Nuerical Studies of a Nonlinear Heat Equation with Square Root Reaction Ter Ron Bucire, 1 Karl McMurtry, 1 Ronald E. Micens 2 1 Matheatics Departent, Occidental College, Los Angeles, California 90041 2 Physics Departent, Clar Atlanta University, Atlanta, Georgia 30314 Received 12 Decber 2007; accepted 23 February 2008 Published online in Wiley InterScience (www.interscience.wiley.co). DOI 10.1002/nu.20361 Interest in calculating nuerical solutions of a highly nonlinear parabolic partial differential equation with fractional power diffusion and dissipative ters otivated our investigation of a heat equation having a square root nonlinear reaction ter. The original equation occurs in the study of plasa behavior in fusion physics. We begin by exaining the nuerical behavior of the ordinary differential equation obtained by dropping the diffusion ter. The results fro this sipler case are then used to construct nonstandard finite difference schees for the partial differential equation. A variety of nuerical results are obtained and analyzed, along with a coparison to the nuerics of both standard and several nonstandard schees. 2008 Wiley Periodicals, Inc. Nuer Methods Partial Differential Eq 00: 000 000, 2008 Keywords: Micens discretization; nonstandard finite difference schee; nonlinear heat equation; nuerical solutions; positivity I. INTRODUCTION We becae interested in the topic of this article after reading a paper by Wilhelsson et al. [1] on the study of a highly nonlinear parabolic partial differential equation. This equation odels the plasa physics of a burning fuel for the generation of energy by eans of nuclear fusion. The particular for of this equation they considered can be expressed as: T t = 1 2 (T 5/2 ) + 1 (T 5/2 ) + (1 r 2 )(at 2 bt 1/2 ) (1.1) 10 r 2 10r r where a and b are positive paraeters, and the boundary conditions are T(1, t) = 0, T(0, t) <. (1.2) Correspondence to: Ron Bucire, Matheatics Departent, Occidental College, 1600 Capus Road, Los Angeles, CA 90041 (e-ail: ron@oxy.edu) Contract grant sponsor: Departent of Energy 2008 Wiley Periodicals, Inc.
2 BUCKMIRE, McMURTRY, AND MICKENS The variable T is the absolute teperature and therefore satisfies the positivity condition T(r, t) 0 for 0 r 1 and t 0. The initial condition can tae any fors; a realistic analytic possibility is T(r,0) = A(r + B)(r 1) 2 (1.3) where A>0 and 0 <B<1. It should be noted that Eq. (1.1) has both nonlinear diffusion and reaction ters. Further, the T 1/2 ter, in the reaction function, appears with a negative coefficient and, as a consequence, gives rise to dissipation. Prior to now, our efforts [2 7], along with those of Pedro Jordan [8], have not been successful in constructing positivity-preserving nonstandard finite difference schees for the full equation given in Eq. (1.1). Thus, in order to better understand the dynaics of Eq. (1.1), we undertae in this article the study of soe related, siplified differential equations having only the square root ter. The first toy equation to be exained is the first-order, nonlinear ordinary differential equation dt dt = λt 1/2, T(t 0 ) = T 0. (1.4) where λ is a positive paraeter. This equation neglects both nonlinear diffusion and the a(1 r 2 )T 2 ter in the reaction function. Next, we exaine a nonlinear partial differential equation having linear diffusion but also containing the square-root ter, i.e., where D>0, T t = D 2 T x 2 λt 1/2, (1.5) T(x,0) = f(x) and T(0, t) = T(1, t) = 0. and f(x)is a given initial condition. Our ajor reason for studying these two toy equations is the belief that their analysis can provide fundaental understandings into how one should proceed with the construction of finite difference schees for Eq. (1.1). In the next section, we construct several nonstandard finite difference schees for Eq. (1.4). The corresponding nuerical solutions are obtained and copared to both a standard discretization and the exact solution. An iportant feature of the exact solution is that fro an initial positive value at t = 0, the solution goes to zero in finite tie. Because the exact solution is nown, an exact finite difference schee can be written down and this will allow us to also ae coparisons between the nuerical solutions of our constructed discretizations and the nuerical values fro the exact solution. Thus, by studying reasonable nonstandard finite difference schees for Eq. (1.4) and carrying out coparative analyses to the exact nuerical solutions of this equation, valuable insight can be obtained as to how one should select the best nuerical approxiation techniques for when the differential equation being solved does not possess a nown exact solution. One exaple of such an equation is the nonlinear partial differential equation given in Eq. (1.5). In the third section, we show how to construct positivity preserving schees for Eq. (1.5) and carry out nuerical experients for a particular initial and boundary value proble. We also find an iportant functional relationship between the tie and space tie-sizes. Nuerical results for different approxiations are also given. Finally, in the last section, we present a suary of our results detailed earlier in the paper. We also present plans for future wor. Nuerical Methods for Partial Differential Equations DOI 10.1002/nu
NUMERICAL STUDIES OF A NONLINEAR HEAT EQUATION 3 Certain expressions are used frequently in this paper. Therefore, it is useful to use the following abbreviations: FD: IVP: NSFD: ODE: PDE: finite difference initial value proble nonstandard finite difference ordinary differential equation partial differential equation. II. THE SIMPLIFIED ODE In this section of the article we analyze the siplified ODE found in Eq. (1.4). It is a separable, first-order ODE and an explicit expression can be found for its solution. In Section IIA, we give this solution. Next, in Section IIB we derive the exact FD schee, a standard FD schee and several NSFD schees. Section IIC presents a short coparative analysis of the nuerical solutions for these discretizations. A. Exact Solution With T 0 > 0, the IVP can be solved to give dt dt = λt 1/2, T(t 0 ) = T 0, (2.1) T(t)= 1 4 [ 2T 1/2 0 λ(t t 0 ) ]2. (2.2) Because dt 0, it follows fro Eq. (2.2) that T(t)reaches zero at soe tie t, i.e. T(t ) = 0, dt and that for t>t, it ust be that T(t) = 0. The value for t is easily obtained fro Eq. (2.2) and is given by ( ) 1 [2T t 1/2 ] = 0 + λt 0, (2.3) λ or for t 0 = 0, 1/2 t 2T0 = λ. (2.4) As a consequence of these results, we conclude that the exact solution to the IVP given by Eq. (2.1), is T(t)= { 1 4 [ 2T 1/2 0 λ(t t 0 ) ]2, 0 t 0 t<t 0, t t. (2.5) For Eq. (2.1), T(t)= 0 is a singular solution and the coposite solution given in Eq. (2.5) is also a classical solution since both it and its first derivative are continuous. For ore details, see Kaplan [9, pp. 328]. Nuerical Methods for Partial Differential Equations DOI 10.1002/nu
4 BUCKMIRE, McMURTRY, AND MICKENS Note that (tae t 0 = 0) t is the tie scale for the proble. By choosing T 0 as a scaling for T(t), a scaled version of Eq. (2.1) can be written as ds ds = 2S1/2, S(0) = 1, (2.6) where S = T T 0 and s = t t and the exact solution to the scaled proble given in (2.6) is S(s) = { (1 s) 2, 0 s<1 0, s 1. (2.7) An interesting observation is that Eq. (2.1) can also be regarded as a special case of Chrystal s equation which arises in nonlinear poroacoustics. See, for exaple, Section VI6 of Jordan [10]. B. Discretizations An exact FD schee for the siplified ODE can be constructed fro the general solution given in Eq.(2.5). This involves discretizing the exact solution given in Eq.(2.5) by applying the following transforations: t t +1 t 0 t T 0 T T(t) T +1 where t = h, h = t, and T = T(t ). The resulting exact standard FD schee can then be obtained as follows: and this can be rewritten as T +1 = 1 4 = 1 4 = 1 4 [ 2T 1/2 λ(t +1 t ) ] 2 [ 2T 1/2 λh ] 2 [ 4T 4T 1/2 λh + λ 2 h 2] T +1 = T (λh)t 1/2 + λ2 h 2 4, T +1 T h = λt 1/2 + λ2 h 4. (2.8) Observe that in the above expression an extra ter appears on the right-side copared to the standard forward-euler approxiation of (2.1) which is T +1 T h = λt 1/2. (2.9) Nuerical Methods for Partial Differential Equations DOI 10.1002/nu
NUMERICAL STUDIES OF A NONLINEAR HEAT EQUATION 5 A first NSFD schee [4, 5, 7] can be derived by anipulating the right-side of (2.1), i.e. writing it as dt dt and then discretizing this expression to give = λt 1/2 = λ T T 1/2 (2.10) T +1 T h ( ) T +1 = λ. (2.11) T 1/2 Solving for T +1 gives ( ) T 1/2 T +1 = T λh + T 1/2. (2.12) Since λh > 0 the ter in the bracet is always less than one in agnitude and thus it follows that 0 T +1 T. (2.13) and it can be concluded that the solution to Eq. (2.12) onotonically decreases to zero. A second NSFD schee can be constructed by use of the following discretization ( ) T 1/2 2T+1 T 1/2, (2.14) T +1 + T which gives T +1 T h ( ) = λt 1/2 2T+1. (2.15) T +1 + T This equation is quadratic in T +1. Solving for the non-negative solution gives the expression T +1 = (λh)t 1/2 + T 2 + (λh)2 T, (2.16) and this can be re-written as T +1 T h = λt 1/2 + { T 2 + (λh) 2 T T h }. (2.17) In the nuerical calculations, the FD schees in Eqs. (2.12) and (2.16) were denoted as NSFD(1) and NSFD(2). C. Nuerical Experients We now have four FD schees which can be used to obtain nuerical solutions to the IVP given in Eq. (2.1). They are (i) the exact schee, Eq. (2.8); (ii) the standard schee, Eq. (2.9); (iii) NSFD(1), the nonstandard schee of Eq. (2.12); and (iv) NSFD(2), the nonstandard schee of Eq. (2.16). In the nuerical experients, the following paraeter values were selected: t 0 = 0, T 0 = 1, λ = 1, and h = W/N where N = 100 and W is the axiu value of the t variable; thus Nuerical Methods for Partial Differential Equations DOI 10.1002/nu
6 BUCKMIRE, McMURTRY, AND MICKENS FIG. 1. Coparison of NSFD(1), NSFD(2), the standard schee, and the exact schee for Eq. (2.1). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.co.] W = O(1) and, in general, was chosen to be W = 4 for our nuerical siulations. Note that for these choice of paraeter values, t = 2. Inspection of Figs. 1 and 2 allows the following conclusions to be ade: (i) All four FD schees give good nuerical representations of the actual solution to Eq. (2.1). (ii) The largest nuerical errors occur in the NSFD(1). (iii) The error in the NSFD(2) and standard FD schees are essentially the sae except for t values near t = 2. (iv) All schees give a nuerically zero solution for t greater than about t. Note that the standard schee goes to zero (at least coputationally) at t = t, while NSFD(2) does so at a slightly higher value than t, and NSFD(1), the worst of the three schees, achieves zero for its solution at a still larger value of t. Thus, in ters of accuracy, the three schees are raned as follows: standard (ost accurate), NSFD(2), and NSFD(1) (least accurate). Although the results listed in (iv) ay coe as soething of a surprise, it should be ept in ind that we have not tried to optiize the NSFD schees with regard to their nuerical accuracy. Our ain goal in this article is to see what viable ethods exist, so that when NSFD schees are constructed for a nonlinear PDE containing a T 1/2 ter, the positivity conditions will hold for the discretization. Nuerical Methods for Partial Differential Equations DOI 10.1002/nu
NUMERICAL STUDIES OF A NONLINEAR HEAT EQUATION 7 FIG. 2. Plot of the differences between the NSFD(1), NSFD(2), the standard schee, and the exact FD schee. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.co.] It should be indicated that the replaceent given by Eq. (2.14) is one of any possible fors. Another possibility is T 1/2 T +1. (2.18) T +1 +T 2 III. HEAT PDE WITH T 1/2 TERM The wor of the last section provides hints as to how the (uch) siplified PDE of Wilhelsson et al. T = D 2 T t x λt 1/2 ; 0 x 1, t>0 (3.1) 2 T(x,0) = f(x)= given, T(0, t) = T(1, t) = 0, (3.2) could be discretized. Note that a standard FD schee for Eq. (3.1) is given by the expression T +1 T t [ T = D +1 2T + T 1 ) ] λ ( T 1/2 ( x) ) (3.3) 2 Nuerical Methods for Partial Differential Equations DOI 10.1002/nu
8 BUCKMIRE, McMURTRY, AND MICKENS where T can tae a variety of fors such as ( ) T 1/2 = ( ) T 1/2, (3.4a) ( ) T 1/2 T+1 = + T + T 1, (3.4b) 3 ( ) T 1/2 T +1 + T + T 1 =. (3.4c) 3 In the above discretizations, we use the notation t t = ( t), x x = ( x), and T(x, t) T. Thus, and are, respectively, the discrete tie and space variables, and T is an approxiation to T(x, t ). Solving Eq. (3.3) for T +1 gives where R = T +1 = DR ( T +1 + T ) 1 + (1 2DR)T (λ t)( T ) 1/2 (3.5) t.ift ( x) 2 satisfies a positivity condition, i.e. T 0 (-fixed, all relevant ) (3.6) then T +1 is not necessarily non-negative. To obtain an assured positivity preserving schee, we apply what was learned in the previous section and use the following discretization T +1 T t [ T = D +1 2T + T ] [ ] 1 T +1 λ ( x) 2 ( ) T 1/2 where ( T ) taes one of the fors given in Eq. (3.4) or any such equivalent expression. Exaination of this last equation shows that it is linear in T +1 ; therefore solving for it gives T +1 (3.7) = DR [ DR ( T +1 + T ) ] [ ( ) 1 + (1 2DR)T T 1/2 ] (λ t) + ( ) T 1/2. (3.8) Inspection of Eq. (3.8) shows that positivity of the evolved solutions is certain if the following condition holds: As in previous wor [5, 7], we let 1 2DR 0. (3.9) 1 2DR = γdr, γ 0, (3.10) where γ is a non-negative nuber. This gives us, first, a relationship between the tie and space step-sizes, i.e. t = Nuerical Methods for Partial Differential Equations DOI 10.1002/nu ( x)2 (2 + γ)d, (3.11)
NUMERICAL STUDIES OF A NONLINEAR HEAT EQUATION 9 and allows the following representation for this NSFD schee: T +1 = DR [ T +1 + γt + T ] [ ( ) T 1/2 ] 1 (λ t) + ( ) T 1/2. (3.12) To use this schee, the following steps should be carried out: (i) Select values for D, λ, and x. (ii) Deterine t fro Eq. (3.11). (iii) Select a set of boundary values and initial conditions. (iv) Use the NSFD schee of Eq. (3.12) to calculate the nuerical solutions of Eq. (3.1). We have carried out siulations using FD schees. They are indicated by the following notations: (a) Standard: Eq. (3.3) with T = T. (b) NSFD(1): Eq. (3.12) with T given by Eq. (3.4a). (c) NSFD(2): Eq. (3.12) with T given by Eq. (3.4b). (d) NSFD(3): Eq. (3.12) with T given by Eq. (3.4c). The initial condition was selected to be with the boundary conditions T(x,0) = sin(πx), 0 x 1, (3.13) T(0, t) = T(1, t) = 0, t>0. (3.14) Typical results fro our nuerical experients are given in Figs. 3 5. The representative nuerics for NSFD(2) and NSFD(3) are presented in, respectively, Figs. 3 and 4. Note that in each case, the solution decreases onotonically with an increase in tie as, as expected, the solutions are sooth and positive. No nown exact solutions of Eq. (3.1) exist; consequently we cannot ae a coparison with such a solution. However, in Fig. 5 we show the difference between the standard schee and NSFD(1). These differences are sall and decrease with tie. IV. DISCUSSION AND CONCLUSION Our priary goal in studying the discretizations given in Sections I and II was to gain insight that could aid us in the forulation of iproved FD schees for ore coplex differential equations such as Eq. (1.1). The ajor difficulty is how to construct discrete odels that also satisfy a condition of positivity as required by the physical principles operating as constraints on the structure of the atheatical (usually differential) equations. This issue is iportant and its iportance derives fro the fact that any nuerical instabilities arise fro violation of soe physical principle by the FD equations [5 7]. In this article, we have deonstrated one possible echanis for dealing effectively with ters of the for T α where 0 <α<1. The case when α<1 Nuerical Methods for Partial Differential Equations DOI 10.1002/nu
10 BUCKMIRE, MCMURTRY, AND MICKENS FIG. 3. Plots of the NSFD(2) schee at various ties. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.co.] FIG. 4. Plots of the NSFD(3) schee at various ties. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.co.] Nuerical Methods for Partial Differential Equations DOI 10.1002/nu
NUMERICAL STUDIES OF A NONLINEAR HEAT EQUATION 11 FIG. 5. Plot of the differences between the standard schee and the NSFD(1) schee. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.co.] presently offers no fundaental probles within the fraewor of the current NSFD schee ethodology [5 7]. The wor presented in Sections II and III illustrate one possibility for this resolution. Clearly, alternative ethods ay also exist to eliinate these issues. The ajor conclusions fro the calculations and constructions we have given here are: (i) positivity can be satisfied in FD schees where fractional power ters appear; (ii) the study of rather eleentary or toy odel differential equations can provide insight into what should be done for ore coplex ODEs and PDEs; (iii) currently, no principle exists to restrict possible discretizations for ters such as T α, 0 <α<1. This last point is one of the topics of research that we are currently studying. Finally, based on the wor done in this article, we are extending these results to the full version of Eq. (1.1). The authors would lie to than the H. Wilhelsson research group for providing such an interesting partial differential equation and for their extensive analytical and nuerical studies on it. R. Bucire and K. McMurtry would also lie to than Occidental College and particularly the Matheatics Departent. Nuerical Methods for Partial Differential Equations DOI 10.1002/nu
12 BUCKMIRE, McMURTRY, AND MICKENS References 1. H. Wilhesson, M. Benda, B. Etlicher, R. Jancel, and T. Lehner, Non-linear evolution of densities in the presence of siultaneous diffusion and reaction processes, Phys Scripta 38 (1988), 863 874. 2. R. Bucire, Application of Micens finite differences to several related boundary value probles, R. E. Micens, editors, Advances in the applications of nonstandard finite difference schees, World Scientific Publishing, Singapore, 2005, pp. 47 87. 3. R. E. Micens and A. Sith, Finite-difference odels of ordinary differential equations: influence of denoinator functions, J Franlin Inst 327 (1990), 143 145. 4. R. E. Micens, Difference equation odels of differential equations, Math Coput Model 11 (1988), 528 530. 5. R. E. Micens, Nonstandard finite difference schees for differential equations, J Difference Equations Appl 8 (2002), 823 847. 6. R. E. Micens, Applications of nonstandard finite differences, World Scientific, Singapore, 2000. 7. R. E. Micens, Nonstandard difference odels of differential equations, World Scientific, Singapore, 1994. 8. P. M. Jordan, 2004 2005, private counications (with R. E. Micens). 9. W. Jordan, Ordinary differential equations, Addison-Wesley, Reading, MA, 1958. 10. P. M. Jordan, Finite-aplitude acoustic traveling waves in a fluid that saturates a porous, Phys Lett A 355 (2006), 216 221. Nuerical Methods for Partial Differential Equations DOI 10.1002/nu