NUMERICAL METHODS FOR FIRST ORDER ODEs

Transcription

1 COMPUTATIONAL METHODS FOR FLOW IN POROUS MEDIA Sprng 2009 NUMERICAL METHODS FOR FIRST ORDER ODEs Lus Cueto-Felgueroso 1 PROBLEM STATEMENT Consder a system of frst order ordnary dfferental equatons of the form du dt = f(t, u), 0 t T, u(t = 0) = u 0, (1) where u and f are vectors wth N components, u = u(t), and f s n general a nonlnear functon of t and u When f does not depend explctly on t, we say that the system (1) s autonomous We dscretze the tme doman t [0, T ] as 0 = t 0 < t 1 < < t n < < t M 1 < t M = T, (2) and seek numercal methods that approxmate u at at tmes {t n, n = 0,, M}, n the sense that u n u(t = t n ), (3) wth the ntal condton u 0 = u(t = 0) = u 0 The two man famles of numercal methods for ODEs are one-step and multstep methods (Fgure 1) 21 General form 2 LINEAR MULTISTEP METHODS I ADAMS-BASHFORTH The explct Adams methds (Adams-Bashforth) can be wrtten as where β j = ( 1) j 1 k 1 =j 1 u n+1 = u n + t ( j 1 k β j f n j+1, (4) ) γ, and γ = ( 1) 1 The order of these methods s p = k Some examples are gven below 0 ( ) s ds (5)

2 2 NUMERICAL METHODS FOR ODES U n+1 t n+1 t n+1 U 2 t n U 1 t n 1 U n t n t n 2 Fgure 1 Schematc of soluton update n one-step (left) and multstep (rght) methods 211 k=1 (Forward Euler) 212 k=2 u n+1 = u n + tf n (6) u n+1 = u n + t 2 ( 3f n f n 1) (7) 213 k=3 214 k=4 u n+1 = u n + t ( 23f n 16f n 1 + 5f n 2) (8) 12 u n+1 = u n + t ( 55f n 59f n f n 2 9f n 3) (9) 24 3 LINEAR MULTISTEP METHODS II ADAMS-MOULTON 31 General form The mplct Adams methods (Adams-Moulton) can be wrtten as k u n+1 = u n + t β j f n j+1 (10) The order of these schemes s p = k + 1 Some examples are gven below j=0 311 k=0 (Backward Euler) u n+1 = u n + tf n+1 (11)

3 LCF k=1 (Trapezodal rule) u n+1 = u n + t 2 ( f n+1 + f n) (12) 313 k=2 314 k=3 315 k=4 u n+1 = u n + t ( 5f n+1 + 8f n f n 1) (13) 12 u n+1 = u n + t ( 9f n f n 5f n 1 + f n 2) (14) 24 u n+1 = u n + t ( 251f n f n 264f n f n 2 19f n 3) (15) LINEAR MULTISTEP METHODS III BACKWARD DIFFERENTIATION The Backward Dfferentaton Formulas (BDF) are mplct methods, based on one-sded dfferences that approxmate du/dt drectly The general form s k α u n +1 = tβ 0 f n+1 (16) The order of these schemes s p = k Some examples are gven below 406 BDF1, k= 1 (Backward Euler) =0 u n+1 u n = tf n+1 (17) 407 BDF2, k= BDF3, k= 3 u n un un 1 = t 2 3 f n+1 (18) u n un un un 2 = t 6 11 f n+1 (19)

4 4 NUMERICAL METHODS FOR ODES 51 General defnton 5 ONE-STEP METHODS: RUNGE-KUTTA METHODS One step of an s-stage Runge-Kutta scheme can be wrtten as wth stage values u n+1 = u n + t b k, k = f (t n + c t, u ), (20) =1 u = u n + t a j k j (21) In the above expressons, t s the tme step, and A = {a j } R s s, b R s and c R s are the characterstc coeffcents of each gven Runge-Kutta scheme, whch can be compactly wrtten usng the so-called Butcher tableau c 1 a 11 a 12 a 1s c 2 a 21 a 22 a 2s c s a s1 a s2 a ss b 1 b 2 b s b1 b2 bs The consstency vector c defnes the ponts (n tme) at whch the method computes approxmatons to the ntal value problem, so that the stage values can be seen as u u (t n + c t) The row sum condton c = a j, = 1,, s, (22) s usually adopted to smplfy the order condtons for hgh-order methods Explct schemes are characterzed by {a j = 0, j } The second set of coeffcents { b, = 1,, s} corresponds to the embedded scheme, whch s used for error estmaton Thus, a second approxmaton ûu n+1 can be defned usng the same coeffcents A = {a j } R s s, and c R s, and stage values u, = 1,, s, as ûu n+1 = u n + t b k, k = f (t n + c t, u ) (23) =1 The dfference between ûu n+1 and u n+1 gves an estmate of the error ncurred by the numercal approxmaton, thus provdng a crteron for tme step adaptvty

5 LCF 5 52 Dagonally mplct schemes Among mplct RK schemes, the most popular ones for the tme ntegraton of PDEs are dagonally mplct Ther Butcher s tableaux typcally take the form γ γ γ 0 0 c 3 a 31 a 32 γ 0 1 a s,1 a s,2 a s,3 γ b 1 b 2 b 3 γ b1 b2 b3 bs In partcular, these schemes are referred to as Explct frst step, Sngle dagonal coeffcent, Dagonally Implct Runge-Kutta (ESDIRK) methods Each stage value of an ESDIRK scheme s at least second-order accurate 521 Implementaton The stage value computaton n a DIRK scheme reads u = u n + t a j k j (24) Gven that the 1 prevous k s have been prevously computed, (24) can be wrtten as u = E + ta k, 1 E = u n + t a j k j (25) The above expresson s, n general, a nonlnear system of equatons The p + 1 Newton teraton assocated to (25) s gven by ( k p) I ta u p u = E + ta k p (26) where p u = u p+1 u p The embedded scheme uses the same raw nformaton as the orgnal one, but n ths case t s processed usng the second set of weghts, { b, = 1,, s} Thus, at the end of each step of the RK ntegrator, we have u n+1 = u n + t b k =1 ûu n+1 = u n + t b k (27) =1 and the error estmate s gven by some sutable norm of the dfference between these two solutons, r n+1 = u n+1 ûu n+1

6 6 NUMERICAL METHODS FOR ODES 53 Addtve Runge-Kutta schemes: mplct-explct (IMEX) Consder systems of ordnary dfferental equatons that can be wrtten n addtve form as du N dt = f [ν] (t, u) (28) ν=1 where f [1], f [2],, f [N] denote certan terms or components of f, whose dstnctve propertes are worth beng taken nto account separately The above expresson (28) s n prncple qute loose n terms of the consderatons that lead to such splttng In general, t may be advantageous to explot the addtve structure of the system (28) when ether f or the unknowns u themselves present components wth sgnfcantly dfferent tme scales In our PDE numercal soluton context, the former case typcally corresponds to stff-nonstff a pror decompostons of the equatons, whereas the latter could apply to grd-nduced stffness The dea behnd addtve schemes s to use, for each component, the ntegrator that best suts ts partcular characterstcs In the general, N-component case, the ntegraton of (28) can be carred out through the applcaton of N dfferent Runge-Kutta methods, one for each of the components A step of an s stage, N part Addtve (ARK N ) or Parttoned (P RK N ) Runge-Kutta scheme, defned by ts generalzed Butcher tableau s gven by c 1 a [1] 11 a [1] 1s a [N] 11 a [N] 1s c s a [1] s1 a [1] ss a [N] s1 a [N] ss b [1] 1 b [1] s b [N] 1 b [N] s b[1] 1 b[1] s b[n] 1 b[n] s wth stage values where u n+1 = u n + t u = u n + t N =1 ν=1 N ν=1 b [ν] f [ν] (t + c t, u ) (29) a [ν] j f [ν] (t + c j t, u j ) (30) The Butcher coeffcents {a [ν] j c j = }, {b[ν] k=1 a [ν] jk ν = 1,, N (31) }, { b [ν] }, ν = 1,, N and {c } are constraned by certan accuracy and stablty requrements The order condtons of the combned scheme nclude those specfc to each elemental method, and also certan couplng condtons The growth of the number of couplng condtons for ncreasngly hgher order and number of components N s such that the

7 LCF 7 practcal desgn of ARK N methods has been typcally restrcted to N = 2 (ARK 2 ) In ths latter case, the system (28) s conceptually wrtten as du dt = f s (t, u) + f ns (t, u) (32) where the rght hand sde of (28) has been genercally splt nto stff (f s ) and nonstff (f ns ) terms Two dfferent Runge-Kutta schemes, specfcally desgned and coupled, are appled to each term, and the mportant case n our context s the mplct-explct (IMEX) approach, whch acknowledges the fact that the stff part s more effcently dealt wth by means of an mplct ntegrator, whereas the nonstff part can be straghtforwardly ntegrated usng an explct scheme In partcular, many problems of practcal nterest are modeled by partal dfferental equatons whose semdscretzaton can be expressed n the form of (32), where f s (t, U) s lnear but stff, and f ns (t, U) s nonlnear but nonstff The resultng system of ODE s can be very effcently ntegrated usng the IMEX approach The combned ntegrators are referred to as IMEX ARK 2 methods or, when the stff terms are lnear, lnearly mplct Runge-Kutta schemes A popular famly of IMEX ARK 2 schemes take the form γ 2γ c 3 a [E] 31 a [E] a [E] s,1 a [E] s,2 a [E] s,3 0 b [E] 1 b [E] 2 b [E] b[e] 1 b[e] 2 3 γ b[e] 3 b[e] s γ γ γ 0 0 c 3 a [I] 31 a [I] 32 γ 0 1 a [I] s,1 a [I] s,2 a [I] s,3 γ b [I] 1 b [I] 2 b [I] b[i] 1 b[i] 2 3 γ b[i] 3 b[i] s In the above expresson, the superscrpts [E] and [I] have been used n reference to the explct and mplct components of the addtve ARK 2 ntegrator, respectvely The stage order of the mplct ntegrator s two 531 Implementaton One step of an s-stage two-part addtve Runge-Kutta scheme, ARK 2, defned by ts Butcher coeffcents (A [I], A [R], b [I], b [E], b [I], b [E], c), s gven by where k [I] u n+1 = u n + t ( =1 b [I] k [I] + b [E] k [E] and k [E] are the dscrete counterparts of the stff and nonstff operators n (32), f s and f ns, k [I] = f h s (t, u ) k [E] = f h ns (t, u ) (34) and the stage values are defned as u = u n + t ( ) a [I] j k[i] + a [E] j k[e] Restrctng our analyss on ARK 2 pars that use DIRK schemes for the mplct part, the above expresson can be rearranged to obtan ) (33) (35)

8 8 NUMERICAL METHODS FOR ODES 1 ( ) u = u n + t a [I] j k[i] j + a [E] j k[e] j + ta [I] k[i] (36) We are nterested n the lnearly mplct case, for whch the above expresson s a lnear system of equatons of the form where k [I] ( ) I ta [I] K u = u n + t 1 ( ) a [I] j k[i] j + a [E] j k[e] j = Ku After solvng u from (37), we can compute k [I] (37) = f s (t, u ), and k [E] = f ns (t, u ) The error estmator s constructed agan n terms of the soluton provded by the embedded scheme, ûu n+1 = u n + t =1 ( b[i] k [I] + and gven by some sutable norm of the dfference between the orgnal and embedded solutons, r n+1 = u n+1 ûu n+1 b [E] k [E] ) (38)

9 LCF 9 61 Heun s thrd-order method 6 SAMPLE RUNGE-KUTTA METHODS /3 1/ /3 0 2/3 0 1/4 0 3/4 62 Kutta s thrd-order method /2 1/ /6 2/3 1/6 63 Kutta s fourth-order method (assocated to Smpson s frst quadrature rule) /2 1/ /2 0 1/ /6 1/3 1/3 1/6 64 Fourth-order scheme assocated to Smpson s second quadrature rule /3 1/ /3 1/ /8 3/8 3/8 1/8

10 10 NUMERICAL METHODS FOR ODES 65 Fehlberg s method /4 1/ /8 3/32 9/ / / / / / / / /2 8/ / / / / / / /50 2/55 25/ / /4104 1/ Dormand and Prnce s method /5 1/ /10 3/40 9/ /5 44/45 56/15 32/ / / / / / / / / / / / / / / / / / / / / / / / / /2100 1/40 67 Bjl s DIRK method

11 LCF 11 7 IMPLEMENTATION OF EXPLICIT RUNGE-KUTTA METHODS 71 A frst, non-pde example Consder N>1 dogs {d j, j = 1,, N}, that are located at the N vertces of a polygon At t = 0, each dog starts chasng ts neghbor counterclockwse; e d1 d2 d(n 1) dn The relatonshp d chases d( + 1), d d( + 1), s understood n the sense that, for all t 0, the velocty vector assocated to dog ponts towards dog ( + 1) Accordngly, the equatons of moton can be wrtten as dx dt = u dy dt = v = 1, 2,, N, wth ntal condtons x (0) = x 0 and y (0) = y 0 The velocty vector v (t) = (u, v ) s gven by u = c x +1 x r v = c y +1 y r In the above expressons, c s a characterstc dog speed (whch s assumed to be constant n tme and equal for all dogs), and r s the dstance between consecutve dogs, r = (x +1 x ) 2 + (y +1 y ) 2 Assumng c = 1, and that the dogs are ntally located along the unt crcumference, x 0 = cos(θ ) y 0 = sn(θ ), θ = π N + 2π ( 1), = 1,, N, N we may ntegrate the trajectores (x(t), y(t)) usng several Runge-Kutta schemes We wll stop the computaton when the smallest dstance between consecutve dogs s less than δ = 10 4 The tme steps wll be chosen accordng to t n+1 = t n+1 t n = κ mn(r n ) where mn(r n ) s the mnmum dstance between consecutve dogs at tme level tn, and κ s some constant The exact trajectory of the frst dog s gven by where r = e a(θ π/n), θ [ π N, ), (39)

12 12 NUMERICAL METHODS FOR ODES cos 2π a = N 1 sn 2π (40) N The trajectores of the other dogs are dentcal, but shfted 2π N Ths problem s solved by the code dogsrkm, whch can be found n the folder example_rk Each step of an s-stage explct RK method works as follows: - For every stage, we need to compute the stage value usng the f s evaluated at prevous stages, as u = u n + t a j k j (41) Once we compute the stage value, u, we store f(t + c t, u ) In the code, ths corresponds to for stage=1:nstage; accum= u0; for jstage= 1:stage-1; accum= accum + dt*ark(stage,jstage)*f(:,jstage); [f,r]= odefun(t+crk(stage)*dt,accum); F(:,stage)= f; - Once all the stage values and ther assocate f s have been computed, we advance the soluton as In the code, ths reads u n+1 = u n + t b k, k = f (t n + c t, u ) (42) =1 u= u0; for stage= 1:nstage; u= u + dt*brk(stage)*f(:,stage); The code may work wth RK methods of orders 1 to 4 (ther Butcher s tableaux are gven n ButcherTm Convergence results and a sample smulaton wth N = 6 dogs are shown n Fgure 2 The convergence study was carred out usng the code dogsrk_convm

13 LCF k= k= 2 05 error k= 3 0 k= t Fgure 2 Left, convergence of varous explct RK methods n the chasng dogs problem Rght, trajectores for N = 6 dogs 72 Applcaton to PDEs Let s solve the problem u t + u x u µ 2 = 0, µ > 0, t > 0, x [0, 1], (43) x2 usng fnte dfferences and explct Runge-Kutta schemes The ntal condton s u(x, t = 0) = exp( 100(x 05) 2 ), (44) and we wll enforce perodc boundary condtons Ths example s coded n exp_adem (folder explct_ade) We may follow the same framework of the prevous example, once we consder the sem-dscrete problem du dt = f(t, u) = Ku, K = (D 1 µd 2 ), (45) where D 1 and D 2 are the dfferentaton matrces One step of the RK update then reads for stage=1:nstage; accum= u0; for jstage= 1:stage-1; accum= accum + dt*ark(stage,jstage)*f(:,jstage); F(:,stage)= K*accum; u= u0; for stage= 1:nstage; u= u + dt*brk(stage)*f(:,stage);

14 14 NUMERICAL METHODS FOR ODES 81 Lnear case 8 IMPLEMENTATION OF IMPLICIT RUNGE-KUTTA METHODS In the case of (dagonally) mplct RK methods, snce the evaluaton of the stage values u nvolves f tself, we need to solve a system of equatons Let us start wth the same lnear advecton-dffuson equaton of the prevous example, ntal condton u t + u x u µ 2 = 0, µ > 0, t > 0, x [0, 1], (46) x2 u(x, t = 0) = exp( 100(x 05) 2 ), (47) and perodc boundary condtons Remember that the general form of the RK update s gven by wth stage values u n+1 = u n + t b k, k = f (t n + c t, u ), (48) =1 u = u n + t a j k j (49) In the present case, snce we consder DIRK schemes only, the stage values wll be gven by wth u = u n + t a j k j, (50) k j = Ku j, K = (D 1 µd 2 ), (51) where D 1 and D 2 are the dfferentaton matrces Thus, the computaton of stage values may be wrtten as 1 u = u n + t a j k j + ta Ku (52) Rearrangng the above expresson, we arrve at a system of lnear equatons of the form 1 (I ta K) u = u n + t a j k j, (53) that needs to be solved n order to get u The code mp_adem (folder mplct_ade) solves ths problem usng backward Euler and Bjl s DIRK method One step of a DIRK scheme s coded as

15 LCF 15 u0= u; for stage=1:nstage; accum= u0; for jstage= 1:stage-1; accum= accum + dt*ark(stage,jstage)*f(:,jstage); %Solve system of equatons Impmat= eye(n)-dt*ark(stage,stage)*k; u= Impmat\accum; %Compute f(u_) F(:,stage)= K*u; %Fnal update u= u0; for stage= 1:nstage; u= u + dt*brk(stage)*f(:,stage); Note that, n the present case, the matrx K could have been precomputed and nverted only once at the begnnng

16 16 NUMERICAL METHODS FOR ODES 82 Nonlnear case: fully mplct vs mplct-explct Our model problem for the nonlnear case s the Kuramoto-Svashnsky equaton u t + ( ) 1 x 2 u2 + 2 u x u x 4 = 0 (54) Ths equaton s solved n [0, 32π] wth perodc boundary condtons The ntal condton s gven by u(x, t = 0) = cos(x/16) (1 + sn(x/16) (55) In ths equaton, the low-order (advectve) term s nonlnear, whereas the hgher-order terms are lnear The fact that a fourth-order term s present makes the equaton very stff There are two man strateges that could be followed: fully mplct tme steppng, where the three terms are advanced mplctly, and mplct-explct, where the advectve term s advanced explctly and the hgher-order terms mplctly The latter strategy has the advantage that we solve lnear systems of equatons In the former, we need to use Newton teratons to solve the resultng nonlnear systems The fully nonlnear strategy s mplemented n the code mp_ksm (folder mplct_ks) One step of a DIRK scheme s mplemented as u0= u; for stage=1:nstage; accum= u0; for jstage= 1:stage-1; accum= accum + dt*ark(stage,jstage)*f(:,jstage); %Solve system of nonlnear equatons r= 1; whle(r>10ˆ-6); %Jacoban Jac= eye(n) + dt*ark(stage,stage)*(d1*dag(u) + D2 + D4); %Resdual R= u - accum + dt*ark(stage,stage)*(d1*(05*u*u) + D2*u + D4*u); %Update deltau= -Jac\R; u= u+deltau; r= norm(deltau); F(:,stage)= -(D1*(05*u*u) + D2*u + D4*u); u= u0; for stage= 1:nstage; u= u + dt*brk(stage)*f(:,stage); The mplct-explct strategy s mplemented n the code mex_ksm (folder mex_ks) We use one of the the ARK 2 methods developed n [6]The advantage of an IMEX formulaton s that the systems that we need to solve are lnear, and therefore we do not need several Newton teratons as n the fully mplct scheme One step of the IMEX-RK scheme reads

17 LCF 17 Fgure 3 Schematc of soluton update n one-step (left) and multstep (rght) methods u0= u; for stage=1:nstage; accum= u0; for jstage= 1:stage-1; accum= accum + dt*(ae(stage,jstage)*ke(:,jstage)+ AI(stage,jstage)*KI(:,jstage)); f stage>1; u= IKmat*accum; KI(:,stage)= -(D2+D4)*u; KE(:,stage)= -D1*(05*u*u); u= u0; for stage= 1:nstage; u= u + dt*( be(stage)*ke(:,stage) + bi(stage)*ki(:,stage) ); Snce we have precomputed and nverted the matrx of the lnear system of equatons, the tme steppng actually does not requre solvng a system of equatons Of course ths s possble because we are usng a constant tme step t Otherwse, we would need to recompute and nvert K Fgure 3 shows the smulated evoluton of u(x) n space-tme, usng the mplct-explct code, mex_ksm

18 18 NUMERICAL METHODS FOR ODES 91 Lnear multstep methods Multstep methods can be wrtten as 9 Regons of absolute stablty k α j u n j+1 = t j=0 k β j f n j+1 (56) The boundary of the regon of absolute stablty of a multstep method s, based on the above defnton, gven by z = j=0 k α j e ( j+1)θ j=0 (57) k β j e ( j+1)θ j=0 where z = λ t, s the magnary unt, and θ vares between 0 and 2π The regons of absolute stablty of several Adams and BDF methods are plotted n Fgures 4 and 5, respectvely, usng the codes n the folder stab_regons A numercal method s A-stable f ts regon of absolute stablty contans the left half-plane, Re(λ) t < 0 It can be shown that: - An explct lnear multstep method cannot be A-stable - The order of an A-stable lnear multstep method cannot exceed 2 - The second-order A-stable mplct lnear multstep method wth the smallest error constant s the trapezodal method 92 Runge-Kutta methods Recall that the regon of absolute stablty s determned by the complex values z = λ t for whch, when the method s appled to the test equaton du dt = λu, (58) we get u n+1 u n One step of an explct Runge-Kutta method appled to the above test equaton can be wrtten as [ ] u n+1 = 1 + zb T (I za) 1 1 u n (59) Expandng the nverse operator, we can rewrte the above expresson as or u n+1 = [ 1 + zb T ( I + za + + z k A k + ) 1 ] u n, (60) u n+1 = R(z)u n (61)

19 LCF 19 Fgure 4 Regons of absolute stablty of Adams methods Left, Adams-Bashforth Rght, Adams-Moulton Fgure 5 Regons of absolute stablty of backward dfferentaton formulas Left, k = 1 4 Rght, k = 4 6

20 20 NUMERICAL METHODS FOR ODES Fgure 6 Regons of absolute stablty of explct Runge-Kutta methods wth p = s 4 For a Runge-Kutta method of order p, we get R(z) = 1 + z + z zp p! + j=p+1 z j b T A j 1 1 (62) In partcular, the regon of absolute stablty of an explct pth-order Runge-Kutta method wth s stages and order p = s 4, s gven by 1 + z + z zp 1 (63) p! Note that all p-stage explct RK methods of order p have the same regon of absolute stablty For an s-stage method of order p < s the stablty regon depends on the method s coeffcents The stablty regons of explct RK methods wth p = s 4 are plotted n Fgure 6, usng the code regons_rkm 10 Stablty of the method of lnes The practcal relevance of determnng the regon of absolute stablty of a gven tme ntegraton scheme s that the egenvalues of our spatal dscretzaton matrx, multpled by t, have to fall nsde that regon The codes n the folder stab_mol assemble the dscretzaton matrces for the model equaton u t + u x + µ 2 u 2 x 2 + µ 3 u 3 x 3 + µ 4 u 4 = 0, (64) x4 and plot the scaled egenvalues together wth the regons of absolute stablty of varous ODE solvers Fgure 7 shows the stablty regons for the Adams-Bashforth and explct Runge-Kutta methods,

21 LCF 21 Fgure 7 Stablty of the method of lnes t-scaled egenvalues for the model problem (65), and regons of absolute stablty of common explct schemes Left, Adams-Basforth methods Rght, Runge-Kutta methods The egenvalues correspond to three dfferent values of the dffusvty, µ together wth the t-scaled egenvalues of a fnte dfference dscretzaton of the model problem u t + u x u µ 2 = 0, (65) x2 for several values of the dffusvty µ We therefore compute the egenvalues {λ j } of the dscretzaton matrx K, K = D 1 + µd 2, (66) and multply them by t As µ ncreases, the scaled egenvalues advance quckly nsde the left halfplane, whch n practce means that we need small tme steps (the problem becomes stff) In that stuaton, A-stable methods provde sgnfcant advantages, snce the tme step can be chosen based on accuracy requrements, rather than based on stablty restrctons BIBLIOGRAPHY 1 Harer E, Nørsett SP, Wanner G Solvng ordnary dfferental equatons (vols 1 and 2) Sprnger-Verlag, Berln, (1993) 2 Butcher JC Numercal methods for ordnary dfferental equatons John Wley & Sons, Chchester, (2008) 3 Ascher UM, Petzold LR Computer methods for ordnary dfferental equatons and dfferental-algebrac equatons SIAM, Phladelpha, (1998) 4 Araújo AL, Murúa A, Sanz-Serna JM Symplectc methods based on decompostons SIAM J Numer Anal 34, , (1997) 5 Ascher UM, Ruuth SJ, Spter RJ Implct-explct Runge-Kutta methods for tme-dependent partal dfferental equatons Appl Numer Math 25, , (1997) 6 Kennedy CA, Carpenter MH Addtve Runge-Kutta schemes for convecton-dffuson-reacton equatons Appl Numer Math 44, , (2003)