-3 | 0.5827 | 5.5376e-4 | 0.7787 |
Table 3.
$ N $ | $ \alpha=0.4 $ | $ \alpha=0.6 $ | $ \alpha=0.8 $ | |||||
$ Max\_err $ | rate | $ Max\_err $ | rate | $ Max\_err $ | rate | |||
$ 2^{5} $ | 1.1780e-2 | 6.1861e-3 | 3.7802e-3 | |||||
$ 2^{6} $ | 6.0418e-3 | 0.9633 | 3.1668e-3 | 0.9660 | 2.0794e-3 | 0.8623 | ||
$ 2^{7} $ | 3.0601e-3 | 0.9814 | 1.6014e-3 | 0.9837 | 1.0853e-3 | 0.9380 | ||
$ 2^{8} $ | 1.5400e-3 | 0.9906 | 8.0511e-4 | 0.9921 | 5.5376e-4 | 0.9708 | ||
$ 2^{9} $ | 7.7251e-4 | 0.9953 | 4.0364e-4 | 0.9961 | 2.7944e-4 | 0.9867 | ||
$ 2^{10} $ | 3.8687e-4 | 0.9977 | 2.0209e-4 | 0.9981 | 1.4039e-4 | 0.9931 |
Table 4.
$ N $ | $ \alpha=0.4 $ | $ \alpha=0.6 $ | $ \alpha=0.8 $ | |||||
$ Max\_err $ | rate | $ Max\_err $ | rate | $ Max\_err $ | rate | |||
$ 2^{5} $ | 9.3428e-3 | 2.5627e-3 | 9.3566e-4 | |||||
$ 2^{6} $ | 2.3654e-3 | 1.9818 | 6.5273e-4 | 1.9731 | 2.5124e-4 | 1.8969 | ||
$ 2^{7} $ | 5.9358e-4 | 1.9946 | 1.6398e-4 | 1.9929 | 6.4435e-5 | 1.9632 | ||
$ 2^{8} $ | 1.4888e-4 | 1.9953 | 4.1050e-5 | 1.9981 | 1.6243e-5 | 1.9880 | ||
$ 2^{9} $ | 3.7600e-5 | 1.9854 | 1.0266e-5 | 1.9995 | 4.0700e-6 | 1.9967 | ||
$ 2^{10} $ | 9.7725e-6 | 1.9439 | 2.5667e-6 | 1.9999 | 1.0181e-6 | 1.9992 |
Table 5.
$ N $ | $ \alpha=0.4 $ | $ \alpha=0.6 $ | $ \alpha=0.8 $ | |||||
$ Max\_err $ | rate | $ Max\_err $ | rate | $ Max\_err $ | rate | |||
$ 2^{5} $ | 8.1920e-3 | 1.9021e-3 | 8.5782e-4 | |||||
$ 2^{6} $ | 1.3564e-3 | 2.5944 | 3.6421e-4 | 2.3848 | 2.0153e-4 | 2.0897 | ||
$ 2^{7} $ | 2.2375e-4 | 2.5999 | 6.9143e-5 | 2.3971 | 4.5568e-5 | 2.1449 | ||
$ 2^{8} $ | 3.6784e-5 | 2.6048 | 1.3105e-5 | 2.3995 | 1.0109e-5 | 2.1724 | ||
$ 2^{9} $ | 5.9417e-6 | 2.6301 | 2.4831e-6 | 2.3999 | 2.2188e-6 | 2.1877 | ||
$ 2^{10} $ | 8.5465e-7 | 2.7975 | 4.7049e-7 | 2.3999 | 4.8440e-7 | 2.1955 |
Table 6.
The CPU time consumed by FD and RFD with different mesh sizes with
$ N $ | $ 2^5 $ | $ 2^6 $ | $ 2^7 $ | $ 2^8 $ | $ 2^9 $ | $ 2^{10} $ | |
FD (CPU) | 216.2954 | 435.1960 | 882.8253 | 1751.2672 | 3460.3362 | 7030.6999 | |
RFD(CPU) | 0.0156 | 0.0624 | 0.1872 | 0.7488 | 2.8080 | 14.0713 |
Table 7.
Take
$ N $ | $ \alpha=0.4 $ | $ \alpha=0.6 $ | $ \alpha=0.8 $ | |||||
$ Max\_err $ | rate | $ Max\_err $ | rate | $ Max\_err $ | rate | |||
$ 2^{5} $ | 9.2288e-3 | 2.1343e-3 | 9.6882e-4 | |||||
$ 2^{6} $ | 1.5290e-3 | 2.5936 | 4.0781e-4 | 2.3878 | 2.2141e-4 | 2.1296 | ||
$ 2^{7} $ | 2.5250e-4 | 2.5982 | 7.7390e-5 | 2.3977 | 4.9476e-5 | 2.1619 | ||
$ 2^{8} $ | 4.1777e-5 | 2.5955 | 1.4667e-5 | 2.3996 | 1.0907e-5 | 2.1815 | ||
$ 2^{9} $ | 7.0163e-6 | 2.5739 | 2.7790e-6 | 2.3999 | 2.3867e-6 | 2.1922 | ||
$ 2^{10} $ | 1.2826e-6 | 2.4516 | 5.2652e-7 | 2.4000 | 5.2036e-7 | 2.1974 |
Table 8.
$ N $ | $ \alpha=0.4 $ | $ \alpha=0.6 $ | $ \alpha=0.8 $ | |||||
$ Max\_err $ | rate | $ Max\_err $ | rate | $ Max\_err $ | rate | |||
$ 2^{5} $ | 2.0114e-3 | 3.4273e-3 | 5.1455e-3 | |||||
$ 2^{6} $ | 7.1710e-4 | 1.4880 | 1.3922e-3 | 1.2997 | 2.3995e-3 | 1.1006 | ||
$ 2^{7} $ | 2.4927e-4 | 1.5245 | 5.5229e-4 | 1.3339 | 1.1033e-3 | 1.1209 | ||
$ 2^{8} $ | 8.5555e-5 | 1.5428 | 2.1578e-4 | 1.3558 | 5.0211e-4 | 1.1358 | ||
$ 2^{9} $ | 2.9144e-5 | 1.5537 | 8.3468e-5 | 1.3703 | 2.2670e-4 | 1.1472 | ||
$ 2^{10} $ | 9.8709e-6 | 1.5619 | 3.2075e-5 | 1.3798 | 1.0172e-4 | 1.1562 |
Table 9.
$ N $ | $ \alpha=0.4 $ | $ \alpha=0.6 $ | $ \alpha=0.8 $ | |||||
$ Max\_err $ | rate | $ Max\_err $ | rate | $ Max\_err $ | rate | |||
$ 2^{5} $ | 2.3712e-4 | 1.9954e-4 | 1.2946e-4 | |||||
$ 2^{6} $ | 6.1882e-5 | 1.9380 | 5.2396e-5 | 1.9291 | 3.5332e-5 | 1.8735 | ||
$ 2^{7} $ | 1.5706e-5 | 1.9782 | 1.3317e-5 | 1.9762 | 9.2562e-6 | 1.9325 | ||
$ 2^{8} $ | 3.9412e-6 | 1.9946 | 3.3471e-6 | 1.9923 | 2.3634e-6 | 1.9695 | ||
$ 2^{9} $ | 9.8622e-7 | 1.9986 | 8.3790e-7 | 1.9981 | 5.9543e-7 | 1.9889 | ||
$ 2^{10} $ | 2.4661e-7 | 1.9997 | 2.0955e-7 | 1.9995 | 1.4920e-7 | 1.9967 |
Table 10.
The CPU time consumed by FD and RFD with different mesh sizes with
$ N $ | $ 2^5 $ | $ 2^6 $ | $ 2^7 $ | $ 2^8 $ | |
FD (CPU) | 2082.2857 | 4202.6201 | 8426.9712 | 16835.3939 | |
RFD(CPU) | 0.0468 | 0.0624 | 0.1872 | 0.8112 |
Table 11.
$ N $ | $ \alpha=0.4 $ | $ \alpha=0.6 $ | $ \alpha=0.8 $ | |||||
$ Max\_err $ | rate | $ Max\_err $ | rate | $ Max\_err $ | rate | |||
$ 2^{5} $ | 9.1847e-3 | 2.1075e-3 | 9.0565e-4 | |||||
$ 2^{6} $ | 1.5275e-3 | 2.5880 | 4.0685e-4 | 2.3730 | 2.1443e-4 | 2.0785 | ||
$ 2^{7} $ | 2.5217e-4 | 2.5987 | 7.7358e-5 | 2.3949 | 4.8728e-5 | 2.1377 | ||
$ 2^{8} $ | 4.1476e-5 | 2.6040 | 1.4667e-5 | 2.3990 | 1.0858e-5 | 2.1660 |
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The absolute error of solutions obtained by the two formats with
The absolute error of solutions obtained by the two formats with
The absolute error of solutions obtained by FD and RFD