Implicit and Semi-implicit Numerical Schemes for the Gradient Flow of the Formation of Biological Transport Networks

Transcription

1 Iplicit and Sei-iplicit Nuerical Schees for the Gradient Flow of the Foration of Biological Transport Networks Di Fang, Shi Jin, Peter Markowich and Benoît Perthae January 14, 19 Abstract Iplicit and sei-iplicit tie discretizations are developed for the Cai-Hu odel describing the foration of biological transport networks. The odel couples a nonlinear elliptic equation for the pressure with a nonlinear reaction-diffusion equation for the network conductance vector. Nuerical challenges include the nonlinearity and the stiffness, thus an eplicit discretization puts severe constraints on the tie step. We propose an iplicit and a sei-iplicit discretizations, which decays the energy unconditionally or under a condition independent of the esh size respectively, as will be proven in 1D and verified nuerically in D. 1 Introduction Biological transport networks have drawn etensive research interests due to their ubiquitous eistence in living organiss and rich phenoenon observed in nature, two coon eaples of which are the leaf venations and blood flow. It has been hypothesized in the literature that the branching structures of the network is governed by an optiization of the energy consuption of the living syste, as a consequence of natural selection see for eaple [, 19, 17, 6, 3, 7]). However, the internal echanis ay appear rather counter-intuitive fro a biological point of view, siply due to the global nature of the optiization approach in general. One naturally questions how one blood vessel, for eaple, with only the knowledge of local inforation, is able to participate in the global energy optiization task of the entire network. The link in between was ade transparent by Hu and Cai in 13 [15]. In their work, a global energy Research supported by NSF grants no. DMS-15184, DMS-11791: RNMS KI-Net, and NSFC grant No Departent of Matheatics, University of Wisconsin-Madison, Madison, WI 5376, USA di@ath.wisc.edu). School of Matheatical Sciences, Institute of Natural Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 4, China shijin-@sjtu.edu.cn) Matheatical and Coputer Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal , Kingdo of Saudi Arabia; Faculty of Matheatics, University of Vienna, Oskar- MorgensternPlatz 1, 19 Vienna, Austria peter.arkowich@kaust.edu.sa; peter.arkowich@univie.ac.at) Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, F-755 Paris, France Benoit.Perthae@sorbonne-universite.fr). 1

2 functional approach is suggested considering both aterial and etabolis costs, but nevertheless the corresponding gradient flow ends up driven by the wall shear stress on the tube walls, which is a local inforation and has been observed in eperients that can be sensed by the tissue [3, 18, ]. This odel unifies the global picture with the local one, and is referred to later as the Cai-Hu odel. The original Cai-Hu odel is in ODE for, which unfortunately cannot describe the growth or initiation of a network. A generalized PDE version is henceforth proposed and studied in [13, 14, 1, 11, 4, 1]. To ai at the understanding of the growth and foration of these biological networks, we consider the PDE version in this work, which is in fact a lot ore challenging in ters of nuerical siulations. In the PDE odel, one has a coupled syste in ters of the conductance vector t, ) R d with space diension d ) whose direction coincides with the allowable flow direction and whose odulus indicates how strongly the tissue passes the flow at position, and the pressure pt, ) R as follows: 1.1) 1.) [r)i + ) p] = S, t = D + c p p) α γ 1), where r) denotes the background pereability, I is the identity atri, S) is the distribution of fluid sources, D > the diffusive constant usually very sall), c > the activation constant driving the network adaptation, α > is the etabolic constant, and γ [1/, 1] is the etabolic rate of the bio-organis according to Murray s Law [1]. To be specific, it has been shown that for blood vessels, γ = 1/ while γ 1/, 1] for leaf venation [15]. Notice that choosing the tie scale allows to fi a paraeter for instance α = 1 as later). Also note that the relationship between the pressure pt, ) and the conductivity vector t, ) follows Darcy s Law in porous edia written as the Poisson equation 1.1), where r)i + is the pereability tensor. The second equation 1.) is a reaction diffusion equation derived as a gradient flow by iniizing the energy cost functional consued by the syste. Now let us ake eplicit the total energy functional 1.3) E) := 1 D + α γ γ + c p[] + c r) p[] ) d, Ω where p[] is the unique solution of the Poisson equation 1.1) subject to appropriate boundary conditions, the first ter represents the diffusive energy, the second the etabolic cost consued by the network according to the celebrated Murray s Law in atheatical biology) to keep the tissue alive, and the last two ters correspond to the energy consuption incurred by the fluid itself the forer for the network while latter for the background. Again the pressure pt, ) here is given according to Kirchoff s law 1.1), where the flu follows Darcy s Law. We point out that this is clearly a non-conve optiization proble in general, where the first two ters are the conve part in the physical cases γ 1/ while the coupling with the Poisson equation gives rise to the non-conveity. We reark that the reaction ter in 1.) illustrates a copetition between the contribution by activation forces by flow inside the network) c p p) and the etabolic part α γ 1). And it is eactly this copetition that gives rise to interesting branching phenoena and nontrivial equilibria. Although odel 1.1)-1.) ay appear standard at a first glance in the for of a Poisson equation coupled with an diffusion-reaction equation, it is by no eans easy to copute in

3 practice. The ajor nuerical difficulty of this syste lies in the stiffness of basically all ters in 1.). Unlike the rather standard reaction-diffusion equation such as the Allen-Cahn equation) where the reaction ter is sooth, we point out that with the physical paraeter γ 1/, 1), the activation ter α γ 1) in 1.) is not sooth and in fact stiff since can be close to zero to be ade ore transparent in later discussions with the 1D case). This feature akes our proble ore difficult than well-studied reaction diffusion equations where the soothness of the reaction ter is guaranteed. Hence a direct application of the currently-developed tricks for reaction diffusion equations, such as linear penalization [9, 8, 1] and its resulting Integration Factor IF) ethod or Eponential Tie Differencing ETD) ethod [9, 8] reain not clear. Before proceeding any further we suarize the ain nuerical difficulties of this syste as follows 1. The diffusion ter D is stiff as in any reaction-diffusion syste. This is actually not a real difficulty at all, since the stiffness of the diffusion can be reoved by treating the ter eactly or iplicitly. Different approiations on the teporal integral involving reaction ter in this approach result in either Integration Factor IF) ethods or Eponential Tie Differencing ETD) ethods [9, 8].. The etabolis ter γ 1) is nonlinear and non-differentiable in general for physical ost relevant cases 1 γ 1, and is also stiff since can be close to zero. This ter is in fact uch harder to treat than the diffusion. 3. The activation atri in D) c p p has eigenvalues and c p, which akes the activation ter stiff when p is large. This is precisely the case inside the network. In the literature, a nuber of efforts have been devoted to construct effective nuerical schees for this syste. [14] and [1] develop nuerical schees with only the diffusion ter iplicit forulated in either a forward Euler or a Crank Nicolson fashion and other ters eplicit, and [] presents a tie-splitting strategy, where the stability of the forer reain unclear and the latter has severe tie-step constraints due to the stiffness. The goal of this paper is to develop efficient iplicit and sei-iplicit schees with good stability properties. First, to effectively take care of the stiffness, we propose a fully iplicit schee for both 1D and D Cai-Hu odels. For 1D, we prove that the energy dissipation equation is preserved which gives the unconditional stability of the iplicit solver. However, as any iplicit solver to nonlinear equations, the schee gives rise to a nonlinear algebraic proble and hence requires the Newton iterations, which akes the ipleentation for ulti-diensional cases utterly costly and coplicated. Even worse, the convergence of the Newton solver becoes unclear for the D case. To achieve a practical algorith with only linear algebraic solvers needed, sei-iplicit treatent is desired. However, as is well known, unlike the fully iplicit treatent, it is usually difficult to construct a sei-iplicit solver that still decays the energy functional physical energy) of the gradient flow. As the second and ain part, we propose a sei-iplicit schee for the Cai-Hu odel in both 1D and D that does not result in nonlinear algebraic probles. Moreover, we prove for 1D that this schee indeed decays the physical energy as long as a condition independent of the esh size is satisfied. With this new solver, one only needs to deal with linear algebraic probles that can be easily solved either by conjugate gradient CG) or preconditioned conjugate gradient PCG). This akes the coputation in D efficient and feasible. Though we only prove the decay of 3

4 the physical energy in 1D for the sei-iplicit solver, nuerically one observes that the physical energy decays in tie for both 1D and D cases. The rest of the paper is organized as follows: In Section, we briefly revisit the 1D Cai-Hu odel, and propose an iplicit algorith whose unconditional stability is proved. Motivated by the practical calculation for higher diensional cases, we henceforth propose a sei-iplicit schee, and prove that it decays the physical energy provided the stability condition is satisfied in 1D. Section 3 presents both the fully iplicit and sei-iplicit schees for D in analogy to the 1D case. In Section 4, nuerical tests are conducted in 1D with both the iplicit and sei-iplicit solvers, and D with the sei-iplicit one. We observe nuerically in both cases the energy decreases in tie. The D eaples present branching phenoena as occurred in leaf venations in nature. Section 5 contains conclusions. The 1D Cai-Hu Model and Nuerical Approiations.1 The 1D Cai-Hu Model As has been shown in [14], the 1D Cai-Hu odel has the for ).1) t D c B) = 1 + ) γ 1) with proper boundary conditions see [14] for the detailed derivation), where the paraeters are already chosen as r) = α = 1 for siplicity and B) is the given function B) := Sy)dy. The energy in the 1D setting is then siplified as.) E) = 1 D + 1 γ γ + c B) 1 + d. Ω As has been pointed out in the introduction, the nonlinear reaction ter is also stiff. To see it, siply defines ) c B).3) H) = 1 + ) γ 1), where 1 γ 1), since 1 γ 1, and.4) H c B) 1 3 γ 1, if, 1+ ) = ) 3 1 γ), if =. Note that H ) is very large as approaches zero.. An Iplicit Treatent and Energy Decay Consider the 1d setting in [14], naely, the interval, 1) and hoogenous Neuann Boundary Conditions for. We propose the following schee:.5) j n j t = D [ j+1 j + j 1 + n j+1 n j + n ] j 1 4

5 .6) c Bj + j + n j ) 1 + j ) )1 + n j ) ) 1 = 1, N 1 = N+1. j ) γ n j )γ γ j n j ). Note that this schee is second order in tie, and the diffusion ter is treated by the Crank- Nicolson ethod. We define the discrete energy as [ ].7) E n = 1 D n j n ) 1 j 1 + γ n j γ + c Bj 1 + n, j ) which is a discrete analog of the energy functional.). We shall prove in the following theore that our schee decays the discrete energy and is hence unconditionally stable. Theore.1. The schee.6) satisfies the discrete energy dissipation equation.8) E E n = 1 t and hence decays the energy.9) E E n. j n j ), Reark.. This result iplies that the schee is unconditionally stable, i.e. there is no restriction on t. It shows the schee preserves in the discrete fashion the energy equation given in Lea 1 of [13].1) d dt E = Ω ) d. t In fact, one should siply view the schee as a discretization of the energy equation. Proof. The lea follows a straightforward energy estiate, naely, ultiply j n j on both sides of the schee.6) and su over j, where I) = II) = j n j ) = I) + II) + III), t D j+1 j + j 1 + n j+1 n j + n ) j 1 j n j ), c B j j ) n j ) 1 + It follows fro suation by parts that j ) )1 + n j ) ), III) = 1 [ j ) γ n j ) γ]. γ.11) j+1 j + ) j 1 j = j j j+1 ) j j j j ) j 1 j 5

6 .1) = j j j 1 ). Siilarly applying suation by parts to other ters, one finds I) = D + D j+1 j + ) j 1 j D = D A siple calculation shows Hence, which copletes the proof. j n j+1 n j + n j 1) j D j ) D j 1 + j II) = = 1 c B j n j n ) j 1. j ) n j ) 1 + j ) )1 + n j ) ) c B j 1 + j ) + 1 I) + II) + III) = E + E n, n j+1 n j + n ) j 1 n j j+1 j + ) j 1 n j c B j 1 + n j ). In nuerical ipleentations, we use the Newton ethod to solve the iplicit algebraic equation.6) in every tie step. Note that supposing D =, then the algebraic equation has only one root, and sufficiently away fro the function in consideration is actually onotone. Nuerically, the Newton iteration converges just in a few steps taking the initial guess sufficiently away fro..3 A Sei-iplicit Treatent and Energy Decay To avoid the nonlinear Newton solver, which ay becoe ipractical for higher diension, we propose the following sei-iplicit treatent in preparation for the D practical siulations..13) j n j t D [ j+1 j + ] c B) j 1 = 1 + n j ) ) n j n j γ 1) j, 1 = 1, N 1 = N+1. Note that in.13), the activation part on the right hand side is treated fully eplicitly, while the etabolic ter is treated in a sei-iplicit fashion. Since n j γ 1) is strictly positive, this ter plays the role of stabilizing the schee. Moreover, one only needs to invert a tridiagonal atri where standard fast algoriths can be used. In the following theore, we establish the conditional stability of the sei-iplicit solver in the sei-discrete set-up which guarantees the physical energy does not increase. Note that the sei-discretization is used for a clear presentation, the fully-discrete stability should be siilar, which is oitted here. 6

7 Theore.3. Assue B L, consider the sei-iplicit schee.13) in the sei-discrete set-up, i.e.,.14) n D = c B) t 1 + n ) ) n n γ 1), then the physcial energy as defined in.) decays provided the condition E ) E n ),.15) c B t, is satisfied. Proof. Denote f) = c B) 1 + ), F ) = c B) 1 + ), then clearly one has F ) = f) and.16) f ) = c B) ) 3 c B. In the following, we use, to denote the inner product and for the L nor in Ω. On one hand, ultiplying the schee.14) by n and integrating in, one obtains.17) 1 t n + D D n + D n = f n ), n n γ ) n) d, Ω where integration by parts and the equality a b, a = 1 a 1 b + 1 a b are used. On the other hand, it holds that.18) E ) E n ) = D D n + Ω by the definition of the physical energy.). Notice that 1 γ γ 1 γ n γ d + F ) F n )d, F ) F n ) = f n ) n ) f τ) τ)dτ, n and hence cobining with.17), we have Ω E ) E n ) + 1 t n + D n = n γ ) n) 1 Ω γ γ + 1 γ n γ d Ω n f τ) τ)dτd := I 1 + I 7

8 By.16), clearly I 1 c B n. Net, we shall show that I 1. Case 1: If n, it suffices to show that the function.19) G, y) = γ y γ 1 y 1 γ yγ + 1 γ γ for all, y. Note that I 1 corresponds to the case when = n and y =.) Though one could prove the result by proving the iniu of G, y) is no less than, an easier way is to use the hoogeneity of the inequality and convert the proble into a single variable function. To be specific, note that.) G, y) = γ y ) y 1 γ y )γ + 1 ). γ Denote z = y/, it suffices to show that Az) = z z 1 γ zγ + 1 γ for all z. It clearly holds for z = and for γ = 1/ respectively, hence we only need to consider z > and γ 1/, 1). Straightforward coputation gives A z) = z 1 z γ 1, A z) = γ 1 z γ, where A z) is onotone increasing with a unique zero at z = γ 1 )γ := z, which eans A z) decreases in, z ) and increases in z, ). On one hand, one has A ) = 1 <, which iplies that A z) has at ost one zero. On the other hand, z = 1 is clearly a critical point, and therefore is the unique critical point of Az). One could easily check the second derivatives and the liits on the boundary to see that the critical point z = 1 is the global iniu. It follows that Az) A1) =, G, y). Case : When n <, it suffices to show that for all, y, which clearly holds since G, y) = γ y + γ 1 y 1 γ yγ + 1 γ γ G, y) = G, y) + γ 1 y. In suary, we have E ) E n ) + D n c B 1 t ) n, where the stability condition.15) is used. This copletes the proof. 8

9 3 The D Cai-Hu Model and Nuerical Schees The ideas developed in the 1D case can be etended to treat the spatially discretized D Cai-Hu odel when set on a rectangular grid. For copleteness, we present both the fully iplicit and iplicit-eplicit schees. 3.1 An Iplicit Approach and Energy Decay Siilar as the 1D case, an iplicit schee can be constructed as [ r h h p + h p n) ) h p + n h p n) + n)] = S, 4 3.) l n l t = D h l + n ) c l + 4 h p + n h p n) h p l + h p n ) l α γ 1) l ) n γ 1) n l ) ) γ, l n l where the subscript l = 1, labels the l-th coponent of the vector or h p; the spatial discretizations are all perfored at the grid point with inde-pair k, j) and hence spatial indices are oitted for siplicity; and h denotes the discretized Laplacian operator, and hence the diffusion ter is treated in a Crank-Nicolson atter siilarly as 1D case). This schee is clearly second order in tie. However, this schee gives rise to an nonlinear algebra proble, which in practice is very difficult to solve. Note that a direct application of the Newton iteration ay fail to converge depending on initial guesses. Hence in the net section, we propose a ore practical nuerical ethod. 3. A Sei-iplicit Schee The key goal here is to avoid a nonlinear algebraic proble which requires the Newton iterations, in which case hopefully one only deals with a linear algebra proble that provides enough efficiency in practical calculation. Following the sei-iplicit idea for the 1D Cai-Hu odel, we propose the following schee for ulti-diensional cases: 3.3) D 1 [ri + n n )D 1 p ] = S, 3.4) 1 n 1 t n t = D D 1 + c D p ) n 1 + D p D y p n ) α n γ 1) 1, = D D + c D y p ) n + D p D y p n ) 1 α n γ 1), where D and D y denote the spatial discretizations in and y directions, respectively, which could be chosen as either the forward difference or the central difference, D 1 = D, D y ), and D = D + D y. Note that siilarly to the 1D schee.13), the diffusion ter is treated iplicitly, the etabolis ter sei-iplicitly and the activation ter eplicitly in the sense that 9

10 the pressure p used here is coputed fro the Poisson equation 3.3) using the inforation of n. This sei-iplict schee is a D version of schee.13) that has been proven to decay the physical energy in 1D, and is efficient in actual coputations, since the sei-iplicit treatent of the last ter in 3.4) helps to stabilize the whole procedure and create positive definite atrices to be inverted. For nuerical tests, we use the staggered grids, as is widely used in coputational fluid dynaics to avoid the faous checkerboard proble. 4 Nuerical Results We now illustrate the theoretical results with nuerical siulations. We copare the seiiplicit schee and the fully iplicit which is ore cople due to Newton iterations but avoids the restriction on tie steps. We pay a special attention to the decay of energy as it holds for the continuous odel, and on the ability of the schee to generate patterns. 4.1 The 1D Cai-Hu Model In this section, we present our nuerical results of the 1D Cai-Hu odel.1) using the iplicit solver.6) and the sei-iplicit schee.13), respectively. Denote the nonlinear syste of equations resulting fro the iplicit schee as 4.1) F ) := j n j t where F : R N R N. The Newton ethod reads D [ j+1 j + j 1 + n j+1 n j + n ] j 1 c B j j + n j ) 1 + j ) )1 + n j ) ) + J F k) ) k + 1) k) ) = F k) ), j ) γ n j )γ γ j n j ) =, with J F ) as the N N Jacobian atri of F and k as the k-th step of iteration. In order to solve this linear syste, the Thoas algorith [5] is ipleented since the Jacobian is a tri-diagonal atri. Eaple 4.1 1D Etinction of Solutions). It has been shown in Lea 4 of [14] that for 1/ γ 1, the initial data L, 1) and the paraeters satisfying c B) L Z γ := γ + 1 ) γ 1 1 γ, 1 + γ one has the etinction of solutions, naely, the solution converges to zero as t. In other words, the solution converges to zero in infinite tie for 1/ γ 1 in contrast to the case 1 γ < 1/ where there eists a finite break down tie). In this eaple, we try to observe the vanishing effect nuerically. Consider γ = corresponding to the leaf venation case, D =.1, cb) = Z γ /, and initial datu as 4.), ) = 1sinπ) + 1) cosπ) 3, which is plotted in Fig. 1a. We choose this initial datu so that it is a bit ore coplicated than siple algebraic functions, and here one has different cobinations of onotonicity and conveity 1

11 in the sub-intervals. Both the iplict and sei-iplict solvers are ipleented with t =.1 and =.5 until the final tie T = 1. We first show the results of the iplicit solver: Fig 1 plots the solutions at various tie t =,, 5, 1 and the discrete physical energy defined in.7). It can be seen that the discrete physical energy decays throughout the siulation in tie as is proved in Theore.1. Net, the solutions coputed by the sei-iplicit schee at different tie and its discrete physical energy are plotted in Fig.. In the nuerics one indeed can see the decay of the discrete physical energy as is proved in Theore.3. In both ethods, the solution converges finally to zero and we nuerically recover the theoretical results proved in Lea 4 of [14]. 1 initial data of in 1D 8 Etinction of in 1D, T = a) t = b) t =. Etinction of in 1D, T = Etinction of in 1D, T = c) t = d) t = 1 15 Discrete Physical Energy decays E n at each tie t n) t e) Discrete Physical Energy Figure 1: Eaple 4.1 coputed by the iplicit solver: the solution at different ties t =,, 5, 1. It can be seen that the solution converges to eventually, and the discrete physical energy decreases in tie. Eaple 4. 1D Non-trivial Equilibriu). In this eaple, we want to consider the solution with non-trivial non-vanishing) equilibriu. First, consider the 1D Cai-Hu odel.1) with 11

12 1 initial data of in 1D 8 Etinction of in 1D, T = a) t = b) t =. Etinction of in 1D, T = Etinction of in 1D, T = c) t = d) t = 1 15 Discrete Physical Energy decays E n at each tie t n ) t e) Discrete Physical Energy Figure : Eaple 4.1 coputed by the sei-iplicit solver: the solution at different ties t =,, 5, 1. It can be seen that the solution converges to eventually, and the discrete physical energy decreases. paraeters D =.1, cb) = 1, γ =, which corresponds to the leaf venation case with the initial datu as 4.), and copute the 1D odel until the final tie T =, using both the iplicit and sei-iplicit ethods with t =.1 and =.5. The nuerical results are plotted in Fig. 3. For both schees, we observe the decay of the discrete physical energy. Net, we change the diffusion paraeter D =.1, and repeat above tests. It can be seen in Fig. 4, the decay of discrete energy can still be observed. 1

13 Solution in 1D, T = Solution in 1D, T = a) Iplicit schee: at t = b) Sei-iplicit schee: at t =. Discrete Physical Energy decays. Discrete Physical Energy decays E n at each tie t n) E n at each tie t n) t c) Iplicit schee: discrete energy t d) Sei-iplicit schee: discrete energy Figure 3: Eaple 4. with D =.1. Left panel: coputed by the iplicit solver; right panel: coputed by sei-iplicit solver. The equilibriu does not vanish and the discrete physical energy is observed to decay in tie. 4. The D Cai-Hu Model In this section, we present a nuber of nuerical eaples for the D Cai-Hu odel, using the sei-iplicit solver. The linear algebraic probles resulting fro the schee are solved using a preconditioned conjugate gradient ethod. The discrete physical energy considered here is defined as E n ) at each tie t = t n, with E defined in 1.3). Eaple 4.3. Consider the equation with the Neuann boundary condition for p and Dirichlet boundary condition for. The background pereability r) is chosen as r) =.1, the activation constant c = 5, the diffusivity D =.1, and α = 1. We choose γ =, which corresponds to leaf venations according to [16]. The initial data and source ter are given in Fig. 5. In this eaple we use t =.5, and = y =.. We plot the long-tie behavior of the syste for various tie T =, 4, 6, 8. To be specific, Fig. 6 shows in the log-scale, which corresponds to the structure of the leaf. Note that for =, the log function is not defined, and hence hereafter log + ɛ ) is plotted in nuerics, where ɛ = 4e 16 is of size of achine precision. In this eaple, we also plot the voluetric flu u proportional to the flow velocity in leaf venations), defined as 4.3) u = ri + ) p in Fig. 7. It can be seen that the leaf venation grows as the fluid spreads, and finally achieve a tree-like structure. Nuerically, we see that the discrete physical energy decays with respect to 13

14 Solution in 1D, T = Solution in 1D, T = a) Iplicit schee: at t = b) Sei-iplicit schee: at t =. Discrete Physical Energy decays. Discrete Physical Energy decays E n at each tie t n) E n at each tie t n) t c) Iplicit schee: discrete energy t d) Sei-iplicit schee: discrete energy E n at each tie t n ) 1e E n at each tie t n ) 1e t e) Iplicit schee: zoo in of discrete energyf) Sei-iplicit schee: zoo in of discrete after t = 1 energy after t = 1 t Figure 4: Eaple 4. with D =.1. Left panel: coputed by the iplicit solver; right panel: coputed by sei-iplicit solver. The equilibriu does not vanish and the discrete physical energy is observed to decay in tie for both solvers. tie through out the siulation as is shown 8. In fact, in the last several steps, the energy keeps the decay behavior even if the first five decial places in the energy value no longer change. To be specific, the last seven energy values are , , , , , , Eaple 4.4 Varying D). In this eaple, we investigate how the diffusion coefficients ay affect the foration of the biological network. Consider the sae set-up as in Eaple 4.3, but with different diffusion coefficients D =.5,.1,.,.5,.. The solutions are coputed till tie T = 4. It can be seen fro Fig. 9 that the saller the diffusion coefficient is, the ore detailed structure the network tends to ehibit and also the faster it reaches the equilibriu. 14

15 a) 1,, y) b),, y) c) S, y) Figure 5: Plot of initial data in the,y) plane and the source ter. r=.1, c=5., D =.1, T = 8, dt =.5 r=.1, c=5., D =.1, T = 4, dt = a) t = b) t = 4 r=.1, c=5., D =.1, T = 6, dt =.5 r=.1, c=5., D =.1, T = 8, dt = c) t = 6 d) t = 8 Figure 6: Eaple 4.3 coputed by the sei-iplicit solver: the leaf structure in the log scale at different ties t =, 4, 6, 8. It can be seen that the flow spreads and finally achieves a tree-like structure. 15

16 r=.1, c=5., D =.1, T = 8, dt = r=.1, c=5., D =.1, T = 4, dt = a) t = 5.6 b) t = 4 r=.1, c=5., D =.1, T = 6, dt = r=.1, c=5., D =.1, T = 8, dt = c) t = d) t = 8 1 Discrete physical energy Discrete physical energy Figure 7: Eaple 4.3 coputed by the sei-iplicit solver: the flu u defined in4.3) at different ties t =, 4, 6, 8. It can be seen that the leaf grows and finally achieves a tree-like structure tie t a) Discrete physical energy for t tie t 75 8 b) Zoo in to the tail for 6 t 8 Figure 8: Eaple 4.3 coputed by the sei-iplicit solver: The discrete physical energy decays all the tie, and it can be seen in the zoo-in figure on the right) that it keeps the decay behavior even if the first two decial places no longer change. 16

17 a) leaf structure in log-scale b) flu u in log-scale d) leaf structure in log-scale g) leaf structure in log-scale k) flu u in log-scale ) leaf structure in log-scale n) flu u in log-scale tie t tie t.8 1. f) Discrete Physical energy versus tie tie t..5 i) Discrete Physical energy versus tie tie t 3 4 l) Discrete Physical energy versus tie Discrete Physical Energy j) leaf structure in log-scale h) flu u in log-scale. 35 c) Discrete Physical energy versus tie e) flu u in log-scale. 4 Discrete Physical Energy Discrete Physical Energy.5. Discrete Physical Energy 8 Discrete Physical Energy tie t o) Discrete Physical energy versus tie Figure 9: Eaple 4.4 coputed by the sei-iplicit solver for various D till T = 4. Fro first row to the last row, the values of D are.5,.1,.,.5 and., respectively. It can be seen that the saller the diffusion coefficient is, the ore detailed the network structure becoes.

18 Eaple 4.5 Varying spatial esh size). In this eaple, we test the sei-iplicit schee with different spatial esh size. In this eaple, we first take asyetric initial datu as in Fig. 5, and denote the non-zero region are Ω. We take with a sall perturbation.1uniforω ) on the initial datu in the region Ω. We take D =.1 and the nuber of grid points in each spatial direction as 5, 8 and 1. It can been see in Fig 1, the differences resulting fro the varying of the spatial esh size are of sall agnitude as can be seen in both the log scale and the regular scale. d =., T = a) leaf structure in log-scale b) d =.5, T = c) leaf structure in log-scale d) d =.4, T = e) leaf structure in log-scale f) Figure 1: Eaple 4.5 asyetric initial datu) coputed by the sei-iplicit solver for D =.1 till T = 4 for various grid points in each spatial direction. The first row: 1 by 1; the second row 8 by 8; the third row: 5 by 5. Though the pattern see a bit different in the log scale, it can be seen fro the plots that the differences between the three results are in fact inor in the noral scale. Now, we take syetric initial data and syetric source S. For γ =, D =.1, the solutions are coputed until T = using the nuber of grid points in each spatial direction as 5, 8 and 1, respectively. It can been see in Fig 1, the pattern sees syetric, and varying different esh sizes does not see to break the syetry. It can be seen in the iddle 18

19 colun of Fig. 1 that the differences resulting fro the change of the grid size are inor, which are ore clearly illustrated in the log-scale plots first and last coluns) that the difference is of order 1e 4 arked as dark yellow in the color bar a) leaf structure in log-scale. b).. c) flu u in log-scale d) leaf structure in log-scale e).. g) leaf structure in log-scale h) f) flu u in log-scale i) flu u in log-scale Figure 11: Eaple 4.5 syetric initial datu) coputed by the sei-iplicit solver for D =.1 till T = for various grid points in each spatial direction. The first row: 1 by 1; the second row 8 by 8; the third row: 5 by 5. The syetry of the patterns is not broken by varying of the esh size. Eaple 4.6 Asyetric Grids). An interesting phenoenon been pointed out in [14] that for asyetric grids, the solution could for an asyetric pattern even with syetric initial data and source. In this last eaple, we change to an asyetric grids, naely, d 6= dy. We consider the syetric initial data as in Eaple 4.5 with γ = and D =.1, but use 5 grid points in direction while 8 grid points in y direction. As can be see in Fig. 1, the pattern becoes asyetric in the log-scale plot, but in fact in the regular scale the solution 19

20 does not see to differ uch fro the results of syetric grids as plotted in Fig. 11 iddle colun. We point out that just as the checkerboard proble in coputational fluid dynaics, to avoid soe potential instability caused by the natural grids, the choice of grids should certainly be carefully designed. Here we used the staggered grids, and a better gridding strategy for both finite difference and finite eleent schees of this odel is a very interesting proble, which is left for future study a) in log-scale b) Figure 1: Eaple 4.6 syetric initial data with asyetric grids d dy. The pattern becoes asyetric as well. But as can be seen in the regular scale on the right), the solution does not differ uch fro the results of syetric grids see Fig. 11 iddle colun). 5 Conclusion In this work, we propose two possible approaches to the Cai-Hu odel for biological transport networks the fully iplicit treatent and a sei-iplicit approach. While the fully iplicit one is proven to be unconditionally stable in 1D, it results in a nonlinear algebraic syste that could be difficult and costly to ipleent in higher diension, and convergence of the Newton solver reains unclear. The sei-iplicit schee avoids the Newton iteration, but has a stability constraint on tie step which depends on soe coefficients of the equation still independent of the esh size though). For the D odel, the sei-iplicit schee is of ore practical potential, since it only deals with linear algebraic probles by inverting positive definite atrices that can be solved efficiently by CG or PCG even in higher diensions. Our nuerical eaples show the stabilities and energy decay of both schees. References [1] G. Albi, M. Artina, M. Foransier, and P. A. Markowich. Biological transportation networks: Modeling and siulation. Analysis and Applications, 141):185 6, 16. [] G. Albi, M. Burger, J. Haskovec, P. Markowich, and M. Schlottbo. Continuu Modeling of Biological Network Foration, pages Springer International Publishing, Cha, 17.

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