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# A fast high order method for time fractional diffusion equation with non-smooth data

• * Corresponding author: Aijie Cheng

This work was supported in part by the National Natural Science Foundation of China under Grants 91630207, 11971272

• 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.

Mathematics Subject Classification: Primary: 26A33, 65M06; Secondary: 65M15.

 Citation: • • Figure 1.  The absolute error of solutions obtained by the two formats with $M = 500$, $N = 500$, $\alpha = 0.6$, $r = \frac{3-\alpha}{\alpha}$ for Example 1

Figure 2.  The absolute error of solutions obtained by the two formats with $M = 500$, $N = 500$, $\alpha = 0.6$, $r = \frac{3-\alpha}{\alpha}$ for Example 2

Figure 3.  The absolute error of solutions obtained by FD and RFD $M = 200$, $N = 200$, $\alpha = 0.6$, $r = \frac{3-\alpha}{\alpha}$ for Example 3

Table 1.  The CPU time consumed by FD and RFD with different mesh sizes with $M = 10000$, $\alpha = 0.6$, $r = \frac{3-\alpha}{\alpha}$ for Example 1

 $N$ $2^5$ $2^6$ $2^7$ $2^8$ $2^9$ $2^{10}$ 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.  $M = 10000$, $r = 1$, the $L^{\infty}$ error and convergence rates for Example 1 on graded meshes

 $N$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $Max\_err$ 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.  $M = 10000$, $r = 1/\alpha$, the $L^{\infty}$ error and convergence rates for Example 1 on graded meshes

 $N$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $Max\_err$ 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.  $M = 10000$, $r = 2/\alpha$, the $L^{\infty}$ error and convergence rates for Example 1 on graded meshes

 $N$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $Max\_err$ 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.  $M = 10000$, $r = (3-\alpha)/\alpha$, the $L^{\infty}$ error and convergence rates for Example 1 on graded meshes

 $N$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $Max\_err$ 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 $M = 10000$, $\alpha = 0.6$, $r = \frac{3-\alpha}{\alpha}$ for Example2

 $N$ $2^5$ $2^6$ $2^7$ $2^8$ $2^9$ $2^{10}$ 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 $M = 10000$, $r = (3-\alpha)/\alpha$ the $L^{\infty}$ error and convergence rates for Example 2 on graded meshes

 $N$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $Max\_err$ 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.  $L1$ format, $M = 10000$, $r = (2-\alpha)/\alpha$ the $L^{\infty}$ error and convergence rates for Example 2 on graded meshes

 $N$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $Max\_err$ 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.  $L2-1_{\sigma}$ format, $M = 10000$, $r = 2/\alpha$ the $L^{\infty}$ error and convergence rates for Example 2 on graded meshes

 $N$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $Max\_err$ 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 $M = 100$, $\alpha = 0.6$, $r = \frac{3-\alpha}{\alpha}$ for Example3

 $N$ $2^5$ $2^6$ $2^7$ $2^8$ FD (CPU) 2082.2857 4202.6201 8426.9712 16835.3939 RFD(CPU) 0.0468 0.0624 0.1872 0.8112

Table 11.  $M = 500$, $r = (3-\alpha)/\alpha$, the $L^{\infty}$ error and convergence rates for Example 3 on graded meshes

 $N$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $Max\_err$ 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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