Numerical methods for the generalized Fisher Kolmogorov Petrovskii Piskunov equation

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1 Applied Nmerical Matematics Nmerical metods for te generalized Fiser Kolmogorov Petrovskii Pisknov eqation J.R. Branco a,j.a.ferreira b,, P. de Oliveira b a Departamento de Fí sica Matemática, Institto Sperior de Engenaria de Coimbra, Ra Pedro Nnes Qinta da Nora, Coimbra, Portgal b Departamento de Matemática, Universidade de Coimbra, Largo D. Dinis, Coimbra, Portgal Available online 4 Febrary 6 Abstract In tis paper we stdy nmerical metods for solving integro-differential eqations wic generalize te well-known Fiser eqation. Te nmerical metods are obtained considering te MOL Metod of Lines approac. Te stability and convergence of te metods are stdied. Nmerical reslts illstrating te teoretical reslts proved are also inclded. 6 IMACS. Pblised by Elsevier B.V. All rigts reserved. MSC: 35B35; 35K57; 65M6; 65M1 Keywords: Fiser eqation; Integro-differential eqation; Nmerical approximation; Stability 1. Introdction It is well known tat te diffsion approximation Fick s law to model reaction diffsion problems gives rise to te Fiser eqation t x, t = D x x, t + f x, t, x a, b, t >. 1 In a large nmber of biological and cemical penomena, te reaction term is represented by f= U1, were U>can be dependent of te space variable. Sc an eqation as te steady state soltions = and = 1, te first one being nstable and te second one stable. Te soltion of tis problem evolves into a traveling wave soltion connecting te two steady states wit a speed of propagation c = 4DU [1]. Wen te reaction is very fast, c becomes arbitrarily large. Tis npysical property can be corrected if memory effects are taken into accont in te matematical model. Tis leads to integro-differential eqations of type t x, t = D x x, s ds + f x, t, x a, b, t >, Tis work as been spported by Centro de Matemática da Universidade de Coimbra and POCTI/3539/MAT/. * Corresponding ator. addresses: jrbranco@isec.pt J.R. Branco, ferreira@mat.c.pt J.A. Ferreira, poliveir@mat.c.pt P. de Oliveira /$3. 6 IMACS. Pblised by Elsevier B.V. All rigts reserved. doi:1.116/j.apnm.6.1.

2 9 J.R. Branco et al. / Applied Nmerical Matematics wit D, >, wic ave been stdied for instance in [3 5]. Eq. is known as a generalized Fiser Kolmogorov Petrovskii Pisknov eqation, FKPP, and it is copled wit initial and bondary conditions of type x, = x, x a, b, a, t = a t, b, t = b t, t >. 3 Te parameter is a relaxation parameter and wen, te FKPP eqation is replaced by 1. Te existence and te beavior of soltions of Eq. wit f= U1, U >, and a Heaviside initial condition was considered in [3]. Different models presenting traveling wave soltions wit finite speed of propagation were considered in [6 8]. In tis paper we stdy properties of a class of nmerical metods tat approximate 3. In Section we establis an energy estimate tat improves te information given by te classical estimate known for te Fiser eqation. Tis new estimate enables s to conclde te stability of relative to pertrbations in te initial condition. In Section 3 we se te MOL approac to solve 3 nmerically by sing a nmerical approximation obtained combining te spatial discretization wit a time integration metod. A semi-discrete analoge of te estimate establised for te teoretical model is dedced for te semi-discrete approximation. In Section 4 flly discrete scemes are analyzed and a discrete version of te continos estimate is proved, see Teorem 6. As a conseqence of Teorem 6 te stability and te convergence of discrete scemes are also stdied. Nmerical experiments illstrating te teoretical reslts are presented in Section 5.. Energy estimates for te PDE In tis section we stdy te stability of te soltion of 3 wen te initial condition is pertrbed. Attending to tis fact we assme in Teorem 1 omogeneos Diriclet bondary conditions. Tese conditions are considered nonomogeneos in te rest of te section. Let, denote te inner prodct in L a, b and L te sal norm indced by,. Ifv is defined in [a,b] [,T] we represent v,tby vt. We establis in wat follows an estimate for te energy fnctional Et = t L + D x s ds, t,t]. 4 L Teorem 1. Let be a soltion of 3 wit a t = b t =, t>, satisfying for eac t [,T] x, t [c,d], x [a,b], t t, x s ds L [a,b], were c,d are constants. If f is continosly differentiable and f =, ten te energy E is sc tat 5 6 Et e max{ 1,f max }t L 7 for eac t,t], were f max = max max{ c, d } f. Proof. Mltiplying eac member of by wit respect to, and integrating by parts we obtain t t, t + D s ds, t x x t = f t,t. Considering tat d t dt L = t, t, t and tat

3 d dt we dedce d dt x t L + D s ds = L J.R. Branco et al. / Applied Nmerical Matematics t x s ds s ds, x x,t L = D x x s ds L, s ds + f t,t. 8 L De to te fact tat f =, we ave f t, t f max t L and ten from 8, we conclde te differential ineqality { d Et max 1 } dt,f max Et. 9 Integrating 9 we finally establis 7. and Under te assmptions of Teorem 1, if 3 as a soltion ten is niqe. Moreover satisfies t L e max{ 1,f max }t L, D x s ds L e max{ 1,f max }t L. 11 Let s consider now te classical Fiser eqation 1. It can be sown tat t L e f max t L 1 and no information is available abot x.btif represents te soltion of, we conclde from 11 tat te average in time of its gradient is bonded by e max{ 1,f max }t L, for eac time t,t]. In wat follows te stability beavior of nder pertrbations in te initial condition is considered. Let and ε be soltions of satisfying te same bondary conditions not necessarily omogeneos and initial conditions and + ε respectively. Ten ε, is a soltion of te initial-bondary vale problem t εx, t = D t s e x ε x, s ds + fx,t f ε x, t, x a, b, t,t], ε x, = εx, x a, b, 13 ε a, t = ε b, t =, t >. Te following stability reslt can be stated: Teorem. Let and ε be soltions of 3 wit initial conditions and + ε, respectively. If for, ε 5 and 6 old and te sorce fnction f is continosly differentiable and f =, ten 1 E ε t e max{ 1,f max }t ε L. 14 Proof. Mltiplying eac member of 13 by v ε = ε wit respect to te inner prodct, we obtain t vε t t, v εt + D v ε x s ds, v ε x t = f t f ε t,v ε t. As f f ε, v ε f max v ε L following te proof of Teorem 1 we conclde 14.

4 9 J.R. Branco et al. / Applied Nmerical Matematics In te main reslt of tis section, Teorem, we establis te stability of te initial-bondary vale problem 3 wit respect to pertrbations of te initial condition. In te case of f= U1 wit U>, we ave f max = U>and 14 enables s to conclde tat for eac t, te first member is bonded. If f max <, we obtain lim ε t t + L =, lim t + x εs ds =. 3. Energy estimates for te semi-discrete approximation In tis section we consider te MOL approac to compte a semi-discrete nmerical approximation t to te soltion of 3. Te approximation t is defined by introdcing a discretization of te spatial variable. Or aim is to establis a semi-discrete analoge of Teorems 1 and for t defined by Let s consider in [a,b] agridi ={x j,j=,...,n} wit x = a,x N = b and x j x j 1 =. We discretize te second partial derivative of wit respect to x in sing te second-order centered finite-difference operator D,x defined by D,x v x i = v x i+1 v x i + v x i 1. Te semi-discrete approximation t is a soltion of te following system of ODE s d dt t = A t, t,t], 15 were and A t i = D L D,x x i,sds + f x i,t, i = 1,...,N 1, x,t= a t, x N,t= b t, x i, = x i, i = 1,...,N We denote by L I te space of grid fnctions v defined in I sc tat v x = v x N =. In L I we consider te discrete inner prodct N 1 v,w = v x i w x i, v,w L I. 17 i=1 We denote by L I te norm indced by te above inner prodct. For grid fnctions w and v defined in I we introdce te notations N w,v,+ = w x i v x i, and Let i=1 N 1/ w L I + = w x i. i=1 v 1 = v L I + D xv L I + 1/, v L I, were D x denotes te backward finite-difference operator. We note tat it represents a norm wic can be viewed as a discretization of te Sobolev norm of te space H 1 a, b.

5 J.R. Branco et al. / Applied Nmerical Matematics Let E t be te semi-discrete version of Et defined by E t = t L I + D D x s ds, t >. L I + A semi-discrete analoge of Teorem 1 is ten establised in Teorem 3. Teorem 3. Let t beasoltionof wit a t = b t =, t>, and sc tat x i,t [c,d], for i =,...,N,and t [,T]. If te sorce fnction f is continosly differentiable and f =, ten for te energy E t olds, for eac time t in,t], E t e max{ 1,f max }t L I. 18 Proof. Mltiplying eac member of 15 by t wit respect to te inner prodct, and sing smmation by parts we obtain 1 d t t dt L + D D x s ds,d x t = f t, t, were f tx i = f x i, t, i = 1,...,N 1. Adapting te proof of Teorem 1 to te discrete case it can be sown tat te last eqality is eqivalent to 1 d t dt L I + D D x s ds = D t D x s ds L I + + L I,+ f t, t. As f t, t f max t, we easily conclde 18. L I A semi-discrete version of Teorem is stated in te next reslt: Teorem 4. Let t,,ε t be defined by wit initial conditions given respectively by x i, = x i and,ε x i, = x i + εx i, i =,...,N.If x i,t,,ε x i,t [c,d] for t [,T], i=,...,n, and te sorce fnction f is continosly differentiable and f =, ten E,ε t e max{ 1,f max }t ε L I. 19 Proof. Te difference v t = t,ε t satisfies te following initial-bondary vale problem { dv dt x i,t= D t s e D,x v x i,sds + f x i,t f,ε x i, t, i = 1,...,N 1, v x i, = εx i, i = 1,...,N 1, v x,t= v x N,t=. Replacing in te proof of Teorem 3, t by v t and f t, t by f t f,ε t, t and considering tat f t f,ε t, v t f max v t we conclde te proof. L I In Teorem 4 te stability of te semi-discrete approximation is establised. In wat follows we stdy te accracy of t. LetT t be te trncation error associated wit te spatial discretization defined by 15 and let e x i,t= x i,t x i,t, i =,...,N,be te spatial discretization error. We ave de dt x i,t= D D,x e x i,sds + f x i,t f x i,t + T x i,t, i = 1,...,N 1,

6 94 J.R. Branco et al. / Applied Nmerical Matematics and e x,t= e x N,t=, e x i, =, i = 1,...,N 1, wit T x i,t= D t s e T x i,sds. For te semi-discretization error e t olds te following reslt: Teorem 5. Let t be te soltion of sc tat x i,t [c,d] for t [,T], i=,...,n.ifte soltion of 3 satisfies 5 and te sorce fnction f is continosly differentiable and f =, ten te spatial discretization error satisfies Ee t D s e max{,f max +1}t s T μ L dμ ds. I Proof. Following te proof of Teorem 3 it can be sown tat 1 d dt Ee t D D x e s ds + f max e t L I + 1 T t L I + 1 e t L I. L I + From te last ineqality we obtain { 1 d dt Ee t max 1,f max + 1 } Ee t + 1 T t L I, tat is, d t e max{ 1,f max + 1 }t Ee t e max{ 1,f max + 1 }s T s dt L I ds. 1 Finally, as T s D L I s we conclde from 1. T μ L I dμ, Considering tat te spatial discretization is defined sing te operator D,x, te trncation error satisfies T t L C max I t,t ] 4 x 4 t = O, were C is a positive constant independent of and. Ten we conclde tat, for eac time t, e t L I + D D x e s ds = O 4 L I + and conseqently e t L I = O and D x e s ds = O. 3 L I +

7 J.R. Branco et al. / Applied Nmerical Matematics Being and t independent variables, we ave D x e t L I + = O, 4 wic to te best of or knowledge is a nonstandard estimate for te spatial discretization error even wen niform grids are sed. 4. Energy estimates for te fll discrete approximation Let s integrate te system of ordinary differential eqations 15 sing te implicit Eler metod in te time grid {t n,n=,...,m} sc tat t =, t M = T and t t n = t. We se te rectanglar rle to approximate te integral in 15. Te discretization of te reaction cold be implicit or explicit depending on te stiffness of te reaction. In te following we establis an estimate for te flly discrete version of 4, E = L I + D t l=1 e t t j D x l L I +, were j is obtained sing an implicit or explicit discretization of te reaction term. 1. Implicit discretization of te reaction term In tis case te flly discrete approximation of is defined by te nonlinear system of eqations were x j n x j t = D t l=1 e t t l D,x l x j + f x j, j = 1,...,N 1, 5 l x = a t l, l x N = b t l, l = 1,...,M 1, x j = x j, j = 1,...,N 1. 6 Teorem 6. Let l be defined by 5 6 wit at = b t =, t>, sc tat l x i [c,d], for i =,...,N, and l =,...,M. If te sorce fnction f is continosly differentiable and f =, ten L I + D t e t t j D x j 1 min{1, 1 tf max } L 7 I provided tat 1 tf max >. L I + Proof. a Let s consider in 5 n N. Mltiplying eac member of 5 by wit respect to te inner prodct, and sing smmation by parts we obtain As, e t t j = 1 we ave from 8 = n, D t D x j,d x,+ e t t j D x j 1 e L I + e t t j t n D x j,d x e tn t j,+ + t f D x j + 1 D x L I +,. 8 L I +, 9

8 96 J.R. Branco et al. / Applied Nmerical Matematics L I + D t = n, + t f e t t j D x j, + D L I + t e n t e tn t j D x j L I + D t D x L I +. 3 Considering in 3 te estimates n, 1 L I + 1 n L I, f, f max L I, we conclde 1 tf max L I + D t e t t j D x j L I + n L I + D t n e t e tn t j D x j. 31 L I + b We consider now in 5 n =. Following te proof of 31 we obtain 1 tf max 1 L I + D td x 1 L I + L I. 3 Finally from 31 and 3 we conclde 7. Te factor 1 S I = min{1, 1 tf max } represents te stability amplification factor. If f max < ten S I = 1 and from 7 we obtain L I + D t e t t j D x j L I. 33 L I + Oterwise if f max >, considering tat for t t we ave 1 f max f max S I = 1 tf max = tf max t t f max t, we conclde L I + D t e t t j D x j e β t L 34 I L I + wit f max β = 1 t f max.. Explicit discretization of te reaction term Let s consider now te IMEX sceme obtained by replacing in 5 f by f n, tat is, x j n x j t = D t l=1 e t t l D,x l x j + f n x j, j = 1,...,N 1. 35

9 J.R. Branco et al. / Applied Nmerical Matematics We remark tat wit ũ n = θn for θ [, 1], we ave t f n, = t f ũ n n, t n L I + tf max L I. Ten te stability coefficient S I is replaced by te stability coefficient S IMEX defined by 1 + t S IMEX = 1 tf max provided tat 1 tf max >. We ave S IMEX f max 1 t f max t and we can prove 34 wit β = 1 + f max 1 t f max. Let s stdy now te convergence of te approximation defined by 5, 6. Let e lx i = l x i x i,t l be te global error of te approximation l x i compted sing 5, 6, and let T lx i be te corresponding trncation error. Tese two errors are related by wit e x i = e n x i + D t e t t j D,x e j x i + f x i f x i,t + tt x i, i = 1,...,M 1 36 e x i =, i = 1,...,N 1, e l x = e l x N =, l= 1,...,M. Following te proof of Teorem 6 te next convergence reslt can be proved. Teorem 7. Let l be defined by 5 6 and sc tat l x i [c,d], for all i and for all l. If te soltion of 3 satisfies 5 and te sorce fnction f is continosly differentiable and f =, ten e L I + D t e t t j D x e j n Ŝ j+1 I t T j L 37 I wit Ŝ I = 1 min{1, f max t}. L I + Considering tat 5 is defined approximating te second-order spatial derivative sing centered differences, te integral term sing te rectanglar rle and te integration in time sing te Eler implicit metod, we ave T x i = t t x i,tn D D 4 t e t t j j t j 1 were t n [t n,t ],x i, x i [x i 1,x i+1 ]. Ten j= e t s s x x i,s s t j+1 ds 4 x 4 x i,t j + 4 x 4 x i,t j,

10 98 J.R. Branco et al. / Applied Nmerical Matematics T = max T l l C max t t,t ] t t + x t + 3 t x t + 4 x 4 t. 38 In te last ineqality C denotes a generic positive constant independent of, t and. Using 38 in Teorem 7 we conclde: Corollary 1. Under te assmptions of Teorem 7 and assming f max 1 ten e n L I + D n t e tn t j D x e j C T. 39 L I + If f max > 1 ten wit e n L I + D n t 1 + f max β = f max t. e tn t j D x e j Ce βn t T, 4 L I + Analogos convergence reslts can be establised for te IMEX metod. 5. Nmerical reslts In tis section we present some nmerical reslts tat sow te effectiveness of te estimates presented in Teorems 5 and A semi-discrete approximation Let s consider te semi-discrete system of ordinary differential eqations 15. In order to avoid te integral term and as or aim is to illstrate te beavior of te spatial discretization we rewrite 15 in te following form { dv dt x i,t= 1 v x i,t+ x i,t, i = 1,...,N 1, d dt x i,t= D D 41,xv x i,t+ f x i, t, i = 1,...,N 1, wit te initial bondary conditions v x i, =, i = 1,...,N 1, v x i,t= t s e x i,sds, i =,N, 4 x,t= a t, x N,t= b t, x i, = x i, i = 1,...,N 1. To illstrate te second-order estimates in space and 4 we integrate in time 41 wit a fort-order Rnge Ktta metod. Te nmerical reslts obtained wit x = e x 5, x [, 5], f= U1, U = 1, D =., =.1 and t =.5 are presented in Table 1. Te estimates for te orders p and p exibited in tis table were compted sing p = log maxj=,...,m e j 1 L I1 max j=,...,m e j L I log 1 and p = log max j=,...,m D x e j 1 L I + 1 max j=,...,m D x e j L I + log 1, respectively, were 1 and represent different space step-sizes. We considered M = 5 and te error was estimated sing a reference soltion compted wit t =.1,=.5.

11 J.R. Branco et al. / Applied Nmerical Matematics Table p p Fig. 1. Nmerical soltions compted wit metods 5 and 35 for U = 1, = D =.1 and t = =.1. Fig.. Nmerical soltions compted wit metods 5 and 35 for U = 1, = D =.1 and t = = A flly discrete approximation We present in wat follows some nmerical reslts tat illstrate te qalitative and stability properties of metods 5 and 35. Te comptational experiments ave been obtained wit a reaction term of type f= U1, and wit te initial condition { 1, x [, 5], x =, x ]5, 1].

12 1 J.R. Branco et al. / Applied Nmerical Matematics Fig. 3. Nmerical soltions compted wit metods 5 and 35 for U = 1, =.1, t= =.1 andd =. Fig. 4. Nmerical soltion compted wit metod 5 for U = 1, D =.1, t = =.1 and =.1. In Fig. 1 we plot te nmerical approximations obtained sing metod 5 and metod 35 wit U = 1, =.1 = D =.1 and t = =.1. Te two nmerical soltions exibit te same stability beavior, bt as we can see in Fig. te speed of te nmerical soltion obtained wit metod 5 is greater. In Fig. 3 we plot te nmerical approximations obtained for D =. As expected, we observe in Figs. 1 and 3 tat increasing diffsion leads to a smooter soltion. Te nmerical approximation obtained from 5 wit D = 1 and =.1 is plotted in Fig. 4. Te plots presented in Figs. and 4 illstrate te fact tat te generalized FKPP eqation is replaced by te classical Fiser eqation wen.

13 J.R. Branco et al. / Applied Nmerical Matematics Fig. 5. Nmerical soltions compted wit metods 5 and 35 for D = =.1 = t = =.1 andu = 5. We ave sown in Section 4 tat if te reaction term f is stiff, ten metod 5 is more stable ten metod 35. Tis beavior is illstrated in Fig. 5 were we plot te nmerical soltion obtained wit te previos metods for U = 5 and = t = = D =.1. As can be observed, te nmerical soltion obtained wit metod 35 presents an nstable beavior. Finally we remark tat wen time t increases te discretization of te integral term needs more and more comptational memory and te metod can become very expensive. In order to avoid tis drawback, metod 5 can be rewritten in te following eqivalent form: I D t D,x x i tf x i = 1 + e t I D t n x i te t f n x i e t n 1 x i, n = 1,...,M 1, D,x 1 x i tf 1 x i = x i. 43 De to te discretization of te memory term te IMEX metod 35 can also be comptationally expensive. In order to avoid tis limitation metod 35 can be rewritten in te following eqivalent form: I D t D,x x i n x i + tf n x i + te t = 1 + e t I D t f n 1 x i e t n 1 x i, n = 1,...,M 1, D,x 1 x i = i + tf x i. 44 Finally we remark tat in [], [9] and [1] were considered different metods for eqations of type. References [1] D.G. Aronson, H.F. Weinberger, Mltidimensional nonlinear diffsion in poplation genetics, Adv. Mat [] J. Doglas, B.F. Jones Jr, Nmerical metods for integro-differential eqations of parabolic and yperbolic type, Nmer. Mat [3] S. Fedotov, Traveling waves in a reaction diffsion system: Diffsion wit finite velocity and Kolmogorov Petrovskii Pisknov kinetics, Pys. Rev. E

14 1 J.R. Branco et al. / Applied Nmerical Matematics [4] S. Fedotov, Nonniform reaction rate distribtion for te generalized Fiser eqation: Ignition aead of te reaction front, Pys. Rev. E [5] S. Fedotov, Front propagation into an nstable state of reaction transport systems, Pys. Rev. Lett [6] V. Mendez, J. Camaco, Dynamic and termodynamics of delayed poplation growt, Pys. Rev. E [7] V. Mendez, J. Llebot, Hyperbolic reaction diffsion eqations for a forest fire model, Pys. Rev. E [8] V. Mendez, T. Pjol, J. Fort, Dispersal probability distribtions and te wave front speed problem, Pys. Rev. E [9] I.H. Sloan, V. Tomée, Time discretization of an integro-differential eqation of parabolic type, SIAM J. Nmer. Anal [1] N.-Y. Zang, On flly discrete Galerkin approximation for partial integro-differential eqations of parabolic type, Mat. Comp