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Two-Grid Based Adaptive Proper Orthogonal Decomposition Method for Time Dependent Partial Differential Equations

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Abstract

In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for the numerical solution obtained in the fine grid. Our new method is cheap and easy to be implement. We apply our new method to the solution of time-dependent advection–diffusion equations with the Kolmogorov flow and the ABC flow. The numerical results show that our method is more efficient than the existing POD methods.

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Acknowledgements

The authors would like to thank the anonymous referees for their nice and useful comments and suggestions that improve the quality of this paper.

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Correspondence to Xiaoying Dai.

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This work was supported by the National Key Research and Development Program of China under Grant 2019YFA0709601, the National Natural Science Foundation of China under Grants 91730302 and 11671389, the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences under Grant QYZDJ-SSW-SYS010, and the NSF Grant IIS-1632935 and DMS-1854434.

Appendices

Numerical Experiments for Tuning \(\gamma _i(i=1,2,3)\)

In this section, we do some numerical experiments for tuning the parameters \(\gamma _i(i=1,2,3)\) to find out a good choice.

Since the parameters \(\gamma _1\) and \(\gamma _2\) are both for extracting POD modes from the snapshots obtained from the standard finite element approximation of the original dynamic system, we set them to be the same value, that is, \(\gamma _1 = \gamma _2\). We have tested the Kolmogorov flow with \(\epsilon =0.05\) and \(\epsilon =0.01\) respectively, and the ABC flow with \(\epsilon =0.01\) by using the following different parameters:

  1. (1)

    case 1: \(\gamma _1= \gamma _2 = 1.0-1.0\times 10^{-8}\), set \(\gamma _3\) to be 0.9, 0.99, 0.999,  and 0.9999, respectively.

  2. (2)

    case 2: \(\gamma _3 = 1.0-1.0\times 10^{-8}\), set \(\gamma _1\) and \(\gamma _2\) to be 0.9, 0.99, 0.999,  and 0.9999, respectively.

  3. (3)

    case 3: \(\gamma _1=\gamma _2 = \gamma _3 = 1.0-1.0\times 10^{-2}, 1.0-1.0\times 10^{-4}, 1.0-1.0\times 10^{-6}, 1.0-1.0\times 10^{-8}\), respectively.

In our experiments, we choose \(\eta _0 = 0.005\). The other parameters are same as stated in Sect. 4.

Kolmogorov flow with \(\epsilon = 0.01\)

We first see the results for the Kolmogorov flow with \(\epsilon = 0.01\).

The results for case 1 are shown in Table 4. We can see that for fixed \(\gamma _1\) and \(\gamma _2\), the greater \(\gamma _3\), the more accurate the results, which means \(\gamma _3\) close to 1 may be a good choice. Then, we can see that when \(\gamma _1\) and \(\gamma _2\) are chosen to be too close to 1, no matter how to choose the parameter \(\gamma _3\), the cpu-time cost is too much. These results tell us that setting \(\gamma _1\) and \(\gamma _2\) too close to 1 is not a good choice.

Table 4 Detailed information for results of the Kolmogorov flow with \(\epsilon =0.01\) obtained by setting \(\gamma _1=\gamma _2=1.0-1.0\times 10^{-8}\) and \(\gamma _3=0.9,0.99, 0.999, 0.9999\), respectively

The results for case 2 are shown in Table 5, from which we can see that for fixed \(\gamma _3\), which is very close to 1, setting \(\gamma _1\) and \(\gamma _2\) to be 0.999 can obtain results with high accuracy. Taking into account the cpu time cost, 0.999 is a better choice.

Table 5 Detailed information for results of the Kolmogorov flow with \(\epsilon =0.01\) obtained by setting \(\gamma _3=1.0-1.0\times 10^{-8}\) and \(\gamma _1=\gamma _2=0.9,0.99, 0.999, 0.9999\), respectively

The results for case 3 are shown in Table 6. We see that taking into account both the accuracy and the cpu time cost, these parameters are not as good as those for case 2.

Table 6 Detailed information for results of the Kolmogorov flow with \(\epsilon =0.01\) obtained by setting \(\gamma _1=\gamma _2=\gamma _3=\gamma =1.0-1.0\times 10^{-2},1.0-1.0\times 10^{-4},1.0-1.0\times 10^{-6},1.0-1.0\times 10^{-8}\), respectively

Anyway, by comparing Tables 4,  5, and 6, we can see that the case of \(\gamma _1=\gamma _2=0.999\), \(\gamma _3 = 1.0-1.0\times 10^{-8}\) is the best choice among all these cases, if taking both the accuracy and the cpu-time cost into account.

Kolmogorov flow with \(\epsilon = 0.05\)

We then take a look at the results for the Kolmogorov flow with \(\epsilon = 0.05\), which are shown in Tables 7,  8, and  9.

Table 7 Detailed information for results of the Kolmogorov flow with \(\epsilon =0.05\) obtained by setting \(\gamma _1=\gamma _2=1.0-1.0\times 10^{-8}\) and \(\gamma _3=0.9,0.99, 0.999, 0.9999\), respectively
Table 8 Detailed information for results of the Kolmogorov flow with \(\epsilon =0.05\) obtained by setting \(\gamma _3=1.0-1.0\times 10^{-8}\) and \(\gamma _1=\gamma _2=0.9,0.99, 0.999, 0.9999\), respectively
Table 9 Detailed information for results of the Kolmogorov flow with \(\epsilon =0.05\) obtained by setting \(\gamma _1=\gamma _2=\gamma _3=\gamma =1.0-1.0\times 10^{-2},1.0-1.0\times 10^{-4},1.0-1.0\times 10^{-6},1.0-1.0\times 10^{-8}\), respectively

By comparing Tables 7,  8, and  9, we can also see that the case of \(\gamma _1=\gamma _2=0.999\), \(\gamma _3 = 1.0-1.0\times 10^{-8}\) is the best choice among all these cases if taking both the accuracy and the cpu-time cost into account.

ABC flow with \(\epsilon = 0.01\)

We have also done some numerical experiments for the ABC flow with \(\epsilon = 0.01\). For this example, the results obtained by setting different choice of the parameters \(\gamma _i(i=1,2,3)\) are shown in Tables 10,  11, and 12. From these results, we can draw the same conclusion as those obtained from the Kolmogorov flow, that is, \(\gamma _1=\gamma _2=0.999\), \(\gamma _3 = 1.0-1.0\times 10^{-8}\) is the best choice among all these cases if taking into account both the accuracy and the cpu-time cost.

Table 10 Detailed information for results of the ABC flow with \(\epsilon =0.01\) obtained by setting \(\gamma _1=\gamma _2=1.0-1.0\times 10^{-8}\) and \(\gamma _3=0.9,0.99, 0.999, 0.9999\), respectively
Table 11 Detailed information for results of the ABC flow with \(\epsilon =0.01\) obtained by setting \(\gamma _3=1.0-1.0\times 10^{-8}\) and \(\gamma _1=\gamma _2=0.9,0.99, 0.999, 0.9999\), respectively
Table 12 Detailed information for results of the ABC flow with \(\epsilon =0.01\) obtained by setting \(\gamma _1=\gamma _2=\gamma _3=\gamma =1.0-1.0\times 10^{-2},1.0-1.0\times 10^{-4},1.0-1.0\times 10^{-6},1.0-1.0\times 10^{-8}\), respectively

Numerical Experiments for Different Coarse Grid

Here, we use some numerical experiments to show how the degree of freedom for the coarse grid affects the accuracy and the cpu time cost when the fine grid is fixed. We hope it can provide some useful information about how to choose the coarse grid.

We take the Kolmogorov flow with \(\epsilon = 0.01\) as an example to show the behavior of our two-grid adaptive POD method with different coarse mesh. In our experiments, we fix the fine mesh as what is shown in Section 4, the other parameters are also set as same as those used for obtaining results shown in Table 1. That is, we set \(\gamma _1=\gamma _2 = 0.999\), \(\gamma _3=1.0-1.0\times 10^{-8}\), and \(\eta _0 = 0.005\). The detailed numerical results are listed in Table 13.

In Table 13, ‘N-Refine’ means the number of refinement used to obtain the mesh from the initial mesh, ‘Time step’ means the time step used for the coarse mesh, ‘DOFs-CoarseGrid’ means the degree of freedom corresponding to the coarse grid, ‘DOFs-POD’ means number of POD modes corresponding to the coarse grid, ‘POD-updates’ means the number of update of the POD modes, ‘Average Error’ is computed by averaging the errors of numerical solution for each time instance, ‘Time-CoarseGrid’ is the wall time for the simulation on the coarse grid, ‘Total-Time’ is the wall time for all the simulation.

Table 13 The results of the Kolmogorov flow with \(\epsilon =0.01\) obtained by TG-APOD with different coarse mesh, respectively

To see more clearly, we compare the error of numerical solutions obtained by TG-APOD with different coarse mesh in Fig. 6.

Fig. 6
figure 6

The change of error for the solution of the Kolmogorov flow with \(\epsilon =0.01\) obtained by TG-APOD with different coarse mesh

From Table 13 and Fig. 6, we can see that as the degree of freedom for the coarse mesh increases, the average error will decrease, while the cpu time cost may increase. Therefore, there is a balance for the accuracy and the cpu time cost. For our case, taking into account the accuracy and the cpu time cost, we choose the coarse grid obtained by refining 16 times from the initial mesh.

If we take a detailed look at Table 13, we will find that the cpu time cost is not only affected by the degree of freedom for the coarse grid. In fact, if the degree of freedom for the coarse mesh is not too large, the cpu time will be mainly determined by the number of POD modes updates (the process of basis update requires discretizing the original equation with finite element basis for a period of time, which is expensive), which is mainly determined by the \(\eta _0\). However, as the degree of freedom for the coarse mesh increases, the cpu time cost by calculating the error indicator on the coarse mesh also needs to be considered. Anyway, it is not easy to determine how large the coarse mesh is for the maximum efficiency.

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Dai, X., Kuang, X., Xin, J. et al. Two-Grid Based Adaptive Proper Orthogonal Decomposition Method for Time Dependent Partial Differential Equations. J Sci Comput 84, 47 (2020). https://doi.org/10.1007/s10915-020-01288-9

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