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The Keller-Segel equations have been widely used to model chemotaxis. These equations and in particular, the properties of their solutions have been intensively investigated, see for example [9,26,32,36]. For adapted and expanded versions of the Keller-Segel model we refer to [7,13,14,29,32,33,12]. Moreover, in recent literature improved flux-limited Keller-Segel models, taking into account the finite speed of propagation, have been developed in [1]. For surveys and extended reference lists, see, for example [2,3].
Our starting point is the classical kinetic chemotaxis equation [11]. Scaling it with the so called diffusive scaling leads to the Keller-Segel equation, see [11]. In general, the derivation of Keller-Segel type models, including flux-limited diffusion models and Fokker-Planck type models, from underlying kinetic or microscopic models is discussed for example in [2,12,11]. In particular, using moment closure approaches one may obtain macroscopic equations intermediate between kinetic and Keller-Segel equations, see the above mentioned references or [19,28,6]. Linear moment closure models called the P1-model (in the radiative transfer literature) or the Cattaneo equations are in many situations resonable approximations of the kinetic equation [6]. However, a major drawback of these approximations is that they do not guarantee the positivity of the cell density. Thus, in this paper we consider a non-linear closure method to deal with this problem. Additionally, we also introduce half-moment non-linear closures for the kinetic equations and obtain associated hydrodynamic macroscopic equations similar to [17].
To model chemotaxis on a network, the crucial point is to define suitable coupling conditions. In previous work [5,6] coupling conditions for the scalar Keller-Segel equation and kinetic equation have been discussed. An analytical investigation can be found in [10]. These investigations deal with the case of a network problem for a parabolic system of equations. Conservation of mass and continuity of density and chemo-attractant at the nodes are used as coupling conditions. Moreover, in [8,25] the hyperbolic-parabolic Cattaneo equations on a graph are studied. For general work on kinetic equations on networks, for example for traffic flow equations, we refer to [18] or [27].
In the present work we will derive the coupling conditions for nonlinear half-moment models from the coupling condition for the kinetic model. These coupling conditions guarantee on the one hand the conservation of mass through nodes, and on the other hand, they satisfy a positivity condition for the density. In the present case, as in [8,25], the hyperbolic equations require another approach to the coupling conditions as in the Keller-Segel context. However, in the diffusive limit, when the scaling parameter goes to
Numerical methods for these nonlinear models have to deal with issues like the numerical moment realizability and (as in the linear case) with a large range of the scaling parameter describing different regimes. Thus, it is desirable to use an asymptotic preserving well-balanced (APWB) scheme. These numerical methods work uniformly with respect to the scaling parameter and one obtains in particular a scheme suitable for computations near the diffusive limit. We refer to [22] for the basic idea of such schemes. Combining this with a numerical scheme to solve the coupling problem we obtain results for the full network problem.
The paper is organized as follows. In section 2 we discuss the kinetic model and its nonlinear hydrodynamic full-and half-moment approximations which are derived using non-linear closure function. Section 3 considers the coupling conditions for the kinetic equations and derived conditions for the macroscopic models. Section 4 and 5 contain details about the numerical schemes on an interval and the numerical treatment of the coupling conditions. Finally, the numerical results for tripod and more general networks are shown in Section 6.
Denote with
{∂tf+v∂xf=−λ(f−ρ2)+12αv¯∂xmρ∂tm−D(∂xxm)=γρρ−γmm . | (1) |
Here, we have defined
¯∂xm=∂xm√1+|∂xm|2. |
We assume
λ≥α | (2) |
in order to guarantee positivity of the turning kernel
λ+αv¯∂xm≥0 |
for all
{∂tf+1ϵv∂xf=−λϵ2(f−ρ2)+12ϵαv¯∂xmρ∂tm−D(∂xx)m=γρρ−γmm. | (3) |
Condition (2) turns into
ϵ≤λα. | (4) |
This equation will be used as the starting point for the macroscopic models considered in the next section.
The kinetic equation (3) can be transformed using the even-and odd-parities [30]
r(x,t,v)=12(f(x,t,v)+f(x,t,−v)),j(x,t,v)=12ϵ(f(x,t,v)−f(x,t,−v)) | (5) |
for positive velocities
∂tr+v∂xj=−λϵ2(r−ρ2) | (6) |
∂tj+1ϵ2v∂xr=−1ϵ2(λj−12αv¯∂xmρ) . | (7) |
When
The procedure is shortly revisited. Start with the scaled version of the kinetic equation (6). Integrating the equation for
∂tρ+2∂x∫10vjdv=0. | (8) |
Moreover we observe
{∂tρ−∂x(13λ∂xρ−α3λ¯∂xmρ)=0∂tm−D∂xxm=γρρ−γmm. | (9) |
In addition, from the scaled version for kinetic chemotaxis equation (3), using moment closure approaches one may obtain macroscopic, hyperbolic equations intermediate between kinetic and Keller-Segel equations, see [19,29]. There are two kinds of moment models for chemotaxis, full-moment models and half-moment models, that are corresponding to full-and half-moment closure methods respectively.
Consider the following averaged quantities
ρ(x,t)=∫1−1f(x,t,v)dv ,q(x,t)=∫1−1vf(x,t,v)dv , |
where
ρ≥0, P≥0, q2ρ2≤Pρ≤1. | (10) |
This leads to the equations
{∂tρ+1ϵ∂xq=0∂tq+1ϵ∂xP=−1ϵ2(λq−ϵα3¯∂xmρ)∂tm−D∂xxm=γρρ−γmm . | (11) |
In order to close the model (11), a moment closure is specified by using a suitable approximation of
Therefore, one may assume that the cell density
f(x,t,v)=aexp(vb) | (12) |
with
P=∫Vv2fdv=ρh(u),h(u)=(1−2bu),u:=qρ=coth(b)−1b , |
where
limu→0h(u)=13, u2≤h(u)≤1 |
guaranteeing the realizability condition. The function
{∂tρ+1ϵ∂xq=0∂tq+1ϵ∂x(ρh(qρ))=−1ϵ(λ1ϵq−α3¯∂xmρ)∂tm−D∂xxm=γρρ−γmm . | (13) |
Scaling and denoting
{∂tρ+∂xqϵ=0∂tqϵ+1ϵ2∂x(ρh(ϵqϵρ))=−1ϵ2(λqϵ−α3¯∂xmρ)∂tm−D∂xxm=γρρ−γmm . | (14) |
Under the realizability condition, the model (14) is hyperbolic and converges to the Keller-Segel equations when
We will construct a macroscopic, half-moment model for (3) by applying a half-moment nonlinear closure, which was introduced in [17], see [6] for a linear version. Consider the following quantities
ρ−=∫0−1f(v)dv ,ρ+=∫10f(v)dv ,q−=∫0−1vf(v)dv ,q+=∫10vf(v)dv ,P−=∫0−1v2f(v)dv ,P+=∫10v2f(v)dv . | (15) |
Using
(q±ρ±)2≤P±ρ±≤±q±ρ±≤1 . | (16) |
These conditions again guarantee the existence of a non-negative density for these equations [16,37].
Integrating the kinetic equation (3) leads to the half-moment model
{ϵ∂tρ±+∂xq±=−1ϵλ(ρ±−ρ++ρ−2)±14α¯∂xm(ρ++ρ−)ϵ∂tq±+∂xP±=−1ϵλ(q±∓ρ++ρ−4)+16α¯∂xm(ρ++ρ−)∂tm−D∂xxm=γρρ−γmm . | (17) |
The nonlinear closure for the half-moment models is derived similarly as for the full-moment case. One uses the ansatz
f(v<0)=a−exp(vb−) ,f(v>0)=a+exp(vb+) . | (18) |
Plugging (18) in (15) leads to
P−=ρ−h−(u−) ,P+=ρ+h+(u+) ,h−(u−)=u−(−1−2b−)−1b− ,h+(u+)=u+(1−2b+)+1b+ ,u−:=q−ρ−=1exp(b−)−1−1(b−) ,u+:=q+ρ+=exp(b+)exp(b+)−1−1(b+) , |
such that
limu±→±12h±(u±)=13 ,(u±)2≤h±(u±)≤±u± . | (19) |
Condition (19) guarantees the realizability condition (16). We note that an explicit maximum-entropy Eddington factor approximating the above closure is obtained by using the Kershaw closure [34]
h−(u−)=23(u−)2−13u− ,h+(u+)=23(u+)2+13u+ . | (20) |
Finally, the half-moment model for chemotaxis reads
{∂tρ±+1ϵ∂xq±=−1ϵ2[λ(ρ±−ρ++ρ−2)∓ϵα4¯∂xm(ρ++ρ−)]∂tq±+1ϵ∂x(ρ±h±(q±ρ±))=−1ϵ2[λ(q±∓ρ++ρ−4)−ϵα6¯∂xm(ρ++ρ−)]∂tm−D∂xxm=γρρ−γmm . | (21) |
Moreover, by introducing new quantities
ρ=ρ++ρ− ,ˆρ=ρ+−ρ−ϵ ,qϵ=q++q−ϵ ,ˆq=q+−q− , | (22) |
the half-moment model can be rewritten as
{∂tρ+∂xqϵ=0∂tqϵ+1ϵ2∂xP(ρ,qϵ,ˆρ,ˆq)=−1ϵ2(λqϵ−α3¯∂xmρ)∂tˆρ+1ϵ2∂xˆq=−1ϵ2(λˆρ−12α¯∂xmρ)∂tˆq+∂xˆP(ρ,qϵ,ˆρ,ˆq)=−1ϵ2λ(ˆq−ρ2)∂tm−D∂xxm=γρρ−γmm | (23) |
with
P(ρ,qϵ,ˆρ,ˆq):=P+(q+ρ+)+P−(q−ρ−)=ρ+ϵˆρ2h+(ϵqϵ+ˆqρ+ϵˆρ)+ρ−ϵˆρ2h−(ϵqϵ−ˆqρ−ϵˆρ),ˆP(ρ,qϵ,ˆρ,ˆq):=1ϵP+(q+ρ+)−1ϵP−(q−ρ−)=ρ+ϵˆρ2ϵh+(ϵqϵ+ˆqρ+ϵˆρ)−ρ−ϵˆρ2ϵh−(ϵqϵ−ˆqρ−ϵˆρ). | (24) |
The half-moment model (23) is hyperbolic and has again the Keller-Segel equations as macroscopic diffusive limit when
We concentrate in the following on the discretization of the equations for the movements of the cell density. The equation for the chemoattractant
First, let us recall the framework of the APWB scheme in [24,20,21] for the Cattaneo model
{∂tρ+∂xqϵ=0∂tqϵ+aϵ2∂xρ=−1ϵ2(λqϵ−c(∂xm)ρ) | (25) |
with
∂tρ+∂x(c(∂xm)λρ−aλ∂xρ)=0. |
By introducing the diagonal variables
∂tU+√aϵ∂xU=(c(∂xm)2√aϵ−λ2ϵ2)U+(c(∂xm)2√aϵ+λ2ϵ2)V∂tV−√aϵ∂xV=−(c(∂xm)2√aϵ−λ2ϵ2)U−(c(∂xm)2√aϵ+λ2ϵ2)V. | (26) |
Using
{U∗(Δx)=V∗(Δx)+A+B1+B(U∗(0)−V∗(Δx))+A+B−21+BV∗(Δx)V∗(0)=U∗(0)+A+B1+B(V∗(Δx)−U∗(0))−A+B−21+BV∗(Δx)A=(1−λΔxϵ√a)exp(Δxac(∂xm))B=(1+λΔxϵ√a)exp(Δxac(∂xm)) | (27) |
the well-balanced Godunov scheme for (26) is given as
{Un+1i=Uni−l(Un+1i−U∗i−12)=Uni−l(Un+1i−Vn+1i)−lAni−12+Bni−121+Bni−12(Vni−Uni−1)+lAni−12+Bni−12−21+Bni−12VniVn+1i=Vni+l(V∗i+12−Vn+1i)=Vni+l(Un+1i−Vn+1i)+lAni+12+Bni+121+Bni+12(Vni+1−Uni)−lAni+12+Bni+12−21+Bni+12Vni+1 | (28) |
with
{Un+1i=2l22l+111+Bni+12Vni+1+(l+12l+1−l22l+1Ani+12+Bni+121+Bni+12)Uni+(l2l+1−2l(l+1)2l+111+Bni−12)Vni+l(l+1)2l+1Ani−12+Bni−121+Bni−12Uni−1Vn+1i=l(l+1)2l+121+Bni+12Vni+1+(l2l+1−l(l+1)2l+1Ani+12+Bni+121+Bni+12)Uni+(l+12l+1−2l22l+111+Bni−12)Vni+l22l+1Ani−12+Bni−121+Bni−12Uni−1. | (29) |
It is easy to check that for small enough values of
Δt≤λΔx24(a+Δx). |
Furthermore, as mentioned in [21], (28) converges to a centered discretization of the Keller-Segel equation, when
Based on the above considerations and the schemes developed in the above cited references we consider the following relaxation system for (14)
{∂tρ+∂xz=0∂tz+aϵ2∂xρ+λϵ2z−α3ϵ2¯∂xmρ=1β(qϵ−z)∂tqϵ+1ϵ2∂xw+λϵ2q−α3ϵ2¯∂xm(1h(ϵqϵρ)w)⏟ρ=0∂tw+a∂xq=1β(ρh(ϵqϵρ)−w) . | (30) |
For
System (30) is split into two into separate sub systems. The right hand side treated with a simple implicit numerical scheme and we obtain for
ρ(1)=ρn,q(1)ϵ=z(1)=qnϵ,w(1)=ρnh(ϵqnϵρn) , | (31) |
which is used as the initial condition for the system given by the left hand side. By introducing the new variables
U=12ρ+ϵ2√az ,V=12ρ−ϵ2√az ,ˉU=ϵ2qϵ+12√aw ,ˉV=−ϵ2qϵ+12√aw , | (32) |
the left hand side of system (30) can be rewritten as
{∂tU+√aϵ∂xU=(c12√aϵ−λ2ϵ2)U+(c12√aϵ+λ2ϵ2)V∂tV−√aϵ∂xV=−(c12√aϵ−λ2ϵ2)U−(c12√aϵ+λ2ϵ2)Vc1(∂xm)=α3¯∂xm{∂tˉU+√aϵ∂xˉU=(c22√aϵ−λ2ϵ2)ˉU+(c22√aϵ+λ2ϵ2)ˉV∂tˉV−√aϵ∂xˉV=−(c22√aϵ−λ2ϵ2)ˉU−(c22√aϵ+λ2ϵ2)ˉVc2(ρ,q,∂xm)=ah(ϵqϵρ)c1 . | (33) |
Now, we can apply the APWB schemes presented in 3.1 for each subsystem in (33). Remark that for
Remark 1. We note that under similar conditions as in the linear case, it can be shown, that the scheme preserves positivity in
Following the same procedure as for the full-moment model in the previous section, we consider the following relaxation system for the half-moment model (23)
{∂tρ+∂xz=0∂tz+aϵ2∂xρ+λϵ2z−α3ϵ2¯∂xmρ=1β(qϵ−z)∂tqϵ+1ϵ2∂xw+λϵ2qϵ−α3ϵ2¯∂xm(2w−ϵ(h+−h−)ˆρh++h−)⏟ρ=0∂tw+a∂xqϵ=1β(P−w) {∂tˆρ+1ϵ2∂xˆz+λϵ2ˆρ−α2ϵ2¯∂xmρ=0∂tˆz+a∂xˆρ+λϵ2ˆz=1β(ˆq−ˆz)∂tˆq+∂xˆw+λϵ2(ˆq−ρ2)=0∂tˆw+aϵ2∂xˆq=1β(ˆP−ˆw) | (34) |
with
{∂tρ+∂xz=0∂tz+aϵ2∂xρ+λϵ2z−α3ϵ2¯∂xmρ=1β(qϵ−z)∂tqϵ+1ϵ2∂xw+λ−σϵ2qϵ−2wα¯∂xm3ϵ2(h++h−)=−σϵ2qϵ−α¯∂xm(h+−h−)3ϵ(h++h−)ˆρ∂tw+a∂xqϵ=1β(P−w) {∂tˆρ+1ϵ2∂xˆz=−λϵ2ˆρ+α2ϵ2¯∂xmρ∂tˆz+a∂xˆρ+λϵ2ˆz=1β(ˆq−ˆz)∂tˆq+∂xˆw+σϵˆq=−λ−σϵ2ˆq+λ2ϵ2ρ∂tˆw+aϵ2∂xˆq=1β(ˆP−ˆw) . | (35) |
Using again an implicit scheme for the right hand side of (35) and setting
{ρ(1)=ρnq(1)ϵ=ϵ2ϵ2+σΔtqnϵ−ϵΔtα3(ϵ2+σΔt)¯∂xmn(h+,n−h−,nh+,n+h−,n)ˆρ(1)z(1)=q(1)ϵw(1)=P(ρ(1),q(1)ϵ,ˆρ(1),ˆq(1)) {ˆρ(1)=ϵ2ϵ2+λΔtˆρn+Δtα2(ϵ2+λΔt)¯∂xmnρnˆq(1)=ϵ2ϵ2+(λ−σ)Δtˆqn+Δtλ2(ϵ2+(λ−σ)Δt)ρnˆz(1)=ˆq(1)ˆw(1)=ˆP(ρ(1),q(1)ϵ,ˆρ(1),ˆq(1)) . | (36) |
This relaxation scheme ensures that system (35) has the correct asymptotic limit as
The transport part in (35) includes four independent subsystems with (36) as the initial data. Introducing the characteristic variables
U1=12ρ+ϵ2√az ,ˉU1=ϵ2qϵ+12√aw ,U2=ϵ2ˆρ+12√aˆz ,ˉU2=12ˆq+ϵ2√aˆw ,V1=12ρ−ϵ2√az ,ˉV1=−ϵ2qϵ+12√aw ,V2=ϵ2ˆρ−12√aˆz ,ˉV2=−12ˆq+ϵ2√aˆw | (37) |
we use the APWB scheme for the left hand side of (35). With
In this section we state a hierarchy of coupling conditions for the kinetic and the nonlinear half-moment model proposed in the previous section. In [6] coupling conditions have been derived for the kinetic model and linear macroscopic approximations. These conditions can be used as well for the nonlinear half moment model discussed in the previous section. Finding conditions for the nonlinear full moment model is much harder since the signs of the eigen values can change with the states of the system. First the coupling conditions for the kinetic model are recalled to derive thereof the nonlinear half moment coupling conditions.
Consider the kinetic equation
∂tf+1ϵv∂xf=−λϵ2(f−ρ2)+12ϵαv¯∂xmρ . | (38) |
As stated in [6], the coupling conditions for (38) should assign a value to all
f+=Af−, |
where
N∑i=1ai,j=1 ∀j=1,…,N . |
Moreover, positivity requires
[f+1f+2f+3]=[01/21/21/201/21/21/20][f−1f−2f−3] . | (39) |
This can be generalized to the cases
Remark 2. Proving existence, uniqueness or stability of the kinetic equation with the above coupling conditions on the graph are interesting further issues, which would deserve a deeper analysis.
Since the kinetic coupling conditions (39) are linear and independent of
{[ρ+1ρ+2ρ+3]=[01/21/21/201/21/21/20][ρ−1ρ−2ρ−3][q+1q+2q+3]=−[01/21/21/201/21/21/20][q−1q−2q−3]. | (40) |
Recall that all properties of (39) are inherited by the above coupling conditions. For example the total mass in the system is conserved since
3∑i=1qϵ,i=1ϵ3∑i=1(q+i+q−i)=0 . |
In particular, the coupling conditions for the half-moment model yield the positivity of the densities, i.e.
Here only the case of a junction coupling three edges will be treated, but all procedures easily extend to arbitrary junctions. Consider the left hand side of (35) which includes four independent subsystems of eight variables
{∂tρ+∂xz=0∂tz+aϵ2∂xρ+λϵ2z−α3ϵ2¯∂xmρ=0 {∂tqϵ+1ϵ2∂xw+λ−σϵ2qϵ−α3ϵ2¯∂xm2(h++h−)w=0∂tw+a∂xqϵ=0{∂tˆρ+1ϵ2∂xˆz=0∂tˆz+a∂xˆρ+λϵ2ˆz=0{∂tˆq+∂xˆw+σϵ2ˆq=0∂tˆw+aϵ2∂xˆq=0 . | (41) |
For the additional numerical quantities we need further coupling conditions. Note that for
[P+1P+2P+3]=[01/21/21/201/21/21/20][P−1P−2P−3]. | (42) |
Inserting (22) into (40), we express the coupling conditions in the variables of (41)
[ρ1+ϵˆρ1ρ2+ϵˆρ2ρ3+ϵˆρ3]=[01/21/21/201/21/21/20][ρ1−ϵˆρ1ρ2−ϵˆρ2ρ3−ϵˆρ3][ϵq1+ˆq1ϵq2+ˆq2ϵq3+ˆq3]=−[01/21/21/201/21/21/20][ϵq1−ˆq1ϵq2−ˆq2ϵq3−ˆq3] | (43) |
and with (42) we obtain
[P1+ϵˆP1P2+ϵˆP2P3+ϵˆP3]=[01/21/21/201/21/21/20][P1−ϵˆP1P2−ϵˆP2P3−ϵˆP3] . | (44) |
This leads to
[ϵz1+ˆz1ϵz2+ˆz2ϵz3+ˆz3]=−[01/21/21/201/21/21/20][ϵz1−ˆz1ϵz2−ˆz2ϵz3−ˆz3][w1+ϵˆw1w2+ϵˆw2w3+ϵˆw3]=[01/21/21/201/21/21/20][w1−ϵˆw1w2−ϵˆw2w3−ϵˆw3] . | (45) |
Since on each edge there are four waves with positive speed, we need exactly twelve equations (43), (45) for three edges, as provided.
Remark 3. We have not investigated the positivity preserving property of the scheme for the density
In this section we investigate the proposed models in several numerical test cases for different values of
Test 1. We consider the interval
f(x,v,0)=F(v)ρ(x,0)=12ρ(x,0) , |
with
ρ(x,0)={1 if 0≤x≤10 if 1≤x≤2 . |
The remaining initial values for the hydrodynamic equations can be derived from the kinetic initial condition. At
In the Figures 2 and 3 the densities at
Test 2. In the second test we investigate the positivity preserving of the nonlinear moment models which is not guaranteed in the case of the linear models. We use non-equilibrium initial conditions
f(x,0,v)={aexp(vb)if−1≤v≤00if0≤v≤1 | (46) |
for the kinetic equation (3).
ρ−(x,0)=∫0−1fdv=ρ0 ,q−(x,0)=∫0−1vfdv=−0.9×ρ0 ,ρ+(x,0)=∫10fdv=0 ,q+(x,0)=∫10vfdv=0 , | (47) |
with
ρ0(x)={1 if 0≤x≤110−6 if 1≤x≤2,m(x,0)=0, ∀x∈[0,2] . |
The initial conditions for the full-moment models are
ρ(x,0):=ρ++ρ−=ρ0,qϵ(x,0):=q++q−ϵ=−0.9ϵρ0 . |
The solutions are plotted in Figure 4, for
This test case focuses on the coupling conditions for the non-linear half moment model proposed in section 4. We study a junction connecting three outgoing edges, as depicted in Figure 1. The results of the nonlinear half-moment model with the coupling conditions derived in section 4 are compared with the results of the kinetic, linear half and full moment and Keller-Segel model with coupling conditions described in [6]. On each of the edges we consider the interval
ρ1(x,0)=1 ,ρ2(x,0)=4 ,ρ3(x,0)=3 , |
which is consistent with the following values for the kinetic equation
fi(x,v,0)=F(v)ρi(x,0)=12ρi(x,0)i=1,2,3 . |
All other quantities are initially zero.
In Figures 5-8 the densities at
In this last test case we consider a larger network of
In Figure 9 the density at time
In Figure 10 the evolution of the total mass in the network up to
In this paper we developed nonlinear half-moment models for chemotaxis models which approximate very efficiently the solution of a network problem governed by kinetic equations. An APWB scheme is investigated for these nonlinear moment models. We derive from coupling conditions for the kinetic models corresponding conditions for the macroscopic models. Note that this procedure also might be applied to other kinetic models. In the numerical tests we investigated the dynamics for different coupling conditions and for varying values of
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