
Left: Comparison of solutions of the Scharfetter-Gummel (red line) and upwind (blue, dashed line) schemes at time
Citation: Cem Kıncal, Zhenhong Li, Jane Drummond, Peng Liu, Trevor Hoey, Jan-Peter Muller. Landslide Susceptibility Mapping Using GIS-based Vector Grid File (VGF) Validating with InSAR Techniques: Three Gorges, Yangtze River (China)[J]. AIMS Geosciences, 2017, 3(1): 116-141. doi: 10.3934/geosci.2017.1.116
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Taxis-diffusion and aggregation equations are widely studied in the context of biological populations (see [10,15,16,21] for instance). They describe cell communities which react to external stimuli and form aggregates of organisms (pattern formation), such as bacterial colonies, slime mold or cancer cells. The Patlak-Keller-Segel model [18] is the most famous system and we are interested in the following generalization
{∂u∂t−∂∂x[∂u∂x−φ(u)∂v∂x]=0,x∈(0,1),t>0,∂u∂x−φ(u)∂v∂x=0,for x=0 or 1,u(x,0)=u0(x)≥0,x∈[0,1]. | (1) |
Here,
v:=v(x)≥0,∂v∂x∈L∞(0,1), | (2) |
or the case of the generalized Keller-Segel (GKS in short) equation, where
v(x,t)=∫K(x,y)u(y,t)dy,K(x,y) a smooth, symmetric kernel. | (3) |
Depending on the modeling choice for
u(x,t)≥0, | (4) |
∫10u(x,t)=∫10u0(x)dx, | (5) |
g(u)=μ+v,g′(u)=1φ(u), | (6) |
where
ddtE(t)≤0,E(t)=∫10[G(u)−κuv]dx, | (7) |
where
κ=1(FP case),κ=12(GKS case). | (8) |
The aims of our work are first to recall two points of view for the derivation of the above energy inequality, second to use them for the construction of conservative, finite volume numerical schemes preserving energy dissipation to solve equation 1, third to make numerical comparisons in the case of complex patterns in order to distinguish physical instabilities from numerical artifacts. The two different derivations of the energy dissipation use two symmetrization strategies: the gradient flow or the Scharfetter-Gummel approach. It turns out that they lead to two strategies for discretization of problem 1. We prove that the proposed schemes statisfy properties 4-7 and because we build implicit schemes, there is no limitation on the time step in the fully discrete case.
There exist other works which propose schemes for the resolution of problems in the form 1. For instance, finite elements methods are used, see [25] and references therein. Optimal transportation schemes for Keller-Segel systems are introduced in [5]. The papers [8] and [9] propose a finite-volume method able to preserve the above properties, including energy dissipation, at the semi-discrete level or with an explicit in time discretization, using the gradient flow approach, see also [4]. The symmetrization using the Scharfetter-Gummel approach is used in [20] where properties similar to ours are proved. However, the results do not include sensitivity saturation. To the best of our knowledge, our work is the first to propose implicit in time methods, without time step limitation (CFL condition), for which we are able to prove that, under generic conditions, the energy decreases at both semi-discrete and discrete level. Moreover, we build an alternative to the gradient flow approach applying the Scharfetter-Gummel strategy [26] for the discretization of drift-diffusion equations 1 with a general saturation function
The paper is organized as follows. In Section 2, we present in more details our assumptions for the equation 1. We also explain some modeling choices in particular for the nonlinearity
The standard biological interpretation of 1 ([14,21,23]) provides us with some further properties of the nonlinearities which we describe now.
Chemotactic sensitivity. The function
φ(u)=uψ(u),withψ(u)≥0,ψ′(u)≤0. | (9) |
More precisely, we consider two cases for the smooth function
ψ(u)>0,∀u>0, | (10) |
or
ψ(u)>0for0<u<M,ψ(M)=0. | (11) |
In the case 11 we only consider solutions which satisfy
It is convenient to introduce the notations
g(u)=∫ua1φ(v)dv,G(u)=∫u0g(s)ds, | (12) |
where
1φ∉L1(a,+∞),g(u)⟶u→∞+∞, | (13) |
an assumption which, as we see it later, appears naturally when it comes to the well-posedness of numerical schemes.
Note that under assumption 10 and if
Expression of the drift
{−∂2v∂x2=u−v,x∈(0,1),∂v∂x=0,x=0 or 1. |
This equation leads to 3 using the Green function given by the positive and symmetric kernel
K(x,y)=λ(ex+e−x)(ey+e2−y),x≤y,λ=12(e2−1). | (14) |
Energy dissipation is the most difficult property to preserve in a discretization and methods might require corrections [17]. Therefore, it is useful to recall how it can be derived simply for the continuous equation. We focus on two different strategies, that lead to two different discretization approaches, the gradient flow approach and the Scharfetter-Gummel approach.
Using the notations 12, the equation for
∂u∂t−∂∂x[φ(u)∂(g(u)−v)∂x]=0, | (15) |
so that
(g(u)−v)∂u∂t=(g(u)−v)∂∂x[φ(u)∂(g(u)−v)∂x]=12∂∂x[φ(u)∂(g(u)−v)2∂x]−φ(u)[∂(g(u)−v)∂x]2. |
Consequently, we find, in the Fokker-Planck case
ddt∫10[G(u)−uv(x)]dx=−∫10φ(u)[∂(g(u)−v)∂x]2dx≤0, |
and in the generalized Keller-Segel case
ddt∫10[G(u)−12uv(x,t)]dx=−∫10φ(u)[∂(g(u)−v)∂x]2dx≤0, |
because, thanks to the symmetry assumption on
∫10∫10K(x,y)u(y,t)∂u(x,t)∂t=∫10∫10K(x,y)∂u(y,t)∂tu(x,t)=12ddt∫10∫10K(x,y)u(y,t)u(x,t). |
Inspired from the case of electric forces in semi-conductors, the equation for
∂u∂t−∂∂x[ev−g(u)φ(u)∂eg(u)−v∂x]=0, | (16) |
so that
(g(u)−v)∂u∂t=(g(u)−v)∂∂x[ev−g(u)φ(u)∂eg(u)−v∂x]=∂∂x[(g(u)−v)ev−g(u)φ(u)∂eg(u)−v∂x]−ev−g(u)φ(u)∂eg(u)−v∂x∂(g(u)−v)∂x. |
It is immediate to see that the last term has the negative sign while the time derivative term is exactly the same as in the gradient flow approach.
At the continuous level, these two calculations are very close to each other. However, they lead to the construction of different discretizations. The gradient flow point of view is used for numerical schemes by [10], the Scharfetter-Gummel approach is used in [20].
We give here our notations for the semi-discretization. We consider a (small) space discretization
Ii=(xi−12,xi+12),i=1,…,I. |
The semi-discrete approximation of
ui(t)≈1Δx∫Iiu(x,t)dx,i=1,…,I. |
As for the discretization of
Kij:=K(xi,xj),i=1,…,I,j=1…,I. |
Integration on the interval
The mass conservative form of 1 leads to a finite volume semi-discrete scheme
{dui(t)dt+1Δx[Fi+1/2(t)−Fi−1/2(t)]=0,i=1,…,I,t>0,F1/2(t)=FI+1/2(t)=0. | (17) |
We use the definition 8 for
Ei(t)=G(ui(t))−κui(t)vi(t). |
The semi-discrete energy is then
Esd(t):=ΔxI∑i=1Ei(t). |
Using the form 15 of equation 1, we define the semi-discrete flux as
Fi+1/2(t)=−φi+1/2Δx[g(ui+1)−vi+1−(g(ui)−vi)],i=1,…,I−1. | (18) |
The precise expression of
Then, the semi-discrete energy form is obtained after multiplication by
ddtΔxI∑i=1Ei(t)=−I∑i=1(g(ui)−vi)[Fi+1/2−Fi−1/2]=I−1∑i=1Fi+1/2[(g(ui+1)−vi+1)−(g(ui)−vi)]. |
Therefore, we find the semi-discrete form of energy dissipation
dEsddt=−ΔxI−1∑i=1φi+1/2[(g(ui+1)−vi+1)−(g(ui)−vi)Δx]2≤0. |
We choose to discretize the form 16, defining the semi-discrete flux as
Fi+1/2(t)=−(ev−g(u)φ(u))i+1/2Δx[eg(ui+1)−vi+1−eg(ui)−vi],i=1,…,I−1, | (19) |
where, again, the specific form of the interpolant
As above, the semi-discrete energy form follows upon multiplication by
ddtΔxI∑i=1Ei(t)=−I∑i=1(g(ui)−vi)[Fi+1/2−Fi−1/2]=I−1∑i=1Fi+1/2[(g(ui+1)−vi+1)−(g(ui)−vi)]. |
Summing up, the semi-discrete form of energy dissipation here writes
dEsddt=−ΔxI−1∑i=1{(ev−g(u)φ(u))i+1/2⋅eg(ui+1)−vi+1−eg(ui)−viΔx⋅(g(ui+1)−vi+1)−(g(ui)−vi)Δx}≤0. |
Steady states make the energy derivative vanish which imposes both in the gradient flow and the Scharfetter-Gummel approaches that
g(ui)=vi+μ,i=1,…,I. | (20) |
We recall from [24] that in the GKS case, there are several steady states and the constant ones can be unstable.
For the time discretization, we consider (small) time steps
uni≈1Δx∫Iiu(x,tn)dx,i=1,…,I,n∈N. |
Integration on the interval
To achieve the time discretization, and restricting our analysis to the Euler scheme, we write the time discretization
{un+1i−uni+ΔtΔx[Fn+1i+1/2−Fn+1i−1/2]=0,i=1,…,I,Fn+11/2=Fn+1I+1/2=0. | (21) |
The issue here is to decide which terms (in
We define the energy at the discrete level through
Eni=G(uni)−κunivni,i=1,…,I,n∈N, |
and
En:=ΔxI∑i=1Eni,n∈N. |
The computation made in the semi-discrete case,
I∑i=1(En+1i−Eni)≤I∑i=1(un+1i−uni)(g(uαni)−vβni). |
Here,
G(un+1i)−G(uni)≤g(un+1i)(un+1i−uni). |
Regarding the term in
−I∑i=1[un+1ivn+1i−univni]≤−2I∑i=1vβni(un+1i−uni). |
It is natural to try and balance the terms by choosing a semi-explicit discretization with
I∑i=12vβni(un+1i−uni)−(un+1ivn+1i−univni)=I∑i=1(un+1ivni−univn+1i)=∑i,jKij(un+1iunj−uniun+1j), |
with the last term vanishing due to the symmetry of
However, implicit and explicit time discretizations for
I∑i=12vβni(un+1i−uni)−(un+1ivn+1i−univni)=(1−2β)∑i,jKij(un+1i−uni)(un+1j−unj). |
As a consequence, an explicit (resp. implicit) scheme is suitable for the time discretization of
Finally, we note that the interpolant does not play any role for energy discretization and we can use the simplest explicit or implicit discretization (both in
We consider the full discretization of 18 and define the fully discrete flux in 21 as
Fn+1i+1/2=−φ(u)n+1i+1/2Δx[(g(un+1i+1)−vni+1)−(g(un+1i)−vni)],i=1,…,I−1. | (22) |
At this level, we need to define the form of the interpolant
φ(u)n+1i+1/2:={un+1iψ(un+1i+1)wheng(un+1i)−g(un+1i+1)+vni+1−vni≥0,un+1i+1ψ(un+1i)wheng(un+1i)−g(un+1i+1)+vni+1−vni<0. | (23) |
Proposition 1 (Fully discrete gradient flow scheme). We assume either 10 and 13, or 11 and give the
(ⅰ) the solution
(ⅱ) it satisfies
(ⅲ) the steady states
(ⅳ) the discrete energy dissipation inequality is satisfied
En+1−En≤−ΔtΔxI−1∑i=1φ(u)ni+1/2[(g(un+1i+1)−vni+1)−(g(un+1i)−vni)]2. |
Notice that this theorem does not state a uniform bound in the case 10 and 13.
Proof. (ⅰ) We prove that the scheme satisfies the hypotheses of the theorem in Appendix A. We set
Ai+1/2(un+1i,un+1i+1)=ΔtΔxFn+1i+1/2. |
Then, the simplest case is when
Fn+1i+1/2=−φ(u)n+1i+1/2Δx[(g(ˉUn+1i+1)−vni+1)−(g(ˉUn+1i)−vni)]=−φ(u)n+1i+1/2Δx[C−C]=0. |
Thus
Moreover, the scheme is monotone since
∂1Ai+12(un+1i,un+1i+1)=−Δt(Δx)2un+1i+1ψ′(un+1i)[g(un+1i+1)−vni+1−(g(un+1i)−vni)]+−Δt(Δx)2ψ(un+1i+1)[g(un+1i+1)−vni+1−(g(un+1i)−vni)]−−Δt(Δx)2φ(u)n+1i+12[−g′(un+1i)]≥0, |
where
[x]+={xforx≥0,0forx<0and[x]−={0forx≥0,xforx<0, |
so that
(ⅱ) Positivity of discrete solutions and the upper bound in the logistic case follow from the subsolution and supersolution built in step (ⅰ).
(ⅲ) Preservation of steady states at the discrete level follows immediately from the form we have chosen for the fully discrete fluxes.
(ⅳ) For the energy inequality, we remark that the contribution regarding time discretization is treated in the introduction of the present section. The space term is exactly treated as in the corresponding subsection of Section 4.
In 21, the fully discrete Scharfetter-Gummel flux reads
Fn+1i+1/2=−(evn−g(un)φ(un+1))i+1/2[eg(un+1i+1)−vni+1−eg(un+1i)−vni],i=1,…,I−1. |
As for the gradient flow approach, we need the upwind technique to get a scheme which satisfies the hypotheses in Appendix A. So, we set for
(evn−g(un)φ(un+1))i+1/2:={un+1i+1ψ(un+1i)evni+1−g(uni+1),ife(g(un+1i+1)−vni+1)−e(g(un+1i)−vni)≥0,un+1iψ(un+1i+1)evni−g(uni),ife(g(un+1i+1)−vni+1)−e(g(un+1i)−vni)<0. |
Proposition 2 (Fully discrete Scharfetter-Gummel scheme). We assume either 10 and 13, or 11 and give the
(ⅰ) the solution
(ⅱ) it satisfies
(ⅲ) the steady states
(ⅳ) the discrete energy dissipation inequality is satisfied
En+1−En≤−ΔtΔxI−1∑i=1{(evn−g(un)φ(un))i+1/2⋅[eg(un+1i+1)−vni+1−eg(un+1i)−vni]⋅[(g(un+1i+1)−vni+1)−(g(un+1i)−vni)]}≤0. |
Proof. We argue exactly as for the gradient flow approach.
The upwind scheme is driven by simplicity and, in 21, the fluxes are defined by
Fn+1i+1/2=−1Δx[un+1i+1−un+1i−φ(u)ni+1/2(vni+1−vni)],i=1,…,I−1, |
with
φ(u)n+1i+1/2:={un+1iψ(un+1i+1)whenvni+1−vni≥0,un+1i+1ψ(un+1i)whenvni+1−vni<0, | (24) |
as in 23, but this time depending on the sign of
Proposition 3 (Fully discrete upwind scheme). We assume either 10 and 13, or 11 and give the
(i) the solution
(ii) it satisfies
Proof. As for the gradient flow approach, the above choice makes the scheme monotone, because
ΔtΔx∂1Fi+12(un+1i,un+1i+1)=−ΔtΔx2(−1−un+1i+1ψ′(un+1i)[vni+1−vni]−−ψ(un+1i+1)[vni+1−vni]+)≥0. |
Thus, arguing as for the gradient flow approach and relying on the results in Appendix A, existence and uniqueness of the discrete solution as well as preservation of the initial bounds follow immediately.
Thus, choice 24 enables to prove that the scheme is well-defined, satisfies
We first present the numerical implementation of the Fokker-Planck equation with
We consider a first case with
v=x(1−x)|x−0.5|. |
In Figure 1, we compare the approximate stationary solutions obtained with the upwind scheme (blue, dashed line) and the Scharfetter-Gummel scheme (red line) with the exact stationary solution (black line), which in this case has the form
Left: Comparison of solutions of the Scharfetter-Gummel (red line) and upwind (blue, dashed line) schemes at time
We turn to the equation 1 coupled with 3 for two nonlinear forms of the chemotactic sensitivity function: the logistic form
{∂u∂t−∂∂x[D∂u∂x−χφ(u)∂v∂x]=0,x∈(0,1),t>0,D∂u∂x−χφ(u)∂v∂x=0,for x=0 or 1,u(x,0)=u0(x)≥0,x∈[0,1], | (25) |
in order to emphasize the coefficients driving the instabilities:
We first consider the logistic case with
Figure 2 shows the evolution in time of the density
Evolution in time of solutions to (25) in the logistic case
As for the schemes, Figure 2 shows that the Scharfetter-Gummel and the gradient flow approaches give the same solution; no difference can be spotted. This is not true for the upwind approach. In Figure 3, we compare the solutions to the Scharfetter-Gummel (red line) and the upwind (blue, dashed line) schemes. The upwind solution transitions faster from one metastable structure to the following than the Scharfetter-Gummel one. In fact, as proved above, the latter preserves discrete stationary profiles which, using the no-flux boundary conditions, solve the equation
∂u∂x=χDφ(u)∂v∂x. | (26) |
From 26, it is clear that, in the logistic case, the expected stationary solutions are 0-1 plateaus (or "steps") connected by a sigmoid curve which is increasing or decreasing when
Moreover, in Figure 4c we compare the
Next, we consider an exponentially decreasing form of the chemotactic sensitivity function with
In the context of the Generalized Keller-Segel system, we have presented constructions of numerical schemes which extend previous works [10,20], built on two different views of energy dissipation. Our construction unifies these two views, the gradient flow and Scharfetter-Gummel symmetrizations. Our schemes preserve desirable continuous properties: positivity, mass conservation, exact energy dissipation, discrete steady states. Being correctly tuned between implicit and explicit discretization, they can handle large time steps without CFL condition.
The present work is motivated by experiments of breast cancer cells put in a 3D structure mimicking the conditions they meet in vivo, namely in the extracellular matrix. After a few days, images of 2D sections show that cells have organized as spheroids, a phenomenon believed to be driven by chemotaxis. The spheroids can then be interpreted as Turing patterns for Keller-Segel type models and it is crucial to use appropriate schemes for them to be distinguishable from actual steady states or numerical artifacts. Comparing 2D simulations of such models with these experimental images will be the subject of future work.
In fact, it is important to remark that the schemes we presented here in 1D could be easily extended to rectangular domains, without loss of properties 4-7. However, it remains a perspective to treat more general geometries in a multi-dimensional setting with our approach.
The authors acknowledge partial funding from the "ANR blanche" project Kibord [ANR-13-BS01-0004] funded by the French Ministry of Research.
B.P. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 740623).
We recall sufficient conditions for which an implicit Euler discretization in time can be solved, independently of the step-size. This is the case for a monotone scheme. The proof relies on the existence of sub- and supersolutions, and thus also yields the preservation of positivity and other pertinent bounds as we have used in Section 5.
We consider the problem of finding a unique solution
un+1i−uniΔt+1Δx[F(uni,uni+1,vni,vni+1,un+1i,un+1i+1)⏟Fn+1i+12−Fn+1i−12]=0,i=1,…,I. | (27) |
We write a general proof for a scheme of the form
ui+Ai+12(ui,ui+1)−Ai−12(ui−1,ui)=fi,i=1,…,I, | (28) |
where we consider the problem of finding a solution
Here we assume that the
∂1Ai+12(⋅,⋅)≥0,∂2Ai+12(⋅,⋅)≤0,i=1,…,I, | (29) |
and there are a supersolution
ˉUi+Ai+12(ˉUi,ˉUi+1)−Ai−12(ˉUi−1,ˉUi)≥fi, | (30) |
U_i+Ai+12(U_i,U_i+1)−Ai−12(U_i−1,U_i)≤fi. | (31) |
We build a solution of 28 using an evolution equation
dui(t)dt+ui(t)+Ai+12(ui(t),ui+1(t))−Ai−12(ui−1(t),ui(t))=fi,i=1,…,I. | (32) |
Theorem A.1. Assume 29 and the existence of a subsolution and of a supersolution. Then,
(ⅰ) For a supersolution (resp. subsolution) initial data, the dynamics 32 satisfies
(ⅱ) A subsolution is smaller than a supersolution.
(ⅲ)
Proof. (ⅰ) We prove the statement with the supersolution. We set
zi(t)=dˉui(t)dt,zi(0)≤0,i=1,…,I. |
Since the
Differentiating equation 32, we obtain for
dzi(t)dt+zi(t)+[∂1Ai+12−∂2Ai−12]zi(t)=−∂2Ai+12zi+1(t)+∂1Ai−12zi−1(t). |
The solution cannot change sign and thus for
(ⅱ) Consider
We write for
wi+[Ai+12(u_i,u_i+1)−Ai+12(ˉui,u_i+1)]+[Ai+12(ˉui,u_i+1)−Ai+12(ˉui,ˉui+1)]−[Ai−12(u_i−1,u_i)−Ai−12(ˉui−1,u_i)]−[Ai−12(ˉui−1,u_i)−Ai−12(ˉui−1,ˉui)]≤0. |
For
I∑i=1(wi)++I−1∑i=1Ji+12+I−1∑i=1Ki+12=0, |
with
Ji+12=[Ai+12(u_i,u_i+1)−Ai+12(ˉui,u_i+1)][sgn+(wi)−sgn+(wi+1)],Ki+12=[Ai+12(ˉui,u_i+1)−Ai+12(ˉui,ˉui+1)][sgn+(wi)−sgn+(wi+1)]. |
For each of the these terms, we show that
u_i≥ˉui,andu_i+1≤ˉui+1. |
Then, we have by assumption 29,
[Ai+12(u_i,u_i+1)−Ai+12(ˉui,u_i+1)]≥0⇒Ji+12≥0,[Ai+12(ˉui,u_i+1)−Ai+12(ˉui,ˉui+1)]≥0⇒Ki+12≥0. |
Therefore
(ⅲ) This is clear since the limits are solutions.
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