Finite difference scheme for a general class of the spatial segregation of reaction-diffusion systems with two population densities Текст научной статьи по специальности «Математика»

Вестник КГЭУ, 2017, № 3 (35) УДК 517

FINITE DIFFERENCE SCHEME FOR A GENERAL CLASS OF THE SPATIAL SEGREGATION OF REACTION-DIFFUSION SYSTEMS WITH TWO POPULATION DENSITIES*

A. Arakelyan, R. Barkhudaryan, L. Poghosyan

Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan,

Armenia

[email protected], [email protected], [email protected]

In the present work we deal with the numerical approximation of equations of stationary states for a general class of the spatial segregation of Reaction-diffusion system with two population densities having disjoint supports. We show that the problem gives rise the generalized version of the so-called two-phase obstacle problem and introduce the notion of viscosity solutions for this new model. Then we use quantitative properties of both, solutions and free boundaries, to develop convergent finite difference scheme.At the end of paper we present a computational test and discuss numerical result.

Keywords: free boundary, Two-phase membrane problem, Reaction-diffusion systems, Finite difference.

*Acknowledgments. This work was supported by State Committee of Science MES RA, in frame of the research project No. SCS 13YR-1A0038

MathematicsSubjectClassification 35R35

КОНЕЧНО-РАЗНОСТНАЯ СХЕМА ДЛЯ ОБЩЕГО КЛАССА ПРОСТРАНСТВЕННОЙ СЕГРЕГАЦИИ РЕАКЦИОННО-ДИФФУЗНИОННОЙ СИСТЕМЫ ПЛОТНОСТЕЙ ДВУХ ПОПУЛЯЦИЙ С НЕПЕРЕСЕКАЮЩИМИСЯ НОСИТЕЛЯМИ*

А. Аракелян, Р. Бархударян, Л. Погосян

Институт математики, Национальная академия наук Республики Армения,

г. Ереван, Армения

[email protected], [email protected], [email protected]

В настоящей работе мы рассматирваем численную аппроксимацию уравнений стационарных состояний для общего класса пространственной сегрегации реакционно -диффузионной системы плотности двух популящий с непересекающимися носителями. Мы показываем, что задача сводится к обобщенной версии так называемой двухфазной задачи препятствий и вводим понятие вязкого решения для этой новой модели. Затем, мы используем количественные свойства решений и свободных границ для разработки сходящийся конечно-разностной схемы. В коные работы мы приводим вычислительный тест и обсждаем численный результат.

Ключевые слова: свободная граница, двухфазная мембранная проблема, реакционно-диффузионные системы, конечная разность.

* Благодарности. Данная работа была поддержана Государственным комитетом по науке MES Республики Армения в рамках исследовательского проекта № SCS13YR-1A0038.

Introduction

In recent years there have been intense studies of spatial segregation for reaction-diffusion systems. The existence of spatially inhomogeneous solutions for competition models of Lotka-Volterra type in the case of two and more competing densities have been considered in [8,9,10]. The aim of this paper is to study the numerical solutions of a certain general class of the Spatial segregation of Reaction-diffusion system with two population densities. Here, for the sake of clarity, we state the problem for general m > 2 population densities and then explain the problem for the particular case m = 2.

Let Q ^ R", (" > 2) be a connected and bounded domain with smooth boundary and m be a fixed integer. We consider the steady-states of m competing species coexisting in the same area Q. Let щ(x) denotes the population density of the i th

component with the internal dynamic prescribed by Fi(x,щ) .

We call the m -tuple U = (u,•", um) e (H 1(Q))m a segregated state if

U(x) • u .(x) = 0, a.e.for i Ф j, x e Q. The problem amounts to

- m / 1 Л

Minimized(щ,•••,um) = — | Vu; |2 + F(x,U) \dx, (1)

over the set

S = {(U!,_, Um) e (H1(Q))m : щ > 0, щ • u] =0, щ = фг on SQ], i

where фi e H2(SQ), ф • фj =0, for i Ф j and ф > 0 on the boundary SQ. We assume that

Fi(x, s) = (x, v)dv

where f (x,s) : Qx R+ ^ R is Lipschitz continuous in s, uniformly continuous in x and f (x,0) > 0. In the sequel, we assume that the functional (1) is coercive, which will be needed to provide the existence of minimizers.

Remark 1 By our definition, the functions f (x, s) 's are defined only for non negative values of s (recall that our densities щ 's are assumed non negative); thus we can arbitrarily extend such functions on the negative semiaxis. For the sake of convenience, when s < 0 we will let f (x,s) = —f (x,—s) This extension preserves the

continuity, due to conditions on f defined above. In the same way, each F is extended as an even function.

For the case of two populations the problem will be reduced to:

Minimized(щ,Щ) = J ^f11 Vu |2 + F(x,Щ) W,

Minimize : J(w) = J

'"2) = J^lj I Vu> r +F(x,U) d*, (2)

over the set

S = {(", u2) e (H*(Q))2 : u > 0,u ■ u2 = 0," = fi on SQ}. i

Here fie H 2(SQ) with property = 0,fi > 0 on the boundary SQ.

Throughout the paper we will also assume that the functions F(x,s) are convex with respect to the variable s.

Segregation problem with two population densities Generalized two-phase obstacle problem

This section is devoted to the minimization problem with two population densities. Following [5] it is easy to see that the problem can be treated as a minimization problem subject to the convex set:

Vw I2 +F (x, w+ ) + F (x-w" ) dx, (3)

over the set K = {w e H*(Q): w - (fi-fe) e H^(Q)}.

The corresponding Euler-Lagrange equation for the minimization problem will be

iAw = f( ^ w) ■^{w>0} - f2(x,-w) ^{wO} x eQ

[w = fi-fe, x eSQ,

where XA stands for the characteristic function of the set A . Inspired by the setting of the two-phase obstacle problem (see [12]) we will call the problem (4) as the generalized two-phase obstacle problem. Nowadays, the theory of the two-phase obstacle-like problems (elliptic and parabolic versions) is well-established and for a reference we again address to the book [12]. For the numerical treatment of the same problems we refer to the works [2; 5; 1].

In [2] the authors introduced the so-called Min-Max formulation for the usual Two-phase obstacle problem, which is very useful to define the notion of viscosity solutions. Moreover, it turns out that the introduced viscosity solution is equivalent to the weak solution of the Two-phase obstacle problem. Our aim is to use the same approach for the generalized counterpart. To this end, we need to make some notations.

Let Q be an open subset of R", and for a twice differentiable function u : Q ^ R let Du and D2u denote the gradient and Hessian matrix of u , respectively. Also let the function G(x, r,p, X) be a continuous real-valued function defined on

Qx R x R" x S", with S" being the space of real symmetric " x " matrices.

In the light of the Min-Max form defined in [2] we introduce the following generalized Min-Max variational equation:

fmin(-Aw + f (x,w),max(-Aw- f(x,-w),w)) = 0, in Q

[w = fi-fe = g, on SQ. (5)

Following the above notations we introduce a function G : Q x R x R" x S" ^ R by G( x, r, p, X) = min(-trace(X) + f (x, r),max(-trace(X) - f (x,-r), r)), (6) then the equation in (5) can be rewritten as

G(x, w, Dw, D2w) = 0 in Q. (7)

Below we recall the definition of degenerate ellipticity, and prove that the equation (5) is degenerate elliptic.

Definition 2 We call the equation (7) degenerate elliptic if

G(x, r, p, X) < G(x, s, p,Y ) whenever r < s and Y < X, where Y < X means that X — Y is a nonnegative definite symmetric matrix.

Lemma 3 The equation (6) is degenerate elliptic.

Proof. Let X, Y G Sn and r, s G R satisfy Y < X and r < s . Then using the fact that F(x, t) is convex in t for all x gQ, we have F(x, t) = f'(x, t) > 0, where the derivatives are taken with respect to t. Thus,

— trace(X) + f (x,r) <—trace ( Y) + f (x,s), and

max(—trace(X) — f (x,—r), r) < max(—trace(Y) — f (x,—s), s). Therefore

G(x, r, p, X) = min(—trace (X) + f (x, r),max(—trace(X) — f (x,—r), r)) < min(—trace(Y) + f (x, s),max(—trace(Y) — f (x,—s), s)) = G( x, s, p, Y ).

Now, we are ready to define viscosity solutions for the generalized two-phase obstacle problem. For general background about the theory of viscosity solutions the reader is referred to [6,10] and references therein.

Definition 4 A bounded uniformly continuous function w : Q ^ R is called a

viscosity subsolution (resp. supersolution) to (5), if for each Ç G C2(Q) and local maximum point of w — Ç (respectively minimum) at x0 G Q, we have

min (— Aç(x0) + f( -X^ w(x0)),max(—Aç( xo) — f2( x0,—w( ^X w(x0))) < 0.

(respectively

min(— AÇ(x0) + fi(-x^ w(x0)),max(—AÇ(x0) — f2(x0,—w(X0)), w(x0))) > 0 )

The function w : Q^ R is said to be a viscosity solution of (5), if it is both a viscosity subsolution and supersolution for (5).

Finite difference scheme

We will make the notations for the one-dimensional and two-dimensional cases parallely.

For the sake of simplicity, we will assume that Q = (—1,1) in one-dimensional case and Q = (—1,1)

x (—1,1) in two-dimensional case in the rest of the paper, keeping in mind that the method works also for more complicated domains.

Let N gN be a positive integer, h = 2/N and x = —1 + ih, y = —1 + ih, i = 0,1,..., N.

We use the notation ui and u . (or simply ua, where a is one- or two-dimensional index) for finite-difference scheme approximation to u(xi) and u(xi,y ) ,

g, = — ^2 (i) = A ( x,) — $2 ( x,)

and

Si, j = A(iJ) " kfrjl = ф( xi, yj) " ф2( Хг, У ),

in one- and two-dimensional cases, respectively, assuming that the function ф—ф2 is extended to be zero everywhere outside the boundary SQ.

In this paper we will use also notations и = (иа), g = (S«) (not to be confused with functions U, g). Denote

N = {i: 0 < i < N} or N = {(i, j):0 < i, j < N}, No ={i: I <i <N — 1} or No ={(i, j):l <i, j <N — I},

in one- and two- dimensional cases, respectively, and 3N = N \ No.

In one-dimensional case we consider the following approximation for Laplace operator: for any i e No,

. и l — 2и + им

V; - LkU = i—1 hl—

and for two-dimensional case we introduce the following 5-point stencil approximation for Laplacian:

д J U—i,j + — 4u, j + ui, j—1 + 4, j+i - LbUu =-j-j--j-—

for any (i, j) e No.

Now with the use of the generalized Min-Max variational equation (5) we define appropriate finite difference scheme as follows:

fmin(-Lhua + f1(xa,ua),max(-Lhua - /2(xa,-ua), uj) = 0, a 6 N0, 1 ua = ga, a 6dN. ( }

Theorem 5 The nonlinear system has a unique solution.

For convergence analysis of the difference scheme we apply Barles-Souganidis theory (see [3]) developed for viscosity solutions. To this aim, we define a uniform structured grid on the domain Q as a directed graph consisting of a set of points x e Q, i = l,..,N, each endowed with a number of neighbors K. A grid function is a real valued function defined on the grid, with values u = u(x ). The typical examples of such

grid are 3 -point and 5 -point stencil discretization for the spaces of one dimension and two dimension, respectively. Here we recall degenerate elliptic schemes introduced by Oberman (see [11]).

A function Fh : RN ^ RN, which is regarded as a map from grid functions to grid functions, is a finite difference scheme if

Fh[и] = F;[u,,и, — и. ,...,и, — щк] (i = l,...,N), where {i1,i2,.,iK} are the neighbor points of a grid point i. Denote

F[u] - F[u,,иi — ui} |j=^] - F[и,и — Uj], i = l,...,N,

where и. is shorthand for the list of neighbors Щ Ij=\k .

Definition 6The scheme F is degenerate elliptic if each component F' is nondecreasing in each variable, i.e. each component of the scheme F i is a

nondecreasing function of u and the differences u — u for j = 1,...,K.

' ' lj

Since the grid is uniformly structured, we denote h >0 be the size of the mesh. Then, for the nonlinear system we have

F'[u', u' — uj ] = min(—Lhui + f1(xi, u ),max(—Lhul — f2 (xi,—ui ),ui)), (8)

where

k 1

Lhu = E-ttu — u), ' = 1,..., N. (9)

j=1 h j

Since the functions f (x,s) are monotone non-decreasing with respect to s, we clearly see that F' [ui, u — u^ ] is non-decreasing with respect to u; and ut — u- as well.

Therefore the finite difference scheme is a degenerate elliptic scheme. But we know that the degenerate elliptic schemes are monotone and stable (see [11]). The consistency of the system is obvious. Thus, we show that the nonlinear system fulfills all the necessary properties for the Barles-Souganidis framework, namely it is stable, monotone and consistent and therefore, due to Barles - Souganidis theorem, the solution to the discrete nonlinear system converges locally uniformly to the unique viscosity solution of the generalized two-phase obstacle problem (5). Numerical example

In this section we present a numerical simulation for two competing densities with internal dynamics f. We consider the following minimization problem:

Minimize£^11 Vu; |2 +F(x,ut) jdx, (10)

over the set

S = {(u, u) e (H:(Q))2 : u ^ 0,u ■ u = 0,u = $ on <9Q}.

Figure. Numerical solution with 50x50 discretization points

We take Q = [0,l] x [0,l], and the internal dynamics such that fi( x, y, Ui) = 7( x2 + y2) + (l + Ui)2,

and

f2( x, y, U2) = l0( x2 + y2) + (l + U2)2, with the boundaries ф(x,y) and ф(x,y) defined as follows:

fx 0 < x < l,

Ф ( x,±l) = [ ,

m [0 — l < x < 0,

ф( l , y) = l , ф (—l, y) = 0,

and

f0 0 < x < l ,

Ф2 (x,± l ) = [ < ^ ^ '

[— x — l< x < 0, ф2(—l, y) = l, Ф2О, y) = 0.5,

In Figure 1 we clearly see the zero set between densities Щ, which is due to taken large internal dynamics f (x, y, Щ ) of the system.

In Figure 2 the level sets of the numerical solution are depicted. We see the tangential touch between two competing densities Щ and U2 close to the fixed boundary of Q.

References

[1] Arakelyan A.A finite difference method for two-phase parabolic obstacle-like problem.Armenian Journal of Mathematics 7, 1 (2015), 32-49.

[2] Arakelyan A., Barkhudaryan R., and Poghosyan M. Numerical Solution of the Two-Phase Obstacle Problem by Finite Difference Method.ArXiv e-prints (2014).

[3] Barles G. and Souganidis P.E. Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4, 3 (1991), 271-283.

[4] Bozorgnia F. Numerical solutions of a two-phase membrane problem.Applied Numerical Mathematics 61, 1 (2011), 92-107.

[5] Bozorgnia F. and Arakelyan A. Numerical algorithms for a variational problem of the spatial segregation of reaction—diffusion systems.Applied Mathematics and Computation 219, 17 (2013), 8863-8875.

[6] Caffarelli L.A. and Cabré X. Fully nonlinear elliptic equations, vol. 43 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1995.

[7] Conti M., Terracini S. and Verzini G. Asymptotic estimates for the spatial segregation of competitive systems.Adv. Math. 195, 2 (2005), 524-560.

[8] Conti M., Terracini S. and Verzini G. A variational problem for the spatial segregation of reaction-diffusion systems. Indiana Univ. Math. J. 54, 3 (2005), 779-815.

[9] Conti M., Terracini S. and Verzini G. Uniqueness and least energy property for solutions to strongly competing systems. Interfaces Free Bound. 8, 4 (2006), 437-446.

[10] Crandall M.G., Ishii H. and Lions P.-L. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27, 1 (1992), 1-67.

[11] Oberman A.M. Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems.SIAM J. Numer. Anal. 44, 2 (2006), 879-895 (electronic).

[12] Petrosyan A., Shahgholian H. and Ural'ceva N.N. Regularity of free boundaries in obstacle-type problems, vol. 136. AmericanMathematical Soc., 2012.

Сведения об авторах

Аветик Аракелян - Институт математики, Национальная академия наук Республики Армения, г. Ереван, Армения.

Рафаэль Бархударян - Институт математики, Национальная академия наук Республики Армения, г. Ереван, Армения.

Лусине Погосян - Институт математики, Национальная академия наук Республики Армения, г. Ереван, Армения.

Authors of the publication

Avetik Arakelyan - Institute of Mathematics NAS of Armenia, Baghramyan 24/5, 0019 Yerevan, Armenia.

Rafayel Barkhudaryan - Institute of Mathematics NAS of Armenia, Baghramyan 24/5, 0019 Yerevan, Armenia.

Lusine Poghosyan - Institute of Mathematics NAS of Armenia, Baghramyan 24/5, 0019 Yerevan, Armenia.

Дата поступления 02.09.2017.