Integrating Factor Methods as Exponential Integrators

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1 Integrating Factor Methods as Exponentia Integrators Borisav V. Minchev Department of Mathematica Science, NTNU, 7491 Trondheim, Norway Abstract. Recenty a ot of effort has been paced in the construction and impementation of a cass of methods caed exponentia integrators. These methods are preferabe when one has to dea with stiff and highy osciatory semiinear probems, which often arise after spatia discretization of Partia Differentia Equations (PDEs). The main idea behind the methods is to use the exponentia and some cosey reated functions inside the numerica scheme. In this note we show that the integrating factor methods, introduced by Lawson in 1967, are aso exampes of exponentia integrators with very specia structure for the reated exponentia functions. In order to prove this reation, we use the approach based on bi-cooured rooted trees and B-series. We aso show under what conditions every bi-cooured rooted tree can be express as a inear combination of standard non-cooured rooted trees. 1 Introduction Reaistic modes of many physica processes require effective numerica sovers for a specia cass of partia differentia equations, which after semidiscretization in space can be written in the foowing form u = Lu + N(u(t)), u(t 0 ) = u 0, (1.1) where u : R R d, L R d d, N : R d R d and d is a discretization parameter equa to the number of spatia grid points. Severa interesting probems can be brought to this form. Exampes are Aen-Cahn, Burgers, Cahn-Hiiard, Kuramoto-Sivashinsky, Navier- Stokes, Swift-Hohenberg, noninear Scrödinger equations. Typicay the inear part of the probem wi be stiff and the noninear part wi be nonstiff. Many numerica integrators have been deveoped to overcome the phenomenon of stiffness. Exponentia integrators was introduced in the eary sixties as an aternative approach for soving

2 2 stiff systems. The main idea behind these methods is to integrate exacty the inear part of the probem and then use an appropriate approximation of the noninear part. Thus the exponentia function, and functions which are cosey reated to the exponentia function, appear in the format of the method. This was the reason why, unti recenty, these methods have not been regarded as practica. The atest achievements in the fied of computing approximations to the matrix exponentia, have provided a new interest in the construction and impementation of exponentia integrators [2, 3, 6, 7, 9]. The main requirements imposed on the functions, which appear in the format of an exponentia integrator are, to be anaytic, map the spectrum of L into a bounded region in C and can be computed exacty or up to arbitrary high order cheapy. Suppose that, for a N and λ R, the operators φ [] (λ) : R d d R d d satisfy the above conditions and can be expanded in the form φ [] (λ)(hl) = j 0 φ [] j (λ)(hl)j. The φ [] functions, which are used in practice are associated with the so caed Exponentia Time Differencing methods [3, 4, 10, 11], and can be written expicity as φ [] (λ)(hl) = (λhl) (e λhl 1 ) (λhl) k. (1.2) k! If h represents the stepsize and U i denotes the interna stage approximation of the exact soution for i = 1, 2,..., s then the computations performed in step number n of an exponentia Runge Kutta (RK) method are reated by the equations U i = u n = j=1 j=1 =1 =1 α [] ij φ[] (c i )(hl) hn(u j ) + e c ihl u n 1, β [] j φ [] (1)(hL) hn(u j ) + e hl u n 1, (1.3) where α [] ij and β[] j are the parameters of the method and the vector c = (c 1, c 2..., c s ) T is the abscissae vector. If α [] ij = 0 for a j i

3 the method is expicit and impicit otherwise. Aternativey the computations performed in step number n can be represented in a more Runge Kutta type formuation as foows c α [1] α [2] α [s] β [1]T β [2]T β [s]t, where each eement in row number i of the matrix α [] is mutipied by φ [] (c i )(hl) and each eement in the vector β []T is mutipied by φ [] (1)(hL). The resuting matrices are then added in a component by component sense. Other important cass of methods which are aso used for soving the semiinear probem (1.1) are the Integrating Factor (IF) methods. The idea behind these methods goes back to the work of Lawson [8]. He proposes to ameiorate the effect of the stiff inear part of equation (1.1) by using change of variabes (aso known as Lawson transformation), v(t) = e tl u(t). The initia vaue probem (1.1) written in the new variabe is v (t) = e tl N(e tl v(t)) v(t 0 ) = v 0, (1.4) where v 0 = e t0l u 0. The approach now is to appy an arbitrary s- stage Runge Kutta method to the transformed equation (1.4) and then to transform the resut back into the origina variabe. Thus, a method which satisfy just the nonstiff order conditions wi not suffer from sever order reduction, when it is appied to stiff probems. The aim of this paper is to show that the IF methods are exampes of exponentia RK methods with specia choices for the φ [] functions and the parameters α [] and β []T. We aso prove that in this specia case the nonstiff order theory for the exponentia RK methods reduces to the cassica Runge Kutta order theory, which expains why the IF methods exhibit the expected order. The paper is organized as foows: we briefy survey the nonstiff order theory for the exponentia RK methods in Section 2. Next, in Section 3, we define the structure the matrices α [] and the vectors β []T as we as the form of the functions φ [], which correspond to the IF methods. Finay, in Section 4, we concude with severa remarks and questions of future interest. 3

4 4 2 Nonstiff order conditions The nonstiff order theory for the exponentia RK methods was first constructed in [4] and ater deveoped in [11]. Here we foow the approach suggested in [9]. It is based on bi-cooured rooted trees and B-series. For those not famiiar with these concepts we suggest the monographs [1, 5] for a compete treatment. Let 2T denote the set of a bi-cooured (back and white) rooted trees with the requirement that the vaency of the white nodes is aways one. This correspods to the fact that the first term on the right hand side of (1.1) is inear. Let represents the empty set which remains if the root of the one node tree or is removed. The order of the tree τ 2T is defined as the number of vertices in the tree, and it is denoted by τ. The density γ of the tree is defined as the product over a vertices of the order of the subtree rooted at that vertex. An exponentia Runge Kutta method with eementary weight function a : 2T R, has nonstiff order p, if for a τ 2T, such that τ p, a(τ) = 1/γ(τ). In order to give a practica representation of the eementary weight function a of the numerica soution, it is convenient to introduce some notations. Let for = 0, 1,... and k = 1,..., m the s s matrix φ [k] (c) = diag Define ( φ [k] (c 1 ),..., φ [k] (c s ) A [] = b []T = k=1 k=1 φ [k] (c)α [k], φ [k] (1)β [k]t, C [] = 1 (+1)! C+1. ) and C = diag(c 1,..., c s ). (2.1) The eementary weight function a of the numerica soution can be computed using the foowing non-recursive rue: Attach b [j]t to the root back node. Attach A [j] to a remaining nontermina back nodes. Attach A [j] e to a termina back nodes. Attach C [j] e to a termina white nodes. Attach I to a remaining white nodes.

5 The vaue j is the number of white nodes directy beow the corresponding node, I is the s s identity matrix and e = (1,..., 1) T. Now for each tree mutipy from the root to the eaf as in the case for Runge Kutta methods, then mutipy these expressions in a component by component sense. 3 IF methods as a specia case Appying a standard s-stage Runge Kutta method to the transformed equation (1.4) and then transforming back the resut into the origina variabe, eads us to the foowing φ [] functions φ [] (λ)(hl) = e (λ c )hl. (3.1) Every IF method can be represented in the form (1.3) with φ [] functions given by (3.1) and with a specia choice of the coefficient matrices α [] and the coefficient vectors β []T. This choice reduces the set of a order conditions to a set, which consists ony of the order conditions corresponding to the back trees. To proof of this fact we need the foowing emma. Lemma 1. Let t R\{0, 1, 2,...}, then for j = 0, 1, 2,... j ( 1) k 1 k!(j k)! (k + t) = 1 t(t + 1) (t + j). Proof. The proof of this statement is by induction on j. The foowing theorem defines the structure of the matrices α [] and the vectors β []T for the IF methods. With this structure of the coefficients, to achieve certain nonstiff order, it is sufficient to satisfy ony the back trees. This impies that the transformed differentia equation (1.4) is soved using a Runge Kutta method. Theorem 1. Let a the non-zero coefficients of an exponentia Runge Kutta method (1.3), with φ [] functions given by (3.1) be ocated in coumn number of the matrix α [] and in position number of the vector β []T for = 1, 2,..., s. The method has nonstiff order p iff a order conditions corresponding to the back trees are satisfied. 5

6 6 Proof. It foows directy that if the exponentia Runge Kutta method has nonstiff order p then a order conditions corresponding to the back trees are satisfied. Let us assume that a the order conditions corresponding to the back trees are satisfied. We need to prove that a the remaining order conditions are aso satisfied. From the definition of the φ [] functions (3.1), it foows that for j = 0, 1, 2,..., φ [] j (1) = (1 c ) j j!, φ [] j (c) = 1 j! diag((c 1 c ) j,..., (c s c ) j ).(3.2) Since a order conditions corresponding to the back trees invove ony the coefficients A [0], b [0]T and c, we need to express every other order condition in terms of these coefficients. Having in mind the specia structure of the matrices α [] and the vectors β []T, after substituting (3.2) into (2.1), we obtain for j = 1, 2,..., A [j] = b [j]t = j j ( 1) k k!(j k)! C[0]j k A [0] C [0]k, ( 1) k k!(j k)! b[0]t C [0]k. (3.3) From the fact that a order conditions corresponding to the back trees are satisfied, it foows that A [0], b [0]T and c form a Runge Kutta method. Therefore, C [0] e = A [0] e, C [0] ζ = (A [0] e)ζ, C [0]k ζ = (A [0] e)... (A [0] e)ζ, (3.4) where ζ is an arbitrary vector and the mutipications between the eements in the brackets are in a component by component sense. Now, we are in a position to define a procedure which transforms every cooured tree τ into a inear combination of back trees of order at most τ. Each tree τ can be decomposed as τ = (, τ j, ), where is a cooured tree on the bottom with ess number of white nodes than τ; τ j is ta white tree with j 1 white nodes and is back tree on the top. First appying formua (3.3) and then (3.4), for the order condition corresponding to a tree τ, we obtain the foowing three representations in terms of back trees or trees with

7 7 ess white vertices. If = then τ reduces to If = then τ reduces to τ = = 1 j!. τ = = δ δ k + + δ j, where δ k = ( 1)k for k = 0, 1, 2,..., j. In the genera case when k!(j k)! τ {t,b}, then τ can be represented as τ = = δ δ k + + δ j. (3.5) For each of the trees in the inear combination we appy the same procedure. Thus, after a finite number of steps a the trees in the combination wi be back. From (2.1) it is cear that the order of every singe back tree cannot exceed the order of cooured tree. To compete the proof we need to show that a(τ) = 1/γ(τ) for a cooured trees τ, where τ p. We prove this by induction on the number of steps θ in the transformation process. Let θ = 1. Every cooured tree τ has representation τ = j δ kτ k, where a τ k are back trees. If γ( ) = x 1 x 2 then a(τ k ) = 1 = 1 γ(τ k ) x 1 ( +k)x 2 and by Lemma 1 for t = it foows that a(τ) = = j ( 1) k k!(j k)! a(τ k) = j ( 1) k 1 k!(j k)! x 1 ( + k)x 2 1 x 1 ( + 1) ( + j)x 2 = 1 γ(τ). Assume that a(τ) = 1/γ(τ) for a cooured trees τ with θ steps in the transformation process. Let e a tree with θ + 1 steps in the transformation process. From (3.5) it foows that τ = j δ kτ k,

8 8 where τ k are cooured trees with θ steps in the transformation process and hence a(τ k ) = 1/γ(τ k ). If γ( ) = x 1 x 2 then a(τ k ) = 1 = 1 γ(τ k ) x 1 ( +k)x 2 and by Lemma 1 for t = it again foows that a(τ) = 1/γ(τ). 4 Concusions We have shown that the IF methods are exampes of exponentia Runge Kutta methods with specia structure of the coefficients matrices and the reated φ [] functions. We have aso proven that, in this specia case, the nonstiff order theory for the exponentia RK methods reduces to the cassica Runge Kutta order theory. This expains why the IF methods exhibit the expected order. Other exampes of the φ [] functions, rather than (1.2) and (3.1), arise from the framework of Lie group methods see [9]. The question how to find the best set of φ [] functions is open and needs further investigation. References 1. J. C. Butcher, Numerica methods for ordinary differentia equations, John Wiey & Sons, M. Cavo and C. Paencia, A cass of expicit mutistep exponentia integrators for semiinear probems, preprint, P. M. Cox and P.C. Matthews, Exponentia time differencing for stiff systems, J. Comput. Phys. 176 (2002), A. Friedi, Veragemeinerte Runge Kutta verfahren zur öesung steifer differentiageichungssysteme, Lect. Notes Math. 631 (1978). 5. E. Hairer, C. Lubich, and G. Wanner, Geometric numerica integration, Springer, 2003, Number 31 in Springer Series in Computationa Mathematics. 6. M. Hochbruck and A. Osterman, Expicit exponentia Runge Kutta methods for semiinear paraboic probems, Submitted to SIAM J. Numer. Ana., Exponentia Runge Kutta methods for paraboic probems, To appear in App. Numer. Math., J. Lawson, Generaized Runge Kutta processes for stabe systems with arge Lipschitz constants, SIAM J. Numer. Ana. 4 (1969), B. Minchev, Exponentia integrators for semiinear probems, Ph.D. thesis, University of Bergen, S. Nørsett, An A-stabe modification of the Adams-Bashforth methods, Lect. Notes Math. 109 (1969), K. Strehme and R. Weiner, B-convergence resuts for ineary impicit one step methods, BIT 27 (1987),