Numerical Study of One Dimensional Fishers KPP Equation with Finite Difference Schemes

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1 Aerican Journal of Coputational Matheatics, 07, 7, ISSN Online: 6- ISSN Print: 6-03 Nuerical Study of One Diensional Fishers KPP Equation with Finite Difference Schees Shahid Hasnain, Muhaad Saqib, Daoud Suleian Mashat Departent of Matheatics, Nuerical Analysis, King Abdulaziz University, Jeddah, KSA How to cite this paper: Hasnain, S., Saqib, N. and Mashat, S. (07) Nuerical Study of One Diensional Fishers KPP Equation with Finite Difference Schees. Aerican Journal of Coputational Matheatics, 7, Received: Deceber 30, 06 Accepted: March 8, 07 Published: March 3, 07 Copyright 07 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Coons Attribution International License (CC BY.0). Open Access Abstract In this paper, we originate results with finite difference schees to approxiate the solution of the classical Fisher Kologorov Petrovsky Piscounov (KPP) equation fro population dynaics. Fisher s equation describes a balance between linear diffusion and nonlinear reaction. Nuerical exaple illustrates the efficiency of the proposed schees, also the Neuann stability analysis reveals that our schees are indeed stable under certain choices of the odel and nuerical paraeters. Nuerical coparisons with analytical solution are also discussed. Nuerical results show that Crank Nicolson and Richardson extrapolation are very efficient and reliably nuerical schees for solving one diension fisher s KPP equation. Keywords Forward in Tie and Centre in Space (FTCS), Lax Wendroff, Taylor s Series, Crank Nicolson and Richardson Extrapolation. Introduction Fisher gives introduction to nonlinear evolution equation to inquisitive, the proliferation of an beneficial gene in a population dynaics []. Fisher s equation also specify the logistic diffusion process []. It has the for u βu + αu u () t xx ( ) where β > 0 is a diffusion constant with α > 0 is the linear growth rate []. The reaction diffusion Equation () also express a odel equation for the evolution of a neutron population in a nuclear reactor [] and also appears in cheical engineering applications []. This equation accoodates the effects of linear diffusion along u xx and nonlinear local ultiplication or reaction along u( u) [3] []. It has becoe one of the ost iportant nonlinear equations and occurs in any biological and cheical processes [] [5]. Recently, any researcher are working on this type of odel to understand growth rate DOI: 0.36/ajc March 3, 07

2 and diffusion aspect, for exaple, Abdullaev has studied the stability of syetric travelling waves in the Cauchy proble for a ore general case than Equation () [6]. Also Logan has studied this proble using a perturbation ethod and settled up with an approxiate solution by expanding the solution in ters of a power series and in ters of soe sall paraeter [7]. The finite difference schees and auxiliary conditions of the nuerical odel ust be consistent with the partial differential equations and initial and boundary conditions of the atheatical odel [8] [9]. The nuerical odel is consistent if the truncation error, that is the discrepancy between the finite difference approxiation and the continuous derivatives, tends to zero as the grid spacing get saller and saller. Secondly the solution to finite difference schees ust converge to the solution of the partial differential equations as the grid spacing gets saller and saller [9]. Thus we can say that, difference between the exact solution and approxiated solution ust vanish as the grid spacing tends to zero. In this paper, we started the solution of Fishers equation with various finite difference schees. We discussed Forward in tie and Central in space (FTCS) and Lax-Wendroff schees, which are explicit and conditionally stable. Also FTCS is first order accurate in tie and second order accurate in space and Lax-Wendroff is second order accurate in both space and tie, which shows iproveent in accuracy in later ethod. Crank-Nicolson schee is unconditionally stable and iplicit with second order accuracy in both space and tie [9]. This applies coputational stability for any size of the tie increent [9]. However, the size of t is still liited by accuracy required in the solution. Usage of very large values of tie step results in poor approxiation to solution because of unacceptably large truncation errors produced [0]. We applied Richardson extrapolation ethod to iprove accuracy with no issue of stability. We cae to know this ethod is highly accurate and easy to ipleent with no ess of coputations [0] [] and we can see fro our results that this ethod is excellent agreeent to the exact solution.. Exact Solution x Ω with t 0. To derive the exact solution of the given Equation (), we have the following exact solution [0] [] to above equation, Let us consider Equation (), within doain (, ) ( ) D α 6x 56αt D u( xt, ) ( + e ) subject to initial condition, D α 6x D u( x,0) ( + e ) Boundary Conditions, t > 0, x Ω u( 0, x) u0 ( x)} initial Conditions, x Ω. () 7

3 3. Nuerical Methods We consider the nuerical solution of the non-linear Equation () in a finite doain Ω. The first step is to choose integers n to define step sizes h ( b a) n. Partition the interval [, ] ab into n equal parts of width h. Place a grid on the rectangle R by drawing vertical and horizontal lines through the points with coordinates ( x i ), where x i a + ih for each i 0,,, n also the lines x xi represent grid lines, also we assue tn nt, n 0,, where t is the tie grid step size. We denote the exact and nuerical solutions at the n n grid point ( x, t n) by u and U respectively. We consider four finite different schees as we entioned in keywords in abstract. 3.. FTCS Schee We consider forward in tie and center in space (FTCS) explicit schee by substituting the forward difference approxiation for the tie derivative and the central difference approxiation for the space derivative in Equation (), which leads to the following, u xx n u u i n n u + i ui ut k ui ui + ui h n n n + ( + ) ( ) u u + R u u + u + ku u n n n n n n i i i i i i i k where R, final for of above equation is, h ( ) ( + ) u u + k R ku + R u + u. (3) n n n n i i i i i Since the one diensional diffusion reaction equation is nonlinear and wellposed [0] [], Lax s equivalence theore indicates that consistency and stability of the FTCS finite difference approxiation is necessary and sufficient for FD solution to converge to diffusion reaction equation []. Once convergence has been proved, the solution to the given partial differential equation can be obtained to any desired degree of accuracy []. Make sure the spacing h for spatial and k for tie of the finite difference grid are ade sufficiently sall. The FTCS schee, fro Equation (3), is classified as explicit because the value n of u + i at the ( n + ) th tie level ay be calculated directly fro known n value of u i at previous tie levels. It is a two level ethod because values of u at only two levels of tie are involved in the approxiating finite difference equation []. Accuracy and Stability to FTCS Schee: Accuracy: To find accuracy of the FTCS schee for Fisher-KPP equation, we apply Taylor s series on each ter of the equation. Let us consider the Taylor s series in the following way, 7

4 n ut ui ui ut u( xt, + k) u( xt, ) Apply Taylor s series 3 k k ut kut + utt + uttt + 6 n n n uxx ui+ ui + ui uxx u x+ h t u x t + u x h t Apply Taylor s series h ux huxx uxxx + (, ) (, ) (, ) Take Equation () into account, and apply this schee with Taylor s series on each ter, updated equation is as follows. ( ) ( ) R ( ( ) ( ) u( x h, t)) ku( x, t) ( u( x, t) ) Eq. u x, t+ k u x, t u x+ h, t u x, t + Eq. u u u u k + u k u h + u k 6 Ruxxxx h + 3 ( t xx ( )) tt R xx ttt Now principle part of the truncation error is along with Equation (). So first part of the above equation goes to zero if we consider Equation (). PPTE of u utt Ruxx h + utttk Ruxxxx h +. 6 Which shows that this schee is first order accurate in tie and nd order Okh. accurate in space, such as (, ) Stability: We want to study under what condition the error can be agnified. Many ethods can be used to study this issue. We consider only Von-Neuann stability analysis to explain this ethod on FTCS schee. Consider the schee in the following way, Linear for of the above equation is, ( ) u + u + R δ u + ku u. n n n n n x ( ) n n n n u + u + R δ xu + ku Constant. () According to Von-Neuann stability analysis, let us consider the solution as: n αnk iβh U e e. (5) The Von-neuann stability condition is k e α. (6) Note that, α ( ) k iβ h n α ( n ) k iβ h n α( nk ) iβ( + ) h + n α( nk ) iβ( ) h. U e e U e e U e e U e e 73

5 Also n β h αnk iβh δ xu sin e e β h α ( ) k iβ h δ xu sin e e. Apply above ters to Equation (), we get the following e αk According to Equation (6), β h R sin + k( Constant ). (7) e αk. Take left hand side of above equation along Equation (7), β h β h ( ) R sin + k Constant ( ) R sin k Constant β h k ( Constant) R If we choose Constant 0 ax sin k R. Take right hand side of above equation along Equation (7), β h + + k ( Constant) R If we choose Constant 0 + k R. ( ) R sin k Constant Since R k h, according to Von-Neuann stability analysis on both sides as left and right, which shows that FTCS schee is conditionally stable for Fisher-KPP equation. 3.. Lax Wendroff Schee A nuerical technique proposed in 960 by P.D. Lax and B. Wendroff [3] [] [5] for solving partial differential equations and syste nuerically. In spite of the ipressive developents on nuerical ethods for partial differential equations fro 970s onwards, in which the Lax Wendroff ethod has played a historic role, there are presently (998) substantial research activities aied at further iproveents of ethods [5]. Lax Wendroff s ethod is also explicit ethod but needs iproveent in accuracy in tie. This ethod is an exaple of explicit tie integration where the function that defines governing equation is 7

6 evaluated at the current tie [5]. Purpose of this ethod to achieve good enough accuracy in tie. This ethod can be explained in two steps which is as follows, nd step is n n n n n n n ( + ) R( + ) α ( ) u 0.5 u + u 0.5 u u + u ku u i i i i i i i i ( + ) 0.5α ( ) u u + R u u + u + ku u i i i i i i i Accuracy and Stability to Lax-Wendroff Schee: Accuracy: To find accuracy of the Lax-Wendroff schee for Fisher-KPP equation, we apply Taylor s series on each ter of the discritized schee. Let us consider the following, n ut ui ui ut u( xt, + k) u( xt, ) Apply Taylor s series 3 k k ut kut + utt + uttt + 6 above entioned schee with Taylor s series, we have Eq. ( 0.5ut 0.5u ( u) 0.5uxx ) k + uttk 0.5uxxh + 8 Now principle part of the truncation error along with Equation (). So first part of the above equation goes to zero if we consider Equation (). PPTE of u uk tt 0.5uxxh + 8 Which show that this schee is second order accurate in tie and space, such O k, h. as ( ) Stability: We consider Von-Neuann stability analysis to explain on Lax-Wendroff schee. Consider the schee in the following way, n n n n n ( + ) Rδ α ( ) u 0.5 u + u u ku u i i x Linear for of the above equation is, ( + ) n n n n 0.5 i + i + 0.5Rδx + 0.5α u u u u ku ( Constant) According to Von-Neuann stability analysis as (5), (6) Note that, n n + n α ( ) k iβ h U e e α ( n ) k iβ h U e e α( nk ) iβ( + ) h U e e α( nk ) iβ( ) h U e e Also n β h αnk iβh δ xu sin e e β h α ( ) k iβ h δ xu sin e e 75

7 Apply above ters to Equation (), we get the following β h e R sin e + αk( Constant) e αk αk αk αk ( )( ( h )) e α α β sin (8) k According to above result, the Von-Neuann stability criterion e α is satisfied as long as α, or equivalently, as long as the CFL condition is satisfied. In this respect, the dissipative properties of the Lax-Friedrichs schee are not copletely lost in the Lax-Wendroff schee but are uch less severe [5]. The Lax-Wendroff schee is a two-level schee, but can be recast in a one-level for by eans of algebraic anipulations [5]. This is clear fro expressions where quantities at tie-levels n and n + only appear [5] Crank Nicolson Schee Let us apply an iplicit ethod to iprove the accuracy. The iplicit ethod we consider, is Crank Nicolson. By substituting the value of u n by average value of esh points, the Crank Nicolson ethod is a finite difference ethod used for nuerically solving linear and nonlinear partial differential equations [6] [7] [8]. It is iplicit in tie and can be written in for of an iplicit Runge Kutta ethod, and it is nuerically stable [6] [7] [8]. The ethod was developed by John Crank and Phyllis Nicolson in the id 0th century [8]. For diffusion equations (and any other equations), it can be shown that, the Crank Nicolson ethod is unconditionally stable [9]. However, the approxiate solutions can still contain (decaying) spurious oscillations if the ratio of tie step k ties to the square of space step h, is large (typically larger than / per Von-Neuann stability analysis). For this reason, whenever large tie steps or high spatial resolution is necessary, the less accurate backward Euler ethod is often used, which is both stable and iune to oscillations [0]. In this ethod, consider the followings, n n ui + ui ui n ui ui ut k n ui ui uxx x h δ + discretization to one diensional Fisher KPP equation, R( + + ) n n ( i i )( i i ). u u u u + u + u u + u n n n n i i i i i i i i + αk u + u u u Accuracy to Crank Nicolson Schee: To find accuracy of the CN schee for Fisher-KPP equation,we apply Taylor s series on each ter of the discritized schee. Let us consider the following, 76

8 which iplies n ut ui ui ut u( xt, + k) u( xt, ) Apply Taylor s series 3 k k ut kut + utt + uttt + 6 n n n uxx ui+ ui + ui + ui+ ui + ui uxx u x+ h t + k u x t + k + u x h t + k Apply Taylor s series (, ) (, ) (, ) u( x h, t) u( x, t) u( x h, t) h hk hk xx xx xxt xxxx xxtt xxxxt u hu + khu + u + u + u + 6 ( t α( ) xx ) ( tt xxt α t α t )( ) Eq. u 0.5u u u k + 0.5u 0.5u uu 0.5 u u k 3 kh uxxxx + uttt + k + 6 Now principle part of the truncation error along with Equation (). So first and second part of the above equation goes to zero if we consider Equation (). PPTE of u uttt k huxxxx Which show that this schee is second order accurate in tie and space, such as O( k, h ). Stability: According to Von-Neuann stability analysis, we have linear for of the Equation () is, n n n n k n n u + R α u ( δxu + + δxu) + ( u + + u)( Constant) According to Von-Neuann stability analysis as (5), (6), after soe siplifications, we reached at, ( β h ) ( β h ) αk sin e. (9) + R sin Hence the Von-Neuann stability condition is satisfied for all positive values of R. This shows that CN schee is unconditionally stable. 3.. Richardson Extrapolation Technique Techniques for obtaining ore accurate solutions by using finite difference ethods which have already been discussed, including reducing the truncation error by using central differences, using ethods which iniize nuerical daping and dispersion as well as using ore accurate approxiation to derivative boundary conditions [] []. Other ethods include, ) reducing the size of grid spacing, ) using higher order difference approxiation, and 77

9 3) reducing the discretisation error by extrapolation (Richardson). Richardson extrapolation ethod, which was introduced by Richardson (90) and later on called deferred approach to the liit, ay lead to considerable iproveent of nuerical results which solving the partial differential equation syste by finite difference ethod. Applications of this ethod to equations solved on rectangular regions in space are described in Richardson and Gaunt (97) and Salvadori (95) []. The reduction of the error depends upon there being a reliable estiate of the discretisation error as a function of the grid spacing h. Let φ be the exact solution of the partial differential equation and φ, φ the finite difference solutions at a particular point in the solution doain coputed using grid spacing h, h respectively [] [3]. Richardsons extrapolation forula is as follows, where N ( ) h N N h ( j ) ( j ) h j ( ) ( j ) + ( j ) N h N j h is the approxiations which is even ore accurate then the previous N j and j, 3,. In our case we use j to get the approxi- N h with truncation error O( h ). This gives ore accurate and precise result which lead to convergence [3]. This ethod iproveent the results upto fourth order accuracy in space. Keep in ind, accuracy in tie still upto second order. Also this ethod hold the stable results. Clearly, this is a useful way of increasing the accuracy of the result at a given grid point without having to excessively increase the nuber of calculations required. This ethod ay be of uncertain value near the boundaries which are not straight, such type of probles can be discussed with ixed boundaries with interpolation technique [3]. ation forula ( ). Results Nuerical coputations have been perfored by using the unifor grid [3]. For the test proble, the approxiated and exact values of u( xt, ) and U( xt, ) have been given in Figure with soe fixed paraeters such as α, and k using FTCS. Figure shows the results taken at different tie to aintain stability. Figure 3 shows results by using Crank Nicolson schee; coparison has been done with existence of exact solution. Figure represents results fro Richardson Extrapolation. These values has been taken at soe typical grid point (TGP) as we presents in Table and Table. Soe critical values can be seen fro the following data Table and Table at different location along x-axis. In Table, we copare our results, calculated fro Crank-Nicolson schee with exact solution. Error can be seen fro last colun. 5. Discussion In this paper, the solution of the Fishers equation is successfully approxiated by a various nuerical finite difference schees. Two of the are explicit in nature such as FTCS and Lax-Wendroff. We have to pay attention to paraeter R, 78

10 S. Hasnain et al. Figure. Figure shows results with FTCS, by fixing soe paraeters to be in range of stability. Due to explicit nature of the schee, we choose h and k in such a way that r. Figure. Figure shows results with Lax-Wendroff, by fixing soe paraeters to stabilize. Due to explicit nature of the schee, we choose h and k in such a way that r with acceleration in coputations. Results at different tie levels, also copared with exact solution. which can stabilize the results as we can see fro Figure and Figure. Such schees on nonlinear one diensional PDE, consistency, stability and convergence are ore difficult to prove [] [5] [6]. For instant, Von-Neuann s 79

11 S. Hasnain et al. Figure 3. Shows results with iplicit Crank Nicolson, different α and tie level along coparison with exact solution. Figure. Shows results with Richardson Extrapolation, different tie levels along coparison with exact solution. Table. Estiates the results at different locations with Crank Nicolson schee α Locations uapproxiated U Exact Erroru U e 0

12 Table. Estiates results fro Richardson-Extrapolation with exact solution. Error can be seen fro last colun α Locations u U approxiated Exact Error u U e e e e ethod of stability analysis can not be used other than locally, since it is only applied to linear finite difference schees. In any cases, nuerical experientation, such as solving the finite difference schees by using progressively saller grid spacing and exaining the behaviour of the sequence of the values of u( xt, ) obtained at given points, is the suitable ethod available with which to assess the nuerical odel [6]. The various ethods of obtaining a finite difference nuerical odel corresponding to a particular atheatical odel ay result in either explicit or iplicit finite difference schees. Explicit schees are conditionally stable and iplicit schees are unconditionally stable [6]. Two iplicit schees are also applied to iprove accuracy, stability restrictions and consistency in solution. It can be observed that the coputed results show excellent agreeent with the exact solution. Our ain purpose of this research to iprove accuracy in result of Fisher KPP equation. At the end, we introduce Richardson extrapolation ethod; we observe iproveent in accuracy and this ethod also stable and show excellent agreeent with the exact solution. Accuracy in results are glanced fro figures and tables. Acknowledgeents The authors are gratefully acknowledge for support by Dr Muhaad Fahee Afzaal, Departent of Cheical Engineering, Iperial College London and Vineet K. Srivastava, Scientist, ISTRAC/ISRO, Bangalore, India in understanding atheatical ters and algoriths. Conflict of Interest The authors do not have any conflict of interest in this research paper. References [] Fisher, R.A. (936) The Wave of Advance of Advantageous Genes. Annals of Eugenics, 7, [] Canosa, J.C. (973) On a Nonlinear Diffusion Equation Describing Population Growth. IBM Journal of Research Developent, 7, [3] Arnold, R.A., Showalter, K. and Tyson, J.J. (987) Propagation of Cheical Reac- 8

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