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Computational Complexity Study on Krylov Integration Factor WENO Method for High Spatial Dimension Convection–Diffusion Problems

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Abstract

Integration factor (IF) methods are a class of efficient time discretization methods for solving stiff problems via evaluation of an exponential function of the corresponding matrix for the stiff operator. The computational challenge in applying the methods for partial differential equations (PDEs) on high spatial dimensions (multidimensional PDEs) is how to deal with the matrix exponential for very large matrices. Compact integration factor methods developed in Nie et al. (J Comput Phys 227:5238–5255, 2008) provide an approach to reduce the cost prohibitive large matrix exponentials for linear diffusion operators with constant diffusion coefficients in high spatial dimensions to a series of much smaller one dimensional computations. This approach is further developed in Wang et al. (J Comput Phys 258:585–600, 2014) to deal with more complicated high dimensional reaction–diffusion equations with cross-derivatives in diffusion operators. Another approach is to use Krylov subspace approximations to efficiently calculate large matrix exponentials. In Chen and Zhang (J Comput Phys 230:4336–4352, 2011), Krylov subspace approximation is directly applied to the implicit integration factor (IIF) methods for solving high dimensional reaction–diffusion problems. Recently the method is combined with weighted essentially non-oscillatory (WENO) schemes in Jiang and Zhang (J Comput Phys 253:368–388, 2013) to efficiently solve semilinear and fully nonlinear convection–reaction–diffusion equations. A natural question that arises is how these two approaches may perform differently for various types of problems. In this paper, we study the computational power of Krylov IF-WENO methods for solving high spatial dimension convection–diffusion PDE problems (up to four spatial dimensions). Systematical numerical comparison and complexity analysis are carried out for the computational efficiency of the two different approaches. We show that although the Krylov IF-WENO methods have linear computational complexity, both the compact IF method and the Krylov IF method have their own advantages for different type of problems. This study provides certain guidance for using IF-WENO methods to solve general high spatial dimension convection–diffusion problems.

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Author information

Authors and Affiliations

  1. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Dong Lu & Yong-Tao Zhang

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  1. Dong Lu
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  2. Yong-Tao Zhang
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Correspondence to Yong-Tao Zhang.

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Dedicated to Professor Chi-Wang Shu on the occasion of his 60th birthday.

Research supported by NSF Grant DMS-1620108.

Appendix: Detailed Formulae for AcIIF-WENO Schemes

Appendix: Detailed Formulae for AcIIF-WENO Schemes

(1) For the three dimensional CDR equation, if \({\mathcal {L}}_{12}\), \({\mathcal {L}}_{13}\) and \({\mathcal {L}}_{23}\) commute with each other, then

$$\begin{aligned} {\varvec{\Theta }}_\mathbf{1}= & {} \underset{1\le k_1\le N_1}{\bigotimes }e^{{\mathcal {A}}_{23}\triangle {t_n}}\Bigg (\underset{1\le k_2\le N_2}{\bigotimes }e^{{\mathcal {A}}_{13}\triangle {t_n}}\Bigg (\underset{1\le k_3\le N_3}{\bigotimes }e^{{\mathcal {A}}_{12}\triangle {t_n}}V_1(:,:,k_3)\Bigg )(:,k_2,:)\Bigg )(k_1,:,:),\nonumber \\ {\varvec{\Theta }}_\mathbf{2}= & {} \underset{1\le k_1\le N_1}{\bigotimes }e^{{\mathcal {A}}_{23}(\triangle {t_n}+\triangle {t_{n-1}})}\Bigg (\underset{1\le k_2\le N_2}{\bigotimes }e^{{\mathcal {A}}_{13}(\triangle {t_n}+\triangle {t_{n-1}})}\nonumber \\&\Bigg (\underset{1\le k_3\le N_3}{\bigotimes }e^{{\mathcal {A}}_{12}(\triangle {t_n}+\triangle {t_{n-1}})}V_2(:,:,k_3)\Bigg )(:,k_2,:)\Bigg )(k_1,:,:). \end{aligned}$$
(69)

If \({\mathcal {L}}_{12}\), \({\mathcal {L}}_{13}\) and \({\mathcal {L}}_{23}\) do not commute with each other, then

$$\begin{aligned} \begin{aligned} {\varvec{\Theta }}_\mathbf{1}&=\underset{1\le k_3\le N_3}{\bigotimes }e^{{\mathcal {A}}^{k_3}_{12}\frac{\triangle {t_n}}{2}}\Bigg (\underset{1\le k_2\le N_2}{\bigotimes }e^{{\mathcal {A}}^{k_2}_{13}\frac{\triangle {t_n}}{2}}V_1^*(:,k_2,:)\Bigg )(:,:,k_3),\\ V_1^*&=\underset{1\le k_1\le N_1}{\bigotimes }e^{{\mathcal {A}}^{k_1}_{23}\triangle {t_n}}\Bigg (\underset{1\le k_2\le N_2}{\bigotimes }e^{{\mathcal {A}}^{k_2}_{13}\frac{\triangle {t_n}}{2}}\Bigg (\underset{1\le k_3\le N_3}{\bigotimes }e^{{\mathcal {A}}^{k_3}_{12}\frac{\triangle {t_n}}{2}}V_1(:,:,k_3)\Bigg )(:,k_2,:)\Bigg )(k_1,:,:); \end{aligned} \end{aligned}$$
(70)

and

$$\begin{aligned} \begin{aligned} {\varvec{\Theta }}_\mathbf{2}&=\underset{1\le k_3\le N_3}{\bigotimes }e^{{\mathcal {A}}^{k_3}_{12}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}\Bigg (\underset{1\le k_2\le N_2}{\bigotimes }e^{{\mathcal {A}}^{k_2}_{13}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}V_2^*(:,k_2,:)\Bigg )(:,:,k_3),\\ V_2^*&=\underset{1\le k_1\le N_1}{\bigotimes }e^{{\mathcal {A}}^{k_1}_{23}(\triangle {t_n}+\triangle {t_{n-1}})}\Bigg (\underset{1\le k_2\le N_2}{\bigotimes }e^{{\mathcal {A}}^{k_2}_{13}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}\\&\Bigg (\underset{1\le k_3\le N_3}{\bigotimes }e^{{\mathcal {A}}^{k_3}_{12}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}V_2(:,:,k_3)\Bigg )(:,k_2,:)\Bigg )(k_1,:,:). \end{aligned} \end{aligned}$$
(71)

(2) For the four dimensional CDR equation, if \({\mathcal {L}}_{12}\), \({\mathcal {L}}_{13}\), \({\mathcal {L}}_{14}\), \({\mathcal {L}}_{23}\), \({\mathcal {L}}_{24}\) and \({\mathcal {L}}_{34}\) commute with each other, then

$$\begin{aligned} \begin{aligned} {\varvec{\Theta }}_\mathbf{1}&= {\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}} \limits _{1\le k_2 \le N_2}}e^{{\mathcal {A}}_{34}\triangle {t_n}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}_{24}\triangle {t_n}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}_{23}\triangle {t_n}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}}\limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}_{14}\triangle {t_n}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}_{13}\triangle {t_n}}\\&\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_3\le N_3}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}_{12}\triangle {t_n}}V_1(:,:,k_3,k_4)\Bigg )(:,k_2,:,k_4)\Bigg )(:,k_2,k_3,:)\Bigg )(k_1,:,:,k_4)\Bigg )(k_1,:,k_3,:)\Bigg )\\\&(k_1,k_2,:,:), \end{aligned} \end{aligned}$$
(72)
$$\begin{aligned} \begin{aligned} {\varvec{\Theta }}_\mathbf{2}&={\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_2 \le N_2}}e^{{\mathcal {A}}_{34}(\triangle {t_n}+\triangle {t_{n-1}})}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}} \limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}_{24}(\triangle {t_n}+\triangle {t_{n-1}})}\\&\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}_{23}(\triangle {t_n}+\triangle {t_{n-1}})}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}} \limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}_{14}(\triangle {t_n}+\triangle {t_{n-1}})}\\&\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}_{13}(\triangle {t_n}+\triangle {t_{n-1}})}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_3\le N_3}} \limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}_{12}(\triangle {t_n}+\triangle {t_{n-1}})}V_2(:,:,k_3,k_4)\Bigg )(:,k_2,:,k_4)\Bigg )\\&(:,k_2,k_3,:)\Bigg )(k_1,:,:,k_4)\Bigg )(k_1,:,k_3,:)\Bigg )(k_1,k_2,:,:). \end{aligned} \end{aligned}$$
(73)

If \({\mathcal {L}}_{12}\), \({\mathcal {L}}_{13}\), \({\mathcal {L}}_{14}\), \({\mathcal {L}}_{23}\), \({\mathcal {L}}_{24}\) and \({\mathcal {L}}_{34}\) do not commute with each other, then

$$\begin{aligned} \begin{aligned} {\varvec{\Theta }}_\mathbf{1}&={\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}} \limits _{1\le k_2 \le N_2}}e^{{\mathcal {A}}^{k_1, k_2}_{34}\frac{\triangle {t_n}}{2}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}} \limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}^{k_1, k_3}_{24}\frac{\triangle {t_n}}{2}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}} \limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}^{k_1, k_4}_{23}\frac{\triangle {t_n}}{2}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}} \limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}^{k_2, k_3}_{14}\frac{\triangle {t_n}}{2}}\\&\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}^{k_2, k_4}_{13}\frac{\triangle {t_n}}{2}}V_1^*(:,k_2,:,k_4)\Bigg )(:,k_2,k_3,:)\Bigg )(k_1,:,:,k_4)\Bigg )(k_1,:,k_3,:)\Bigg )(k_1,k_2,:,:), \end{aligned} \end{aligned}$$
(74)
$$\begin{aligned} \begin{aligned} V_1^*&={\mathop {\mathop {\bigotimes }\limits _{1\le k_3\le N_3}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}^{k_3, k_4}_{12}\triangle {t_n}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}^{k_2, k_4}_{13}\frac{\triangle {t_n}}{2}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}} \limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}^{k_2, k_3}_{14}\frac{\triangle {t_n}}{2}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}} \limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}^{k_1, k_4}_{23}\frac{\triangle {t_n}}{2}}\\&\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}} \limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}^{k_1, k_3}_{24}\frac{\triangle {t_n}}{2}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_2 \le N_2}}e^{{\mathcal {A}}^{k_1, k_2}_{34}\frac{\triangle {t_n}}{2}}V_1(k_1,k_2,:,:)\Bigg )\\&(k_1,:,k_3,:)\Bigg )(k_1,:,:,k_4)\Bigg )(:,k_2,k_3,:)\Bigg )(:,k_2,:,k_4)\Bigg )(:,:,k_3,k_4). \end{aligned} \end{aligned}$$
(75)

And

$$\begin{aligned} {\varvec{\Theta }}_\mathbf{2}= & {} {\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_2 \le N_2}}e^{{\mathcal {A}}^{k_1, k_2}_{34}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}^{k_1, k_3}_{24}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}} \Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}^{k_1, k_4}_{23}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}\nonumber \\&\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}}\limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}^{k_2, k_3}_{14}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}\Bigg ( {\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}^{k_2, k_4}_{13}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}V_2^*(:,k_2,:,k_4)\Bigg )\\&(:,k_2,k_3,:)\Bigg )(k_1,:,:,k_4)\Bigg )(k_1,:,k_3,:)\Bigg )(k_1,k_2,:,:),\nonumber \end{aligned}$$
(76)
$$\begin{aligned} V_2^*&={\mathop {\mathop {\bigotimes }\limits _{1\le k_3\le N_3}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}^{k_3, k_4}_{12}(\triangle {t_n}+\triangle {t_{n-1}})}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}^{k_2, k_4}_{13}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_2\le N_2}}\limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}^{k_2, k_3}_{14}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}\nonumber \\&\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_4 \le N_4}}e^{{\mathcal {A}}^{k_1, k_4}_{23}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}\Bigg ( {\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_3 \le N_3}}e^{{\mathcal {A}}^{k_1, k_3}_{24}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}\Bigg ({\mathop {\mathop {\bigotimes }\limits _{1\le k_1\le N_1}}\limits _{1\le k_2 \le N_2}}e^{{\mathcal {A}}^{k_1, k_2}_{34}\frac{(\triangle {t_n}+\triangle {t_{n-1}})}{2}}\nonumber \\&V_2(k_1,k_2,:,:) \Bigg )(k_1,:,k_3,:)\Bigg )(k_1,:,:,k_4)\Bigg )(:,k_2,k_3,:)\Bigg )(:,k_2,:,k_4)\Bigg )(:,:,k_3,k_4). \end{aligned}$$
(77)

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Lu, D., Zhang, YT. Computational Complexity Study on Krylov Integration Factor WENO Method for High Spatial Dimension Convection–Diffusion Problems. J Sci Comput 73, 980–1027 (2017). https://doi.org/10.1007/s10915-017-0398-7

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