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Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D

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Abstract

In this work, we propose efficient and accurate numerical algorithms based on difference potentials method for numerical solution of chemotaxis systems and related models in 3D. The developed algorithms handle 3D irregular geometry with the use of only Cartesian meshes and employ Fast Poisson Solvers. In addition, to further enhance computational efficiency of the methods, we design a difference-potentials-based domain decomposition approach which allows mesh adaptivity and easy parallelization of the algorithm in space. Extensive numerical experiments are presented to illustrate the accuracy, efficiency and robustness of the developed numerical algorithms.

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Acknowledgements

The research of Y. Epshteyn and Q. Xia was partially supported by National Science Foundation Grant # DMS-1112984. Q. Xia would also like to thank N. Beebe, M. Berzins, M. Cuma and H. Sundar for helpful discussions on different aspects of high-performance computing related to this work. Q. Xia gratefully acknowledges the support of the University of Utah, Department of Mathematics.

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  1. Department of Mathematics, The University of Utah, 155 S 1400 E Rm. 233, Salt Lake City, UT, 84112, USA

    Yekaterina Epshteyn & Qing Xia

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5 Appendix

5 Appendix

1.1 5.1 Proof to Theorem 1

Proof

For the reader’s convenience, let us recall proof of Theorem 1 in [13, 41]. First of all, let us assume that there is a density \(u^{i+1}_{\gamma }\) that satisfies the BEP (17). Using \(u^{i+1}_{\gamma }\), we can construct a superposition of the Particular Solution using (12–14) and the Difference Potential using (12–13) and (16):

$$\begin{aligned} u^{i+1}_{j,k,l}=P_{N^+\gamma }u^{i+1}_{\gamma }+G_{h,\Delta t}f^{i},\quad (x_j,y_k,z_l)\in M^0. \end{aligned}$$
(64)

It can be seen that \(u^{i+1}\) defined in (64) is a solution to the following discrete system on \(N^0\):

$$\begin{aligned} \begin{aligned} L_{h,\Delta t}u^{i+1}_{j,k,l}&=\left\{ \begin{array}{ll} f^i_{j,k,l}, &{} \quad (x_j,y_k,z_l)\in M^+,\\ L_{h,\Delta t}[u^{i+1}_{\gamma }],&{}\quad (x_j,y_k,z_l)\in M^-, \end{array} \right. \\ u^{i+1}_{j,k,l}&=0,\quad (x_j,y_k,z_l)\in N^0\backslash M^0, \end{aligned} \end{aligned}$$
(65)

which implies that \(u^{i+1}_{j,k,l},\;(x_i,y_j,z_k)\in M^+\) in (64) is some solution to (11) on \(M^+\), and note that the trace of \(u^{i+1}\) satisfies

$$\begin{aligned} Tr_{\gamma }u^{i+1}=Tr_{\gamma }\left( P_{N^+\gamma }u^{i+1}_{\gamma }+G_{h,\Delta t}f^{i}\right) =P_{\gamma }u^{i+1}_{\gamma }+G_{h,\Delta t}f^i_\gamma , \end{aligned}$$
(66)

from the definition of trace operator. Moreover, from our assumption on the density \(u^{i+1}_{\gamma }\), we have \(P_{\gamma }u^{i+1}_{\gamma }+G_{h,\Delta t}f^i_\gamma =u^{i+1}_{\gamma }\). Thus, \(Tr_{\gamma }u^{i+1}=u^{i+1}_{\gamma }\) and we can conclude that if some density \(u^{i+1}_{\gamma }\) satisfies the BEP (17), it is a trace to some solution of system (11).

Now assume the density \(u^{i+1}_{\gamma }\) is a trace to some solution of system (11): \(u^{i+1}_{\gamma }=Tr_{\gamma }u^{i+1}\) where \(u^{i+1}_{j,k,l}\), \((x_j,y_k,z_l)\in N^+\) is the unique solution to the difference equation (11) subject to certain boundary condition. Then by our assumption on \(u^{i+1}\), \(u^{i+1}\) with zero extension from \(N^+\) to \(N^0\) is also a solution to the system of difference equation (65) on \(N^0\). Since \(P_{N^+\gamma }u^{i+1}_{\gamma }+G_{h,\Delta t}f^i\) is also a solution the system of difference equation (65), we conclude that:

$$\begin{aligned} u^{i+1}\equiv P_{N^+\gamma }u^{i+1}_{\gamma }+G_{h,\Delta t}f^i, \quad (x_j,y_k,z_l)\in N^+, \end{aligned}$$
(67)

by the uniqueness argument. Finally, restriction of the \(u^{i+1}\) from \(N^+\) to the discrete grid boundary \(\gamma \) would give us

$$\begin{aligned} Tr_{\gamma }u^{i+1}=Tr_\gamma \left( P_{N^+\gamma }u^{i+1}_{\gamma }+G_{h,\Delta t}f^i\right) =P_{\gamma }u^{i+1}_{\gamma }+G_{h,\Delta t}f^i_{\gamma }. \end{aligned}$$
(68)

and \(u^{i+1}_{\gamma }=P_{\gamma }u^{i+1}_{\gamma }+G_{h,\Delta t}f^i_{\gamma }\), since \(u^{i+1}_{\gamma }=Tr_{\gamma }u^{i+1}\) by our assumption. In other words, if \(u^{i+1}_{\gamma }\) is a trace to some solution of system (11), it satisfies the BEP (17). \(\square \)

1.2 5.2 Proof to Proposition 1

Proof

The proof is similar to ideas in [13, 41]. Without boundary conditions, the difference equations (11) would admit infinite number of solutions and so will the BEP (17), due to Theorem 1. However, if we provide density values \(u^{i+1}_{\gamma _{ex}}\) on \(\gamma _{ex}\), the difference equation (11) will be well-posed and will admit a unique solution. Hence, the BEP (17) will have a unique solution if \(u^{i+1}_{\gamma _{ex}}\) is given. The solution to BEP (17) is uniquely determined by densities on \(\gamma _{ex}\), thus the solution \(u^{i+1}_{\gamma }\) to BEP (17) has dimension \(|\gamma _{ex}|\), which is the cardinality of set \(\gamma _{ex}\). As a result, the BEP (17) has rank \(|\gamma |-|\gamma _{ex}|=|\gamma _{in}|\).\(\square \)

1.3 5.3 Proof to Theorem 2

Proof

The proof is based on ideas in [13, 41]. First, define the grid function:

$$\begin{aligned} v^{i+1}:=P^{i+1}+G^{i+1}-u^{i+1}_{\gamma },\quad \text{ on } N^0, \end{aligned}$$
(69)

where \(P^{i+1}\) is a solution to the AP (12)–(13) on \(N^0\) with right hand side (16) using density \(u^{i+1}_{\gamma }\), \(G^{i+1}\) is a solution to the AP (12)–(13) on \(N^0\) with right hand side (14), and \(u^{i+1}_\gamma \) is extended from \(\gamma \) to \(N^0\) by zero. By the construction of \(v^{i+1}\), one can see that \(v^{i+1}\) is a solution to the following difference equation:

$$\begin{aligned} \begin{aligned} L_{h,\Delta t}[v^{i+1}]&=\left\{ \begin{array}{ll} f^i-L_{h,\Delta t}[u^{i+1}_{\gamma }],&{}\quad \text{ on } M^+,\\ 0, &{} \quad \text{ on } M^-. \end{array} \right. \end{aligned} \end{aligned}$$
(70)

Thus, we conclude that \(v^{i+1}\) solves the following homogeneous difference equations on \(M^-\):

$$\begin{aligned} L_{h,\Delta t}v^{i+1}=0,\quad \text{ on } M^-. \end{aligned}$$
(71)

In addition, due to the construction of \(v^{i+1},P^{i+1}\) and \(G^{i+1}\), the grid function \(v^{i+1}\) satisfies the following boundary condition:

$$\begin{aligned} v^{i+1}=0,\quad \text{ on } N^0\backslash M^0. \end{aligned}$$
(72)

Next, observe that the BEP (17) and the reduced BEP (19) can be reformulated using grid function \(v^{i+1}\) in (69) as follows:

$$\begin{aligned} v^{i+1}=0, \quad \text{ on } \gamma , \quad (\text{ BEP } (17)), \end{aligned}$$
(73)

and

$$\begin{aligned} v^{i+1}=0, \quad \text{ on } \gamma _{in},\quad (\text{ BEP } (19)). \end{aligned}$$
(74)

Hence, it is enough to show that (73) is equivalent to (74) to prove the equivalence between the BEP (17) and the reduced BEP (19). First, note that if (73) is true, then (74) is obviously satisfied.

Now, assume that (74) is true and let us show that (73) holds. Let us consider problem (71): \(L_{h,\Delta t}v^{i+1}=0\) on \(M^-\), subject to boundary conditions (72) and (74), since the set \(\gamma _{in}\cup (N^0\backslash M^0)\) is the boundary set for set \(M^-\). Then we have the following discrete boundary value problem:

$$\begin{aligned} L_{h,\Delta t}v^{i+1}&=0,\quad \text{ on } M^-, \end{aligned}$$
(75)
$$\begin{aligned} v^{i+1}&=0,\quad \text{ on } N^0\backslash M^0, \end{aligned}$$
(76)
$$\begin{aligned} v^{i+1}&=0,\quad \text{ on } \gamma _{in}, \end{aligned}$$
(77)

which admits a unique zero solution: \(v^{i+1}=0\) on \(M^-\). Since \(\gamma _{ex}{\subset } M^-\), we conclude that \(v^{i+1}=0\) on \(\gamma _{ex}\), as well as on \(\gamma \equiv \gamma _{ex}\cup \gamma _{in}\), which shows that (74) implies (73).

Thus, we showed that (73) is equivalent to (74), and therefore, BEP (17) is equivalent to the reduced BEP (19). Moreover, due to Proposition 1, the reduced BEP (19) consists of only linearly independent equations. \(\square \)

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Epshteyn, Y., Xia, Q. Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D. J Sci Comput 80, 26–59 (2019). https://doi.org/10.1007/s10915-019-00928-z

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