Comparison of Stability of Selected Numerical Methods for Solving Stiff Semi- Linear Differential Equations

Transcription

1 International Journal of Applied Science and Technology Vol. 7, No. 3, Septeber 217 Coparison of Stability of Selected Nuerical Methods for Solving Stiff Sei- Linear Differential Equations Kwaku Darkwah Departent of Matheatics Kwae Nkruah University of Science and Technology Kuasi Ghana Isaac Azure Departent of Business Adinistration Regentropfen College of Applied Science Bolgatanga Ghana Abstract Attention was focused on the explicit Exponential Tie Differencing (ETD) integrators that are designed to solve stiff sei-linear probles. Sei-linear PDEs can be split into a linear part, which contains the stiffest part of the dynaics of the proble, and a nonlinear part, which varies ore slowly than the linear part. The ETD ethods solve the linear part exactly, and then explicitly approxiate the reaining part by polynoial approxiations. The research involves an analytical exaination and coparison of the asyptotic stability properties of soe Exponential Tie Differencing Schees (ETD1, ETD2, ETD2RK1 and ETD2RK2) ethods in order to present the advantage of these ethods in overcoing the stability constraints. Keywords : Differential Equation, Exponential Tie Differencing Method, Exponential Tie Differencing Runge Kutta Method, Integrating Factor Method, Ordinary differential Equation, Partial Differential Equation and Stiff linear differential Equation. Introduction Various probles in the world can be solved when they are odeled and presented in the for of an ordinary differential equation or partial differential equation. However, there are ties where different phenoena acting on very different tie scales occur siultaneously introducing a paraeter called stiff paraeter which soeties akes it difficult to solve. All differential equations with this property are said to be stiff differential equations. According to Curtiss, et al (1952) the earliest detection of stiffness in differential equations in the digital coputer era, was apparently far in advance of its tie. They naed the phenoenon and spotted the nature of stiffness (stability requireent dictates the choice of the step size to be very sall).to resolve the proble they recoended possible ethods such as the Backward Differentiation Forula for nuerical integration. This study looked at how to solve stiff differential equations using the Exponential Tie Differencing Schees aking reference to their asyptotic stabilities. Related Works In 1963, Dahlquist defined the stiff proble and deonstrated the difficulties that standard differential equation solvers have with stiff differential equations. Dahlquist et al (1973), defined a stiff syste as one containing very fast coponents as well as very slow coponents; they represent coupled physical systes having coponents varying with very different tie scales: that is, they are systes having soe coponents varying uch ore rapidly than the others. Another significant person worth entioning in this field of studies is Gear. He ade considerable efforts to develop nuerical integration for stiff probles and hence brought the attention of the atheatical and coputer science counity to stiff probles.(gear,1968). Hairer, et al (1996) studied different ethods of solving differential equations and identified that all the explicit ethods were unable to solve a particular type of differential equation (called stiff differential equation). They defined stiff equation as a proble for which explicit ethods don t work. 6

2 ISSN (Print), (Online) Center for Prooting Ideas, USA According to Du (24), Exponential Tie Differencing schees are tie integration ethods that can be efficiently cobined with spatial spectral approxiations to provide very high resolution to the sooth solutions of soe linear and nonlinear partial differential equations. Du, explained in his paper the stability properties of soe exponential tie differencing schees, he also presented their application to the nuerical solution of the scalar Allen-Cahn equation in two and three diensional spaces. Liverore et al (27) explained that over the last decade there has been renewed interest in applying exponential tie differencing (ETD) schees to the solution of stiff systes. He presented an ipleentation of such a schee to the fully spectral solution of the incopressible agneto hydrodynaic equations in a spherical shell. Quit recently, Hala have carried out soe research on the stability of nuerical ethods. According to Hala (28), the stability of a given ethod for solving a syste of ODE is a theoretical easure of the extent to which the ethod produces satisfactory approxiation. According to hi, stability is related to the accuracy of the ethods and are referred to as errors not growing in subsequent steps. According to Thohura, et al (213), although a nuber of ethods have been developed and any ore basic forulas suggested for stiff equations, until recently there has been little advice or guidance to help a practitioner choose a good ethod for this proble. In case of stiff differential equations, stability requireents force the solver to take a lot of sall tie steps; this happens when we have a syste of coupled differential equations that have two or ore very different scales of the independent variable over which we are integrated. Various authors have looked at solutions of stiff differential equations using the Exponential Tie Differencing (ETD) and Exponential Tie Differencing Runge-Kutta (ETDRK) ethods without checking their stability. This study seeks to copare the asyptotic stability of soe of these ethods and to obtain stability expressions for these nuerical schees. Model Forulation To derive the s-step ETD schees, consider for siplicity a single odel of stiff ODE du t = cu t + F u t, t (1) dt we ultiply equation (1) through by the integrating factor e ct, and then integrate the equation over a single tie step fro t = t n to t = t n+1 = t n + t to get t u t n+1 = u t n e c t + e c t e cτ F u(t n + τ), t n + τ dτ (2) This forula is exact, and the next step is to derive approxiations to the integral in equation (2). This procedure does not introduce an unwanted fast tie scale into the solution and the schees can be generalized to arbitrary order. If we apply the Newton Backward Difference Forula, using inforation about F u(t, t) at the nth and previous tie steps, we can write a polynoial approxiation to F u t n + τ, t n + τ in the for s 1 F u t n + τ, t n + τ G n t n + τ = 1 τ t = G n t n, (3) where is the backward difference operator defined as follows k= G n t n = 1 k k= 1 k k G n k t n k, k F u t n k, t n k, 4 and! q = q q 1 q + 1, = 1,.., s 1 (note that! q = 1. If we substitute the approxiation equation (3) in the integrand equation (2), we get u t n+1 u t n e c t t s 1 = Where q = τ t. We will indicate the integral in equation (5) by 1 1 c t 1 q q e dq G n t n 5 61

3 International Journal of Applied Science and Technology Vol. 7, No. 3, Septeber c t 1 q e g = 1 q dq, 6 and then calculate the g by bringing in the generating function. For z R, z < 1, we define the generating function Γ z = 1 c t 1 q = e 1 = = g z, q z = e c t 1 q 1 z q dq, = e c t 1 z e c t 1 z c t + log 1 z 7 Rearranging equation (7) to for c t + log 1 z Γ z = e c t 1 z 1, and expanding as a power series in z c t z z2 z3 g g 1 z + g 2 z 2 + = e c t 1 z z 2 z 3, we can find a recurrence relation for the g for by equating like powers of z c tg o = e c t 1, (7a) c tg = g g g g 1 = + 1 k g k 8 k= Haven deterined the g, the ETD schees equation (5) then can be given in explicit fors. Substituting equation (4) and equation (6) in equation (5), we deduce the general generating forula of ETD schees of order s s 1 +1 = e c t + t g 1 k k = k= dq, F n k 9 where and F n denote the nuerical approxiation to u(t n and F(u(t n, t n respectively, and the g are given by equation (8). ETD Schees ETD1 Schee Fro equation (7a) above, g can be written as g = ec t 1 c t To obtain the ETD1 schee, we set s = 1 in the explicit generating forula equation (9) to get +1 = e c t + tg F n +1 = e c t + t e c t 1 F c t n, hence the ETD1 schee is given by; +1 = e c t + e c t 1 F n c, 1 ETD2 Schee In the sae anner, setting s = 2 in equation (9) gives us the second-order ETD2 schee +1 = e c t + c t + 1 e c t 2c t 1 F n + e c t + c t + 1 F n 1 c 2 t 11 62

4 ISSN (Print), (Online) Center for Prooting Ideas, USA Generally, for the one-step tie-discretization ethods and the Runge-Kutta (RK) ethods, all the inforation required to start the integration is available. However, for the ulti-step tie-discretization ethods this is not true. These ethods require the evaluations of a certain nuber of starting values of the nonlinear ter F u t, t at the nth and previous tie steps to build the history required for the calculations. Therefore, it is desirable to construct ETD ethods that are based on RK ethods. ETD Runge - Kutta Schees Cox et al (22), constructed a second-order ETD Runge-Kutta ethod, analogous to the iproved Euler ethod given as follows. ETDRK1 Schee Putting s = 1 in equation (9) gives +1 = e c t + e c t 1 F n c. 12 Let a n +1, than it iplies that a n = e c t + ec t 1 F n 13 c The tera n approxiates the value of u at t n + t. The next step is to approxiate F in the interval t n t t n+1, with F = F n + t t n F a n, t n + t F n t + O t 2 and substitute into equation (13) to give the ETD2RK1 schee +1 = a n + e c t c t 1 F a n, t n + t F n c 2 t. 14 ETD2RK2 Schee In a siilar way, we can also for an ETD2RK2 schee analogous to the odified Euler ethod. The first step a n = e c t 2 + e c t 2 1 F n c, is fored by taking half a step of equation (13); then use the approxiation F = F n + t t n F a t 2 n, t n + t 2 F n + O t 2, in the interval t n, t n + t in equation (2) to deduce the ETD2RK2 schee +1 = e c t + c t 2 e c t + c t + 2 F n + 2 e c t c t 1 F a n, t n + t 2 c 2 t (15) Finally, we note that as c in the coefficients of the s-order ETD-RK ethods, the ethods reduce to the corresponding order of the Runge-Kutta schees. Stability Analysis The approach developed by Beylkin, et al (1998) studied the stability for a faily of explicit and iplicit ELP schees, and showed that these schees have significantly better stability properties when copared with known iplicit-explicit schees. The approach developed for the stability analysis of coposite schees, i.e. schees that use different ethods for the linear and nonlinear parts of the equation, coputes the boundaries of the stability regions for a general test proble. That is to analyze the stability of the ETD schees, we linearize the autonoous ODE du t = cu t + F u t, 16 dt about a fixed point u (so that cu + F u =, to obtain du t = cu t + λu t = c + λ u t, 17 dt where u t is the perturbation to u and df u t λ = u t = u du Again F u F = λ u λ But u = Therefore, F( ) = λ 18 63

5 International Journal of Applied Science and Technology Vol. 7, No. 3, Septeber 217 In order to keep the fixed point u stable, we require R c + λ < (note that the fixed points of the ETD ethods are the sae as those of the ODE equation (16), in contrast to the IF ethods which do not preserve the fixed points for the ODE that they discretize. It sees desirable for a nuerical ethod to fulfill this property with respect to capturing as uch of the dynaics of the syste as possible). If both c and λ are coplex, the stability region is four-diensional. But if both c and λ are pure iaginary or pure real, or if λ is coplex and c is fixed and real then the stability region is two-diensional. This study concentrates on two cases to deterine whether the schees are asyptotically stable. The conditions are as follows; c is fixed and negative andλ and c are purely real. c is negative and both c and λ are purely real. Algorith To deterine whether an exponential tie differencing ethod is asyptotically stable, considering the proble du t = cu t + λu t dt Step 1: Solve the proble using any one of the ETD1, ETD2, ETD2RK2 and ETD2RK2 ethods. Step 2: Divide through the +1 solution with to obtain an equation for +1, x = λ t and y = c t, where c and λ are paraeters in the given proble and t is the tie Step 3: Set r = +1 step. Step 4: For a schee to be asyptotically stable then; r = +1 1 Given the proble above, the asyptotic stability of the schees can be deterined as follows; Stability of ETD1 Schee Equation (1) can be written in the for; +1 = e c t + ec t 1 F c 19 n Fro equation (18), F n can be written as; F n = λ 2 Putting equation (2) in to equation (19), gives; +1 = e c t + ec t 1 λu c n +1 = e c t + ec t 1 λ 21 c putting x = λ t, y = c t andr = +1 If then ETD1 is asyptotically stable. Stability of ETD2 Schee Equation (11) can be written in the for; in to the above equation gives; r = e y + x y ey 1 22 r = e y + x y ey c t + 1 e c t 2c t 1 F n + e c t + c t + 1 F n 1 = e c t + c 2 t Substituting equation (18) in to the above equation gives. 64

6 ISSN (Print), (Online) Center for Prooting Ideas, USA +1 c t + 1 e c t 2c t 1 λ + e c t + c t + 1 F n 1 = e c t + c 2 24 t putting x = λ t, y = c t andr = +1 in to the above equation gives; y 2 r 2 y 2 e y + y + 1 e y 2y 1 x r + e y y 1 x = r = e y + ey 1 x + ey y 1 y y 2 x 2 25 If r = e y + ey 1 x + ey y 1 y y 2 x then ETD2 is asyptotically stable. Stability of ETD2RK1 Schee Equation (14) can be written as; F a n, t n + t c 2 t +1 = e c t + ec t 1 F n + e c t c t 1 c Substituting F n = λ in to the above equation gives +1 = e c t + ec t 1 λ + e c t c t 1 F a n, t n + t c c 2 27 t Putting x = λ t, y = c t andr = +1 If in to the above equation, we get r = e y + e y 1 λ c + ey y 1 e y 1 y 2 x 2 r = e y + e y 1 x y + ey y 1 e y 1 y 2 x 2 r = e y + ey 1 y x + ey y 1 e y 1 y 2 x 2 28 r = e y + ey 1 x + ey y 1 e y 1 y y 2 x then ETD2RK2 is asyptotically stable. Stability of ETD2RK2 Schee Equation (15) can be written as +1 = e c t + c t 2 e c t + c t + 2 F n + 2 e c t c t 1 F a n, t n + t + t 2) c 2 t (3) If Let r = +1, x = λ t and y = c t r = e y + 2 ey y 1 e y 2 + y 2 e y + y + 2 y 2 x + 2 ey y 1 e y 2 1 y 3 x 2 (31) r = e y + 2 e y y 1 e y 2 + y 2 e y +y+2 y 2 x + 2 e y y 1 e y 2 1 y 3 x 2 1 (32) then ETD2RK2 is asyptotically stable. Results Coputational Results of Stability of ETD Schees Du et al (24), gave the paraeter values for c, λ and Δt. These values were adopted in this study to copute the values of x and y given that x = λδt y = cδt. 65

7 International Journal of Applied Science and Technology Vol. 7, No. 3, Septeber 217 Following the first condition under the stability analysis, where λ is real and c is fixed, negative and both λ c are purely real; the values of x and y were coputed using the adopted values for the paraeters c, λ Δt and represented in a tabular for below. Table 1: x and y Values Given that c is Fixed and Negative and λ and c are both Purely Real. 66 t c λ x y It can be observed fro Table (1) above that as the values of Δt andλ increase and c reain constant, values of x and y decreases accordingly. Because of the negative values of c, all values obtained for y were also negative. Fro tables (1), the coputed values of x and y were used to carry out the coputations for the r values for ETDI, ETD2, ETD2RK1 and ETD2RK2. Considering the condition that c is fixed and negative and both λ and c are purely real, equations (17), equation (21) and equation (22) coputes the r values for ETD1, in a siilar way, equations (17), equation (24) and equation (25) coputes the r values for ETD2, while equations (17), equation (27) and equation (28) coputes the r values for ETD2RK1. Finally equations (17), equation (3) and equation (31) coputes the r values for ETD2RK2. Below is a suary of the coputed values of r for the schees. Table 2: The r Values of the Schees when Paraeter c is Fixed and Negative and λ is Coplex r VALUESOFSCHEMES ETD1 ETD2 ETD2RK1 ETD2RK Fro Table (2), all values corresponding to the ETD and ETDRK schees are less than one indicating that all schees are asyptotically stable at these points studied. It can also be observed that at each Δt all schees have the sae values and this is true for all values of Δt. Hence none of the schees can be said to be ore asyptotically stable than the other. Again descending down the table, the values of r corresponding to the schees decreases, hence aking the schees ore stable. Considering the second condition under the stability analysis, where c is changing and negative and both λ c are purely real; the values of x and y were coputed using the adopted values for the paraeters c, λ Δt and represented in a tabular below. Table 3: x and y Values Given c is changing and Negative and λ and c are Real t c λ x y Values fro Table (3) show that given the condition that c is changing and negative and both c λ are real, values of x and y decrease as Δt increases. Considering the condition that c is changing and negative and both λ and c are purely real, equations (17), equation (21) and equation (22) coputes the r values for ETD1, in a siilar way, equations (17), equation (24) and equation (25) coputes the r values for ETD2, while equations (17), equation (27) and equation (28) coputes the r values for ETD2RK1. Finally equations (17), (3) and (31) coputes the r values for ETD2RK2. Below is a suary of the coputed values of r for the schees.

8 ISSN (Print), (Online) Center for Prooting Ideas, USA Table 4.: The r Values of the Schees when Paraeter c is changing and Negative and λ is Coplex. r VALUESOFSCHEMES ETD1 ETD2 ETD2RK1 ETD2RK Again in Table (4), all values corresponding to ETD and ETDRK schees are less than one, indicating that all schees are asyptotically stable at these points. It can also be observed that at each Δt all the schees have the sae values suggesting that none of the schees studied is better in ters of asyptotic stability than the other. Descending down the table, the r values of the schees are decreasing, hence suggesting that the schees are becoing ore asyptotically stable. Discussion Fro Table (1), given that Δt 6 1 3, c is fixed and negative and λ is coplex, results obtained for x = λ t and y = cδt showed that both values of x and y increased for every increase in Δt, however all values of y were negative. In Table (2), if c is changing and negative and λ is real, the values of x reained the sae while y values were found to be changing. Fro Table (3), all values corresponding to the various ETD and ETDRK schees are less than one indicating that all schees are asyptotically stable at these points studied. It can also be observed that at each Δt all schees have the sae values and this is true for all values of Δt. Hence none of the schees can be said to be ore asyptotically stable than the order. Again descending down the table, the values of r corresponding to the schees decreases, hence aking the schees ore stable. Again in Table (4), all values corresponding to ETD and ETDRK schees are less than one, indicating that all schees are asyptotically stable at these points. It can also be observed that at each Δt all the schees have the sae values suggesting that none of the schees studied is better in ters of asyptotic stability than the other. Descending down the table, the r values of the schees are decreasing, hence suggesting that the schees are becoing ore asyptotically stable. Conclusion This research suggest that the coparison of the asyptotic stability of ETD1, ETD2, ETD2RK1 and ETD2RK2 schees in solving the stiff sei-linear differential equation (17) was properly executed. This was ade possible when soe paraeters c, λ Δt were adopted and used for coputations. To ensure that the first objective was et, ETD1, ETD2, ETD2RK1 and ETD2RK2 schees were used to solve the stiff sei-linear differential equation (17) to obtain the asyptotic stability expressions; equation (23), equation (24), equation (29) and equation (32). The second objective suggested the following conclusions; When the paraeter c is negative and changing, and λ is coplex, all the schees are asyptotically stable, however as t increases and the paraeter c is changing, the corresponding r values of the schees decreases accordingly aking the ore asyptotically stable. When the paraeter c is negative and fixed and λ is coplex and both are real, all the schees studied are asyptotically stable, however as t increases, corresponding r values of the schees decreases accordingly aking the ore stable. At each t, the r values of all the schees are the sae, that is; at t =.1, r value of ETD1 is.9999, ETD2 is.9999, ETD2RK1 is.9999 and ETD2RK2 is Hence as far as asyptotic stability is concern, none of the schees studied is ore stable than the other, therefore ETD1, ETD2, ETD2RK and ETD2RK2 are efficient schees for solving stiff sei-linear differential equations. It will take several t values to ake the schees ore stable when c is fixed and negative and λ is coplex than when c is changing and negative and λ is coplex. 67

9 International Journal of Applied Science and Technology Vol. 7, No. 3, Septeber 217 References Beylkin,.G. and Keiser,.J.M (1998). A New Class of Tie Discretization Schees for the Solution of Nonlinear PDE s. J. Coput. Phys. Vol. 147, pages Cox,. S.M. and Matthews,. P.C.(22). Exponential Tie Differencing for Stiff Systes, J.Coput. Phys.vol.176, pages Curtiss,. C.F. and Hirchfelder,. J.O. (1952). Integration of Stiff Equations. Proc. Nat. Aca. Sci.vol.48, pages Dahlquist,.G.(1963). A Stability Proble for Linear Multi-Step. BIT Nuer. Math.vol.131, pages Du,. Q. and Zhu. (24), Stability Analysis and Applications of the Exponential Tie Differencing Schees, Journal of Cotational Matheatics, vol.22, No.2 Gear,.C.W. (1968). Autoatic Integration of Ordinary Differential Equations. Counications of the ACM, VOL.14. Pages Hairer,. E. and Wanner,. G. (1996). Solving Ordinary Differential Equations II. Springer-Verlag, Berlin, second edition, pages Hala,.A. (28). Nuerical Methods for Stiff Systes. University of Northenghan. Liverore,.R. and Faires,.J. D. (21). Nuerical Analysis. Wadsworth Group, seventh edition. Sharaban,.T. and Azad,.R. (213). Nuerical Approach for solving Stiff Differential Equation: A Coparative study, Global Journal Inc. (USA), vol 13,Issue 6, version 1., page 5 68