Some new additive Runge Kutta methods and their applications

Transcription

1 Journal of Computatonal and Appled Mathematcs 9 (6) Some new addtve Runge Kutta methods and ther applcatons Hongyu Lu,, Jun Zou Department of Mathematcs, The Chnese Unversty of Hong Kong, Shatn, N.T., Hong Kong Receved 4 November 4; receved n revsed form 4 January 5 Dedcated to Roderck S.C. Wong on the occason of hs 6th brthday Abstract We propose some new addtve Runge Kutta methods of orders rangng from to 4 that may be used for solvng some nonlnear system of ODEs, especally for the temporal dscretzaton of some nonlnear systems of PDEs wth constrants. Only lnear ODEs or PDEs need to be solved at each tme step wth these new methods. 5 Elsever B.V. All rghts reserved. Keywords: Addtve Runge Kutta method; A-stablty; Stffly accurate; Constraned PDEs. Prelmnares In ths paper we shall explore some new addtve Runge Kutta methods that may be used for solvng some nonlnear system of ODEs, especally for the temporal dscretzaton of some nonlnear systems of PDEs wth constrants: for example, the Naver Stokes equatons n ncompressble flow, where the velocty satsfes the dvergence-free constrant, and the mean-feld magnetc nducton system n geodynamo modelng, where the magnetc feld has to satsfy the dvergence-free constrant. Only lnear ODEs or PDEs need to be solved at each tme step wth these new addtve Runge Kutta methods. Correspondng author. E-mal addresses: hylu@math.cuhk.edu.hk (H. Lu), zou@math.cuhk.edu.hk (J. Zou). Department of Mathematcs, The Chnese Unversty of Hong Kong, Shatn, N.T., Hong Kong, supported by a Drect Grant of CUHK. Substantally supported by Hong Kong RGC Grants (Projects 4343 and 448/P) /$ - see front matter 5 Elsever B.V. All rghts reserved. do:.6/j.cam.5..

2 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) We start wth the ntroducton of some basc notatons and results on the exstng Runge Kutta (RK) methods. As usual, we shall represent a standard s-stage RK method (cf. [7]) by the tableau () where A = (a j ) s,j= s the coeffcent matrx, bt = (b ) s = the weght vector, and c = (c ) s = a vector used to specfy the dscrete tmes. We shall use h> to denote the tme stepsze, and t n = t + nh, n =,,,..., for the dscrete tme ponts. When appled to the followng system of frst-order ODEs, y (t) = f(t,y) for t>t ; y(t ) = y, () where y R m and f : (t, ) R m R m s a nonlnear vector-valued functon, scheme () can be wrtten as y (n) = y n + h a j f(t n + c j h, y (n) j ), =,,...,s, (3) y n = y n + h j= = b f(t n + c h, y (n) ), (4) where y n s an approxmaton of y(t n ) and y (n) y(t n + c h). Wthout loss of generalty, t suffces for us to consder only one tme step of scheme (3) (4). So we shall set n = and wrte the ntermedate stage vectors y (n) as Y y(t + c h) and y y(t ). Then scheme (3) (4) can be expressed as Y = y + h a j f(t + c j h, Y j ), =,,...,s, (5) y = y + h j= b f(t + c h, Y ). (6) = In general, the parameters c s n (5) (6) are requred to satsfy the condtons c = a j, =,...,s, j= n order to essentally smplfy order condtons, especally for hgh-order methods. These condtons ndcate that the local truncaton error of each approxmaton n (5) s at least frst-order accurate. Stablty wll be one of our central ssues to be consdered when we construct the new RK methods. Recall that the stablty functon R(z) wth z = λh of an RK method s the approxmate soluton generated by one step of the method for the Dahlqust test problem: y (t) = λy, y =, (7)

3 76 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) wth Re{λ} <. Then the stablty regon of the method s defned to be S ={z C; R(z) }. (8) We know that the stablty functon of the mplct RK method () s gven (cf. [7]) by R(z) = det(i za + zbt ). (9) det(i za) Recall that a method s sad to be A(α)-stable f the sector S α ={z; arg( z) α, z = } s contaned n the stablty regon. An A( π )-stable method s called A-stable. It can be seen that an RK method s A-stable f and only f R(y) for all real y, R(z) s analytc for Re{z} <. () Clearly, an A-stable RK method must be mplct. For very stff problems, one may need schemes wth L-stablty, that s, schemes whch are A-stable and lm R(z) =. () z In many stff stuatons, the L-stablty condton () may be too strong. In fact, t s often suffcent n those very stff cases to requre that lm z R(z) equals some constant less than, for example, α < ; ths wll be used n our subsequent constructon of the addtve methods.. Addtve Runge Kutta methods The focus of our study s on the numercal soluton of the followng ODEs of addtve form: y (t) = f(t,y)+ g(t, y), () where y R m, f : (t, ) R m R m s stff, but lnear wth respect to y, and g(t, y) s nonlnear but not stff. We remark that our subsequent constructon technques can be equally appled to construct smlar schemes for the case where f(t,y)s lnear but non-stff, whle g(t, y) s nonlnear but stff. As we wll show n Secton, system () can also be PDEs wth constrants, where f(t,y) and g(t, y) are both dfferental operators of space varables. In fact, the PDEs wth constrants are the major systems our new addtve RK methods ntend to solve. The addtve methods use the dea of the RK method but am to provde a more effectve way to deal wth the ODEs of the addtve form (). One can already fnd some addtve RK methods n the lterature, whch can acheve favorable results n numercal solutons of certan stff addtve ODEs lke () (see [,5,6,9,]). But all the exstng addtve schemes are not sutable to the PDEs wth constrants, or are applcable but very expensve. When appled to Eq. (), an s-stage addtve RK method s a scheme of the form: y (n) = y (n ) s + h j= a j f(t n + c j h, y (n) j ) + h j= b j g(t n + c j h, y (n) j ), (3)

4 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) where =,,...,s and n =,, 3,...The followng row condtons are always assumed c = a j = b j, (4) j= j= n order to smplfy the order condtons so that t s possble to fnd some reasonable hgh-order schemes. By puttng f(t,y)= org(t, y) = n (), we can get from (3) two standard s-stage RK methods of form () wth the weghted values b = a s or b s. Hence, they are stffly accurate n parlance of [] and one can convenently represent an s-stage addtve RK method by the trple (c, A, B) or the followng tableau (wth weghts as n () no longer necessary): Due to the couplng between blocks A and B n (5), the order condtons and the stablty analyss for the addtve RK methods are much more complcated than the standard (). (5) 3. Stablty analyss The general addtve RK methods were frst ntroduced by Cooper and Sayfy (cf. [5,6]) for solvng the stff problem y (t) = F(t,y) for t>t ; y(t ) = y. (6) They assocated the addtve methods wth a sequence of decompostons of F(t,y)of the form F(t,y)= J (n) y + g (n) (t, y), n =,, 3,..., (7) where {J (n) } s chosen to be ndependent of t and often as an approxmaton to the Jacoban of F evaluated at some sequence of computed values. To study the stablty of the addtve methods, let y be the partcular soluton of the system y (t) = Jy + g(t, y), t > t, (8) whch has the ntal value y(t ) = y. It s known that f the trval soluton of y = Jy s exponentally stable,.e., Re{λ} < for all λ σ[j ], and that g(t, y) =o( y ), then the trval soluton of (8) s also exponentally stable, whch mples that there s an ε > such that f y ε, then y(t) has lmt zero. Usng ths model problem, Cooper and Sayfy establshed the stablty for ther addtve schemes assocated wth system (8). Theorem. Suppose that J s a constant m m matrx. The trval soluton of y (t) = Jy s exponentally stable and g(t, y) =o( y ). Furthermore, we assume that the (c, A, B) RK method s lnearly

5 78 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) mplct,.e., y (n) = y (n ) s + h j= a j Jy (n) j + h j= b j g(t n + hc j,y (n) j ), (9) and the scheme (c, A) s A-stable. Then for any fxed postve h and arbtrary y s (), method (9) unquely defnes a sequence y s (n) for =,,...,sand n=,, 3..., and there exsts a δ > such that f y s () δ, the sequence {y s (n) } has lmt zero. We are nterested n the more general addtve system (), nstead of system (6) wth specal Jacobantype decompostons (7). Clearly the results n Theorem cannot be used for the stablty estmates of the addtve methods (5), especally the smallness condton,.e., g(t, y) =o( y ), whch plays a key role n the proof of the above theorem, s usually not true to system () of our nterest. For our purpose, we shall take a smlar approach to the one n [] to drectly apply the lnear stablty analyss for standard RK methods to the followng test problem assocated wth (): dy dt = λ f y + λ g y, y =, () where λ f and λ g represent the egenvalues of f/ y and g/ y n (). They are complex parameters satsfyng 3 Re{λ f }, Re{λ g } and Re{λ f }? Re{λ g }. We emphasze that the addtve RK schemes proposed n [] are dfferent from the RK schemes of form (3) to be studed n the current paper, and cannot be used for those PDE systems wth constrants, see Secton. By applyng the addtve method (5) to the test problem (), we get after one tme step that y = R(z f,z g ), z f = hλ f, z g = hλ g, () where R(z f,z g ) s the stablty functon. Then we ntroduce our new defnton of the A(α)- and L-stablty for the addtve methods. Defnton. An A(α)-stablty doman of an addtve method (5) n the complex plane of z g = hλ g s defned as the doman where Sα f g ={z g C; R(z f,z g ) for all z f Sα f }, () S f α ={z f C; arg( z f ) α,z f = } {}. (3) An addtve method (5) s sad to be L-stable f t s A-stable (.e. A( π )-stable) and lm R(z f,z g ) =. (4) z f 3 Our subsequent analyss can be extended equally to the case Re{λ f } > Re{λg}.

6 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) If we denote by S g the stablty doman of the explct part (c, B) n (5) as a standard RK method, we have S f g α S g. (5) Ths can be seen by takng λ f = n the test (), and means that n the stablty doman Sα f g, the explct part (c, B) s stable. In the forthcomng dscussons, snce the term f(t,y)n () s stff, t s natural for us to requre that the sem-explct part (c, A) s to be A(α)-stable, and we should have Sα f g. In case the frst part (c, A) s A-stable as a standard RK method, then the L-stablty of the addtve method (5) mples that ts frst part (c, A) s stll L-stable and ths can be verfed drectly. 4. Constructon of the addtve RK methods Our constructon of addtve RK methods s based on the satsfacton of both stablty and accuracy condtons. Correspondng to (5), we ntroduce the followng notatons for =,,..., s,σ=,, 3,... a (σ) = c σ σ j= a j c σ j b (σ) = c σ σ j= b j c σ j. (6) Then the Taylor expanson gves the order condtons for the general method (5) up to fourth order as follows (see [5]). An addtve RK method (3) s of order p 4 f and only f the condtons a () = b () =, =,,...,s, (7) a s (σ) =, b s (σ) =, σ p, (8) = = a s c τ a (σ) =, b s c τ a (σ) =, = = and the followng extra condtons for p = 4, a s = j= a j a j () =, a s c τ b (σ) =, σ + τ p, (9) b s c τ b (σ) =, σ + τ p (3) a s = j= a j b j () =, (3) a s = j= b s = j= b j a j () =, a j a j () =, a s = j= b s = j= b j b j () =, (3) a j b j () =, (33)

7 8 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) b s = j= b j a j () =, b s = j= are satsfed, where σ and τ take all possble postve nteger values. b j b j () =, (34) 5. Sem-mplct addtve RK methods As we can easly see, the addtve RK schemes of the general form (5) are fully mplct, and wll be very expensve when appled for solvng ODEs or for the tme marchng schemes of PDEs. Recall that our target system s of form (), where the lnear part f(t,y) s stff but the nonlnear part g(t, y) s not stff. It wll be more practcal to have schemes that are only mplct n terms of f(t,y)and explct n terms of g(t, y) as ths needs to solve only lnear systems at each ntermedate tme step. To further reduce the computatonal complexty, we would lke to have schemes that are only sem-mplct n terms of f(t,y), that s, the frst coeffcent matrx A s lower trangular whle the second coeffcent matrx B s strctly lower trangular n (5). Besdes, n order to take nto account certan constrant equatons n the PDEs (see Secton ), we requre that the computed value at each ntermedate step,.e., y (n) n (3), s gven mplctly and ths can be satsfed f a = for =, 3,...,s. On the other hand, t s qute necessary for every ntermedate stage y (n) of the method to gve physcal meanngful computaton, that s, y (n) should be the approxmated value of y(t) at tme t = t n + c h. Naturally, we shall take c s = and y s (n) as the fnal computed value of each tme step. In summary, the new addtve RK methods we are lookng for can now be descrbed by the followng tableau: Before startng our constructon, we shall dscuss a bt about the dfference between our new schemes and the exstng ones. The most mportant exstng addtve schemes are the ones developed n [6], whch take a ss = and can therefore be vewed as the combnaton of an (s )-stage dagonally mplct RK (DIRK) method (wth weghts b =a s,=,,...,s ) and an (s )-stage explct RK method (wth weghts b = b s, =,,...,s ). For the mplct part of those schemes, the dagonal entres take as many zeros as possble n order to make the derved scheme more effcent and ease the constructon smultaneously. On the contrary, our new schemes are requred to satsfy the condtons that a = for =, 3,...,s due to the applcaton problems n our mnd (see Secton ). If ths s dffcult, at least we should have a ss =. Now, n the sem-mplct addtve RK method (35), the (c, A) method s n fact an s-stage stffly accurate DIRK method (wth weghts b = a s, =,,...,s) and the (c, B) method s reduced to an (s )-stage explct RK method (wth weghts b = b s,=,,...,s ). Thus, our new addtve RK schemes can be vewed as the combnaton of an s-stage stffly accurate DIRK method and an (s )-stage explct RK method. To our best knowledge, there are no such addtve methods n the lterature. In fact, the constructon of such schemes are not straghtforward at all and we cannot see (35)

8 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) any drect applcaton of the exstng technques for the constructon of explct RK and DIRK methods to acheve our purpose here. Theorem below wll play the key role n our subsequent constructon of the schemes whch meet our needs. We are now gong to construct the addtve schemes of form (35) wth a = for =, 3,...,s.In order to deal wth the stffness of the lnear part f(t,y), we shall requre that the part (c, A) n (5) as a standard RK method s A(α)-stable. The algebrac condtons for a sem-explct RK method (c, A), where the frst dagonal element of A s zero, to be A-stable can be gven n terms of two sets of parameters {α } = and {β } = (see [6,4]). {β } = are defned by s ( τa rr ) = β τβ + τ β. r= Ths gves β = and β s = β s+ = =. Let e, e,...,e s be the natural bass for R s and let e = e + e + +e s be the vector wth unt elements. The terms e T s Ar e, r =,, 3,..., are the sums of the elements n rows s of A, A, A 3,...and for a method of order p t has that e T s Ar e = r!, r =,,...,p. Defne α s = α s+ = = and α r = β r β r e T s Ae + +( )r β e T s Ar e, r =,,...,s, so that α =. Then a method of order p s A-stable ff a rr for r =,,...,s and s y r r=π r ( ) r+j (β r j β j α r j α j ) j= y, where π s the ntegral part of p/ + and the astersk denotes that the terms j = r are halved. Snce n the sem-mplct addtve RK method (35), the (c, B) method s n fact reduced to be an (s )-stage standard explct RK method, the order of such methods cannot exceed s ; f s>5, the order cannot exceed s, see [7]. That s, f we want to construct a fourth-order scheme of ths type, t s at least of 5-stage. By observng Eqs. (6) (34), one can readly see that the order condtons turn out to be too tedous to deal wth. Together wth the A-stable condtons, the constructon becomes extremely dffcult. However, t wll be much easer f we have a s = b s n (35) for =,,...,s, as mplemented n [6], snce the number of the order condtons s almost halved as one can see from (6) (34). But ths conflcts wth our requrement that a ss = and b ss =. Fortunately, we have the followng soluton to overcome ths dlemma. Theorem. Assume that the s-stage addtve RK method (5) s of order p, c s = c s =, and the (s )th ntermedate stage s an approxmaton to y(t) of order p,.e., y (n) s =y(t +nh)+o(h p ). Then the method obtaned by replacng b s,s and b ss wth (b s,s + b ss ) and, respectvely, s stll of order p.

9 8 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) For the proof of ths theorem, we need the followng lemma: Lemma. Suppose Y, Z R M are two vectors and satsfy Z Y h{ A[f(Z) f(y)] + B[g(Z) g(y)] }+O(h p ), (36) where A, B R M M are two constant matrces, p s a postve nteger and f, g : R M R M are often dfferentable and have bounded frst-order dervatves. Then we have Z Y O(h p ). (37) Proof. By Eq. (36), we know that there must exst some constant C such that Z Y Ch Z Y + O(h Z Y ) + O(hp ). (38) Agan, by Eq. (36), we easly see that Z Y O(h). Usng ths and Eq. (38), we have Z Y O(h ). Repeatng ths process, we can derve (37). Proof of Theorem. Settng y (n) f(t n + c h, y Y (n) =. (n), F(Y (n) ) g(t n + c h, y (n) ) =., G(Y (n) ) ) =., y s (n) f(t n + c s h, y s (n) ) g(t n + c s h, y s (n) ) we can rewrte (3) as Y (n) = Y (n ) + h(a I)F(Y (n) ) + h(b I)G(Y (n) ), (39) where Y (k) s a block column vector consstng of s, m-dmenson column vector y s (k), A and B are the coeffcent matrces of the frst and second part of method (5), respectvely, I s the m m dentty matrx, and denotes the Kronecker product. Next, replacng b s,s and b ss wth b s,s + b ss and, respectvely, the obtaned new method for Eq. () s Z (n) = Z (n ) + h(a I)F(Z (n) ) + h( B I)G(Z (n) ), (4) where Z (n), Z (n) have the same usages as Y (n), Y (n), and B s the same as B wth only ts (s, s )- and (s, s)-elements beng b s,s + b s,s and, nstead of b s,s and b s,s of B. By subtracton of Eq. (39) and (4), we obtan Z (n) Y (n) = (Z (n ) Y (n ) ) + h{(a I)[F(Z (n) ) F(Y (n) )] + ( B I)[G(Z (n) ) G(Y (n) )]} + h[( B B) I]G(Y (n) ). (4)

10 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) In order to prove that the new method s also of order p, t s suffcent to show that the local truncaton error of z s (n) s O(h p+ ). Thus, we take n = and z s () = y s () = y. By notng that y (n) s s an approxmaton of y(t) of order p and y s (n) s of order p,wesee [( B B) I]G(Y () ) = b s,s [G(y () s ) G(y() s )] C y () s y() s O(h p ), (4) where C = b s,s L and L s the bound for g, and we have made use of the fact that the local truncaton errors of y () s and y() s are, respectvely, O(h p ) and O(h p+ ), by the assumpton of the theorem,.e., y () s y(t + h) =O(h p ), y s () y(t + h) =O(h p+ ). Hence, we derve by (4) Z () Y () h{ (A I)[F(Z () ) F(Y () )] + (B I)[G(Z () ) G(Y () )] }+O(h p+ ). (43) Ths mples, by Lemma, Z () Y () O(h p+ ). (44) Then we can deduce z s () y(t + h) z s () y s () + y s () y(t + h) Z () Y () + y s () y(t + h) O(h p+ ). (45) The proof s completed. We know from Theorem that for those pth-order addtve RK methods of form (35), where the frst (s ) stages form an (s )-stage addtve RK method,.e., a s = b s = for =,...,s, Theorem says f the frst (s )-stage s of order p, then by replacng b s,s and b ss by (b s,s + b ss ) and, respectvely, the resultng method s stll of order p. Ths provdes an mportant prncple, whch we can use to construct the desred addtve RK methods: To construct an addtve method of form (35) wth a ss = butb ss =, we can frst construct a pth order method that satsfes a s = b s ( =,,...,s), a ss = b ss =, c s = c s = and the frst (s )-stage s a (p )th order lnearly mplct addtve RK method. Then, replacng b s,s and b ss wth (b s,s + b ss ) and, respectvely, we wll acheve a new method of order p, whch s lnearly mplct of form (35) and satsfes that a ss = and b ss =. All our subsequent constructons of the new addtve RK methods are based on the above prncple. Wth ths prncple and the order condtons n Secton 4, we can construct addtve RK methods wth orders rangng from to 4. We shall gnore all the tedous dervatons but present many dfferent examples of such schemes n the subsequent sectons. For most of the schemes, the A( π )-stablty regon of the addtve RK method wll be plotted aganst the stablty regon of the correspondng explct part (c, B) as a standard explct RK method of form () for comparsons. Some numercal experments wth these new RK methods wll be presented to demonstrate ther accuraces, stabltes and effcences. In the sequel, we shall use the descrptons lke addtve RK.m.A.n or RK.m.L.n for some postve ntegers m and n. addtve RK.m.A(L).n ndcates an addtve RK method, whch s mth-order accurate and A-stable (or L-stable). The number n means t s the nth method of ths class lsted n ths paper.

11 84 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) Examples of addtve RK methods wth nonzero dagonal entres Ths secton presents some new addtve RK methods wth nonzero dagonal entres a for =, 3,...,s. The followng tableau gves a new class of addtve RK methods of order whch s A-stable: where <c<, α, β are postve and satsfy α β [αβ (α + β) + ]. If we take α = β β, β > orβ <, then the resultng method s L-stable. We frst take c = /, α = β = and get the followng A-stable one: Its stablty functon R(z f,z g ) as defned n () s R(z f,z g ) = ( z f z f ) + ( z f )z g + z g z f + z f and ts A( π )-stablty doman s plotted n Fg. (left). One can observe that the A( π )-stablty doman of the addtve method,.e. S f g π/, s slghtly smaller than that of the explct part as a conventonal RK method,.e. S g. For addtve stff problem (), f we only use the explct part, whch s a second-order

12 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) Fg.. Stablty doman of addtve (sold lne) and explct RK (dot) methods Fg.. Stablty doman of addtve (sold lne) and explct RK (dot) method. standard RK method, the step length should be chosen to satsfy (λ f + λ g )h S g, where λ f σ{ f y } and λ g σ{ g y }, n order to meet the stablty requrement. But as dscussed earler, the addtve method needs only to satsfy that λ g h S f g π/. Snce Re{λ f } > Re{λ g }, eventhough the stablty doman of the addtve method s smaller than that of ts explct part as n the conventonal RK method, t allows a relatvely much larger range to choose the step length h. Moreover, consderng the effcency, we can see that such an addtve method s defntely superor to the pure mplct method as t does not need to solve any nonlnear equatons. So the addtve methods of ths new type present some advantages over the exstng RK methods for solvng the addtve system (). Below are two more examples wth A-stablty by takng α=/, β=/, c =/ and α=/, β=/, c = /4, respectvely: These two methods have the same stablty functon R(z f,z g ), R(z f,z g ) = ( 4 z f ) + z g + z g z f + 4 z f

13 86 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) and ther A( π )-stablty doman s shown n Fg. (rght). One can see that these two schemes have a relatvely larger stablty regon (n fact, they almost concde wth that of the explct RK method), and thus should be more preferable n solvng those not too stff problems. The followng are two L-stable schemes: The stablty functons of these two schemes are, respectvely, R(z f,z g ) = [ + ( )z f ]+[ + ( )z f ]z g + /zg ( )z f + (3/ )zf, R(z f,z g ) = ( + 7/4z f ) + ( + 7/4z f )z g + /z g 3/4z f + 3/4z f and ther A( π )-stablty domans are gven n Fg. (left for RK..L. and rght for RK..L.). Next, we provde some thrd-order methods that should be at least of 4-stage as we ponted out earler. But one can show that there exsts a 4-stage lnearly mplct RK method of order 3 f and only f a 44 = (the proof s omtted here). Ths does not meet our requrement that a ss = for an s-stage method. So we can only try to fnd some 5-stage thrd-order methods. Ths s possble, and some A-stable schemes are gven below:

14 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) wth ts A( π )-stablty doman as shown n Fg. 3 (left). wth ts stablty regon as shown n Fg. 3 (rght). wth ts A( π )-stablty doman gven n Fg. 4 (left). Below s one example of L-stable schemes, wth ts A( π )-stablty doman gven n Fg. 4 (rght). Fnally, we present some fourth-order methods.a few 6-stage schemes wth a 44 = are gven n Secton 7, whch are applcable n some specal cases. Here, we lst two 7-stage A-stable methods.

15 88 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) Fg. 3. Stablty doman of addtve (sold lne) and explct RK (dot) method Fg. 4. Stablty doman of addtve (sold lne) and explct RK (dot) method. wth ts A( π )-stablty doman plotted n Fg. 5 (left). Its A( π )-stablty doman s plotted n Fg. 5 (rght). 7. Numercal experments In ths secton we carry out some numercal experments to attest the stablty and accuracy of the addtve RK methods constructed n Secton 6. Some notatons are needed. We shall use E(h) to denote the dscrete L -norm error between the computed soluton y h (t) by an addtve RK method and the exact soluton y(t) to system (), whle r(h)

16 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) Fg. 5. Stablty doman of addtve (sold lne) and explct RK (dot) method. measures the asymptotc convergence rate. The two parameters are gven by (wth h >h ) E(h) = h y(t ) y h (t ), r(h) = ln E(h / ) ln h. E(h ) h In our tests of numercal schemes, we wll use the usual practce to compute the numercal solutons n the transent phase where the stff terms f(t,y)n () contrbute to the solutons of the consdered stff problems and outsde ths phase the stff terms de out. The numercal solutons are computed wth a relatvely hgh-order explct method usng much smaller tme steps for accuracy n the transent phase. In all the tests, we take the fourth-order explct RK methods for computatons n the transent phase. Example. The frst model problem s taken to be y = Ay + g(y), y() =[,, ] T, (46) where y =[y,y,y 3 ] T, and A and g(y) are gven by [ ] 9 a A = 9, g(y) = b y [ y ] y, y 3 where two parameters a and b are delberately ntroduced to test dfferent stuatons. Snce σ(a) = {, 4 + 4, 4 4}, the problem s of type (). We shall test the cases wth dfferent a s and b s. Frst, we take a =, b= and, respectvely, to check the stablty of both RK..A. and RK.3.A.3, wth the tme range taken to be [, 5]. () b =, a =. The exact soluton n ths case s y (t) = /e 5t (cos 4t + sn 4t) + /e t, y (t) = /e 5t (cos 4t + sn 4t) + /e t, y 3 (t) = e 5t (sn 4t cos 4t). Fg. 6 s the numercal result of RK..A. to ths case wth h =.3,.,..

17 9 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) x y 3 y y y y y y y y Fg. 6. Numercal results of RK..A. wth h =.3,.,. n turn..8.6 y(t) y(t) y3(t) Fg. 7. True soluton of Eq. (46) (a =,b= ) computed wth the fourth-order RK method. Table Egenvalues of g/ y at dfferent tme ponts n the case a = and b = t λ g We observe that f the step length h gets larger, the numercal solutons vbrate more strongly and even dverge. If only the step length s chosen to satsfy that ah (here a s the λ g n ()) falls nto the A( π )-stablty doman of the method used, t s safe to have the numercal soluton to be damped. () b =, a =, where the theoretcal soluton s computed usng the 4-stage and fourth-order explct RK method (see [8]), whch s plotted n Fg. 7. Below we lst the egenvalues of g/ y at dfferent tme ponts (Table ):

18 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) y.6 y y y y y y y y Fg. 8. Numercal results of RK.3.A.3 wth h =.3,.,. n turn. Table Egenvalues of g/ y at dfferent tme ponts n the case a = and b = t λ g Table 3 Convergence order of the method RK.3.A. h E(h).4e 5.49e 6.93e 7.459e 8 3.3e 9 r(h) Table 4 Convergence order of the method RK.3.A.3 h E(h).7e 5.884e e e e 9 r(h) Fg. 8 shows the numercal behavors of RK.3.A.3 wth h =.3,.,., respectvely. We can also see that f λ g h falls nto the A( π )-stablty doman of the method, t s safe to have the decay result. () b =,a=, the theoretcal soluton computed as n the prevous case. The egenvalues of g/ y at some tme ponts are lsted n Table. Now, wth ths model, the convergence order of some schemes can be computed as n the followng tables. Table 3 s for RK.3.A.. Tables 4 and 5 are for RK.3.A.3 and RK..A., respectvely. To accurately fnd the exact convergence order of the fourth-order scheme RK.4.A., we construct the followng model: y = λy + αy, y() = (47)

19 9 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) Table 5 Convergence order of the method RK..A. h E(h).48e e 6.494e e 7 9.3e 8 r(h).... Table 6 Convergence order of the method RK.4.A. h E(h) 3.693e e 9.478e.76e.96e 4.47e 5 r(h) Table 7 Convergence order of the method RK.4.A. h E(h).e 8.9e e 5.e 9.9e 5 5.8e 6 r(h) whch has the exact soluton λe λt y(t) = α( e λt ) + λ. (48) In ths, we take λ =, α = and snce the stffness s brought n the consderaton of the soluton at the transent phase (whch we may regard as [,]) s taken as the true soluton, whch s equvalent to takng t =, and the computaton s carred out as n [,6] and we arrve at results shown n Table 6. The convergence of RK.4.A. s lsted n Table 7. From all the prevous tables, one can see that the numercal experments have clearly verfed the actual orders of the correspondng addtve RK methods. 8. Methods wth zero dagonal elements In ths secton, we wll gve a few addtve methods of form (35), whch allow some dagonal elements a to be zero, but the last entry a ss =. The followng scheme s the frst one lsted n Secton 6 wth α =, β = c = :

20 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) Ths method can be vewed as a drect combnaton of the Crank Ncolson method and the modfed md-pont formula [], and ts stablty functon as defned n () s R(z f,z g ) = + /z f + ( + /z f )z g + /z g /z f. The A( π )-stablty doman of the addtve method and the stablty doman of ts explct part (c, B) as a standard RK method are shown n Fg. 9 (left). In the remanng part of ths secton, we present some more such addtve methods wth order 3 and 4. where a,b,c,d are 4 real parameters satsfyng a>, b = 3a 6a 3, b + 4a + 8ab 5 3 = ( 6b)c. Takng a =,b = /3,c = 3,d = and a = /3,b=,c= 5/3,d =, we get the followng two schemes: These two methods have the same stablty functon and ther A(π/)-stablty doman s shown n Fg. 9 (rght).

21 94 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) Fg. 9. Stablty doman of addtve (sold lne) and explct RK (dot) method Fg.. Stablty doman of addtve (sold lne) and explct RK (dot) method. and ts A( π )-stablty doman s plotted n Fg. (left). Its A( π )-stablty doman s shown n Fg. (rght).

22 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) Table 8 Convergence order of the method RK.4.nA.5 h E(h).8e 5.44e 6 7.5e e 9.67e 7.68e r(h) Table 9 Convergence order of the method RK.4.nA.6 h E(h) 7.43e e 6.63e 7.6e 8 4.e.56e r(h) Some not A-stable fourth-order methods In ths secton, we present some fourth-order methods that are not A-stable, whch may be used for the nonlnear ODEs or PDEs of addtve form () that are not stff. The followng tables verfy the fourth-order convergence of addtve RK.4.nA.5 (Table 8) and addtve RK.4.nA.6 (Table 9) when appled to the model problem (47) n the non-stff case wth λ =. The followng are two more such schemes of fourth-order that are not A-stable:

23 96 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) Some applcatons In ths secton we dscuss brefly how to apply our new addtve RK methods to solve two mportant PDE systems wth constrants... Knematc magnetc nducton system In the knematc geodynamo modelng, the followng mean-feld dynamo magnetc nducton system s wdely used, B = (β(x) B) + f(x, t; u, B) n Ω (, T), (49) t B = n Ω (,T), (5) wth approprate ntal and boundary condtons [,3], where B s the magnetc feld of the dynamo system, β(x) s the magnetc Reynolds number, and f(, x, t; u, B) s a nonlnear vector-valued functon of flow u and magnetc feld B, whch determnes the key dynamo system so that the magnetc feld can sustan n the correspondng physcal system. Now, as an example, we apply the addtve method RK..A. to the temporal sem-dscretzaton of ths system and obtan the followng scheme advancng from tme t n to t n+ : B n = B n h (β(x) B n ) + h f(x, t n; u n, B n ), (5) B n =, (5)

24 H. Lu, J. Zou / Journal of Computatonal and Appled Mathematcs 9 (6) B n+ = B n h [β(x) (B n + B n+ )]+h f(x, t n+/ ; u n+/, B n ), (53) B n+ =, (54) where B n B(t n ), B n B(t n+/), B n+ B(t n+ ). Clearly, f explct schemes are used, then the dvergence constrants cannot be enforced for both stage values B n and B n+. When mplct schemes as above are used, systems (5) (5) and (53) (54) for the two stage values B n and B n+ are well-posed [,3]... Naver Stokes system The Naver Stokes system s a general model for ncompressble flud dynamcs: u = ν Δu p (u )u + f n Ω (,T), (55) t u = n Ω (,T), (56) where u s the flud velocty, p s the pressure and f s an external source, and ν s the vscosty of the flud. Applyng the addtve method RK..L. to ths system, we obtan the followng scheme: where u n = u n + h (νδu n p n ) + h 5 (νδu n p n ) + h 4 { (u n )u n + f n }, (57) un =, (58) u n+ = u n + h 8 (νδu n p n ) + h (νδu n p n ) + 3h 8 (νδu n+ p n+ ) h{ (u n )u n + f n }+h{ (u n )u n + f t n +/4}, (59) u n+ =, (6) (u n,p n ) (u(t n ), p(t n )), (u n,p n ) (u(t n+/4), p(t n+/4 ). Clearly, f explct schemes are used, then the dvergence constrants cannot be enforced for both stage values u n and u n+. When mplct schemes as above are used, systems (57) (58) and (59) (6) for the two stage values (u n,p n ) and (u n+,p n+ ) are well posed. Acknowledgements The authors wsh to thank Prof. Geng Sun for some helpful dscussons and two anonymous referees for ther constructve comments whch led to a great mprovement of the results and the presentaton of the paper.

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