Weak second order explicit exponential Runge-Kutta methods for stochastic differential equations

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1 CSSE-44 September 8, 015 Weak second order explicit exponential Runge-Kutta metods for stocastic differential equations Yosio Komori, David Coen and Kevin Burrage, Department of Systems Design and Informatics Kyusu Institute of Tecnology Department of Matematics and Matematical Statistics Umeå University Department of Computer Science, University of Oxford Scool of Matematics, Queensland University of Tecnology Kawazu, Iizuka, , Japan Umeå, Sweden Wolfson Building, Parks Road, Oxford, UK Brisbane, Australia Abstract We propose new explicit exponential Runge-Kutta metods for te weak approximation of solutions of stiff Itô stocastic differential equations SDEs. Tese metods ave weak order two for multi-dimensional, non-commutative SDEs wit a semilinear drift term, wereas tey are of order two or tree for semilinear ordinary differential equations. Tese metods are A-stable in te mean square sense for a scalar linear test equation wose drift and diffusion terms ave complex coefficients. We perform numerical experiments to compare te performance of tese metods wit an existing explicit stabilized metod of weak order two.

2 1 Introduction For stiff ordinary differential equations ODEs, tere are some classes of explicit metods tat are well suited. One suc class is te class of Runge-Kutta Cebysev RKC metods. Tey are useful for stiff problems wose eigenvalues lie near te negative real axis. Van der Houwen and Sommeijer [30] ave constructed a family of first order RKC metods. Abdulle and Medovikov [3] ave modified tis class and ave proposed a family of second order RKC metods. Anoter suitable class of metods is te class of explicit exponential Runge-Kutta RK metods for semilinear problems [10, 14, 15, 16, 1, 6]. Altoug tese metods were proposed many years ago, tey ave not been regarded as practical until recently because of te cost of calculations for matrix exponentials, especially for large problems. In order to overcome tis problem, new metods ave been proposed [1, 14, 15, 16]. Similarly, for stocastic differential equations SDEs explicit RK metods tat ave excellent stability properties ave been developed. Abdulle and Cirilli [1] ave proposed a family of explicit stocastic ortogonal Runge-Kutta Cebysev SROCK metods wit extended mean square MS stability regions. Teir metods ave strong order one alf and weak order one for non-commutative Stratonovic SDEs, wereas tey reduce to te first order RKC metods wen applied to ODEs. Abdulle and Li [] ave proposed SROCK metods of te same order for non-commutative Itô SDEs. Komori and Burrage [19] ave developed tese ideas and ave proposed weak second order SROCK metods for non-commutative Stratonovic SDEs. If te metods are applied to ODEs, tey reduce to te second order RKC metods of Abdulle and Medovikov [3]. Komori and Burrage [0] ave also proposed strong first order SROCK metods for non-commutative Itô and Stratonovic SDEs, wic reduce to te first or second order RKC metods for ODEs. Te weak second order SROCK metods given by Komori and Burrage [19] ave te advantage tat te stability region is large along te negative real axis, but tey still ave a drawback, tat is, teir stability region is not so wide. In order to overcome tis drawback, Abdulle, Vilmart and Zygalakis [5] ave proposed a new family of weak second order SROCK metods for non-commutative Itô SDEs, in wic anoter family of second order RKC metods is embedded. On te oter and, Si, Xiao and Zang [8] ave proposed an exponential Euler sceme for te strong approximation of solutions of SDEs wit multiplicative noise driven by a scalar Wiener process. Coen [7] and Tocino [9] ave proposed exponential integrators for second order SDEs wit a semilinear drift term and multiplicative noise. Adamu [6], Geiger, Lord and Tambue [11], and Lord and Tambue [] ave proposed exponential integrators for stocastic partial differential equations wit a semilinear drift term and multiplicative noise. Komori and Burrage [18] ave proposed anoter explicit exponential Euler sceme for non-commutative Itô SDEs wit a semilinear drift term, wic is of strong order one alf and A-stable in te MS. In te present paper, we derive stocastic exponential Runge-Kutta SERK metods for te weak approximation of solutions of non-commutative Itô SDEs wit a semilinear drift term. We will acieve tis on te basis of te derivative free Milstein-Talay DFMT metod proposed by Abdulle et al. [4, 5] and explicit exponential RK metods for ODEs proposed by Hocbruck and Ostermann [15]. In Section we will briefly introduce explicit exponential RK metods for ODEs. In Section 3 we will derive our SERK metods, and in Section 4 we will give teir stability analysis. Section 5 will present numerical results 1

3 and Section 6 our conclusions. Explicit exponential RK metods for ODEs We consider autonomous semilinear ODEs given by y t = Ayt + fyt, t > 0, y0 = y 0,. 1 were y is an R d -valued function on [0,, A is a d d matrix and f is an R d -valued nonlinear function on R d. In order to introduce some exponential RK metods for. 1, we make te following assumption [15]: Assumption.1 For a given time T > 0,.1 satisfies te conditions below. 1 Tere exists a constant C suc tat for all t [0, T ]. e ta C Te nonlinear function f is locally Lipscitz continuous in a local region U wic contains te exact solution y on [0, T ], tat is, {yt t [0, T ]} U. 3 Te solution y is a sufficiently smoot function on [0, T ] and f is sufficiently often differentiable in U. All occurring derivatives of y and f are uniformly bounded in [0, T ] and U, respectively. Remark tat te global error estimation of all exponential RK metods introduced in tis section can be influenced by te constant C [15]. By te variation-of-constants formula, te solution of. 1 is yt n+1 = e A yt n + tn+1 t n e At n+1 s fysds.. Let y n denote a discrete approximation to te solution yt n of. 1 for an equidistant def grid point t n = n n = 1,,..., M wit step size = T/M < 1 M is a natural number. By interpolating fys at fy n only, we obtain te simplest exponential sceme for. 1 [16]: y n+1 = e A y n + ϕ 1 Afy n,. 3 were ϕ 1 Z def = Z 1 e Z I and I stands for te d d identity matrix. Tis is called te explicit exponential Euler metod. Higer order exponential RK metods ave been proposed in [15, 16]. For example, te following is a one-parameter family of second order exponential RK metods: Y 1 = e ca y n + c ϕ 1 c Afy n, { y n+1 = e A y n + ϕ 1 A 1 } ϕ A fy c n + 1 ϕ AfY 1, c. 4

4 were c is a parameter and ϕ Z def = Z e Z I Z. In addition to Assumption.1, let us assume tat tere exists a constant C suc tat n 1 A e ka C for n =, 3,..., M. Note tat te global error estimation of te following family of exponential RK metods can be also influenced by te above constant C [15]. Ten, a two-parameter family of tird order exponential RK metods is given by Y 1 = e c A y n + c ϕ 1 c Afy n, Y = e c3a y n + {c 3 ϕ 1 c 3 A ψa} fy n + ψafy 1, { y n+1 = e A y n + ϕ 1 A γ + 1 } ϕ A fy γc + c n 3 + ϕ A {γfy 1 + fy }, γc + c 3 were c, c 3 and γ are parameters satisfying γc + c 3 = 3 γc + c and ψz def = γc ϕ c Z + c 3 c ϕ c 3 Z. 3 Weak second order SERK metods We sall now derive SERK metods of weak order two by utilizing some results for a well-designed existing stocastic Runge-Kutta SRK metod. For tis, we give a brief introduction to te SRK metod in te first subsection. After tis, we will present SERK metods in te second subsection. 3.1 Te derivative free Milstein-Talay metod Similarly to te case of ODEs, we are concerned wit autonomous SDEs wit a semilinear drift term given by dyt = Ayt + fytdt + g j ytdw j t, t > 0, y0 = y 0, 3. 1 were g j, j = 1,,..., m are R d -valued functions on R d, te W j t, j = 1,,..., m are independent Wiener processes and y 0 is independent of W j t W j 0 for t > 0. In order to deal wit weak approximations for 3. 1, let g 0 y denote Ay + fy and 3

5 let us consider te following DFMT metod [4, 5]: K 1 = y n + g 0 y n, K = K 1 + g j y n ξ j, y n+1 = y n + {g 0 y n + g 0 K } } + 1 {g j y n + g k y n ζ kj g j y n g k y n ζ kj y + {g n + K 1 j + g k y n χ k } y +g n + K 1 j g k y n χ k ξ j, 3. were te χ j and ξ j, j = 1,,..., m are discrete random variables satisfying P χ j = ±1 = 1, P ξ j = ± 3 = 1 6, P ξ j = 0 = 3 and te ζ kj, j, k = 1,,..., m are given by ζ kj def = ξ j ξ j 1/ j = k, ξ k ξ j χ k / j < k, ξ k ξ j + χ j / j > k. Let C L P Rd, R denote te family of L times continuously differentiable real-valued functions on R d, wose partial derivatives of order less tan or equal to L ave polynomial growt. Wenever we deal wit weak convergence of order q, we will make te following assumption [17, p. 474]: Assumption 3.1 All moments of te initial value y 0 exist and g j j = 0, 1,..., m are Lipscitz continuous wit all teir components belonging to C q+1 P R d, R. Ten, we can give te definition of weak convergence of order q [17, p. 37]: Definition 3.1 Wen discrete approximations y n are given by a numerical metod, we say tat te metod is of weak global order q if for all G C q+1 P R d, R, constants C > 0 independent of and δ 0 > 0 exist, suc tat E[GyT ] E[Gy M ] C q, 0, δ 0. In order to consider numerical metods of weak order q, te following teorem is very useful, wic as been originally proposed by Milstein [4] see [5, p. 100] and wic is very often utilized by oter researcers [4, 5, 7]. Teorem 3.1 In addition to Assumption 3.1, suppose tat te following conditions old: 1 for sufficiently large r, te moments E[ y n r ] exist and are uniformly bounded wit respect to M and n = 0, 1,..., M; 4

6 for all G C q+1 P R d, R, te local error estimation E[Gytn+1 ] E[Gy n+1 ] Kyn q+1 olds if yt n = y n, were K C 0 P Rd, R. Ten, te sceme tat gives y n n = 0, 1,..., M is of weak global order q. Te second condition concerning te local error in te teorem provides us order conditions for an SRK metod to be of weak order q [7]. In addition, te DFMT metod is of weak order two [4]. Tese facts give us a way of deriving new SRK metods of weak order two [4, 5]. For tis, we propose a useful lemma to give a sufficient condition for SRK metods based on te DFMT metod to satisfy te second condition in Teorem 3.1. Lemma 3.1 For an approximate solution y n, let y n+1 be given by 3.. For te y n, let ŷ n+1 be given by ŷ n+1 = ỹ n+1 + g 0 Y 1 + g 0 Y ξ j {g j Y 3 + g k Y 3 ζ kj g j Y 3 {g j Y 4 + g k Y 5 χ k } +g j Y 4 g k Y 5 χ k ξ j } g k Y 3 ζ kj and assume tat ỹ n and Y i, i = 1,,..., 5 ave no random variable and satisfy ỹ n+1 + g 0 Y 1 = y n + g 0 y n + g 0 y n g 0 y n + O 3, 3. 3 Y i = y n + a i + O i = 1,, 3, 5, Y 4 = y n + g 0 y n + a 4 + O 3, were a i, i = 1,,..., 5 are vectors independent of. Note tat te symbol O p represents terms x suc tat x Ky n p for an K C 0 P Rd, R and a small > 0. Ten, for all G C r P Rd, R r 3 E [ G ŷ n+1 ] E [ G yn+1 ] = O 3. 5

7 Proof. As g 0 Y 1 + = g 0 Y / 6 j,k,l=1 g j Y ξ j g 0 y n g j y n ξ j + j, g 0 y n [ g j y n, g k y n, g l y n ] ξ j ξ k ξ l m + 3/ g 0 y n g j y n a ξ j + 3/ m g 0 y n [ g j y n, g k y n ] ξ j ξ k g 0 y n [ a 1, g j y n ] ξ j + O, we ave ỹ n+1 + g 0 = 5/ r 1 + O 3 Y 1 + g j Y ξ j { y n + } g 0 y n + g 0 K from 3. 3, were r 1 = 1 { g 0 y n g j y n a + g 0 y n [ a 1 g 0 y n, g j y n ]} ξ j. As were we ave 1 = 1 {g j Y 3 + g k Y 3 ζ kj g j Y 3 g j y n g k y n ζ kj + r + O 3 j, r = j, {g j Y } g k Y 3 ζ kj { g j y n [a 3, g k y n ] + g j y n g k y n a 3 } ζkj, g k Y 3 ζ kj g j Y 3 {g j y n + = r + O 3. } g k Y 3 ζ kj g k y n ζ kj g j y n } g k y n ζ kj 6

8 As = {g j Y 4 + g k Y 5 χ k + g j Y 4 {g j y n + g 0 y n + +g j y n + g 0 y n g k y n χ k } g k Y 5 χ k ξ j } g k y n χ k ξ j + r 3 + O 5/ were we ave r 3 = { g j y n a {g j Y 4 + = 5/ r 3 + O 3. From tese results, From tis and 3., tus, k,l=1 g j y n [g k y n, g l y n a 5 ] χ k χ l } ξ j, g k Y 5 χ k + g j Y 4 {g j 1 y n + K 1 + +g j 1 y n + K 1 g k y n χ k } g k Y 5 χ k ξ j } g k y n χ k ξ j ŷ n+1 y n+1 = r + 5/ r 1 + 5/ r 3 + O 3. G ŷ n+1 G yn+1 = G y n+1 ŷn+1 y n+1 + O 4 = G y n + g j y n ξ j + O ŷn+1 y n+1 + O 4 = G y n [ m ] ŷ n+1 y n+1 + 5/ G y n g j y n ξ j, r + O 3. Consequently, we obtain E [ G ŷ n+1 ] E [ G yn+1 ] = O 3 because E[r 1 ] = E[r ] = E[r 3 ] = E[ξ j r ] = 0 j = 1,,..., m. 7

9 3. SERK metods We sall propose weak second order SERK metods for As a simple case, let us begin wit { y n+1 = Y 1 + ϕ A f Y 1 + } g j Y ξ j fy n + m e A I g j Y ξ j + H Here and in wat follows, we set Y 1, Y and H by Y 1 def = e A y n + ϕ 1 Afy n, Y def H def = 1 + {g j Y 1 + = e A y n + ϕ 1 g k Y 1 ζ kj g j Y 1 {g j Y + } g k Y 1 ζ kj +g j Y g k Y χ k } g k Y χ k ξ j. A fy n 3. 5 If te diffusion terms vanis, 3. 4 is equivalent to. 4 wit c = 1. Teorem 3. Let g 0 y denote Ay + fy and suppose tat 3. 1 satisfies Assumption 3.1 for q =. Suppose also tat g jyg k y j, k = 1,,..., m satisfy te linear growt condition: g jyg k y C 1 + y 3. 6 for a constant C > 0. Ten, 3. 4 is of weak order two. Proof. First, let us consider ŷ n+1 = ỹ n+1 + g 0 K 1 + g j Y ξ j + H, were ỹ n+1 = y n + g 0 y n, K 1 = y n + g 0 y n. By Lemma 3.1, te local error of tis metod is of weak order tree because Y 1 = y n + g 0 y n + O, Y = y n + g 0 y n + 8 Ag 0 y n + O 3. 8

10 Using g 0 y = Ay + fy, we can rewrite tis as follows: ŷ n+1 = y n + Ay n + f y n + AK 1 + K f 1 + g j Y ξ j + 3/ m A g j Y ξ j + H Te last term is te same in 3. 4 and If g j 0 for j = 1,,..., m, ten y n+1 ŷ n+1 = O 3 because 3. 4 and 3. 7 are of order two for semilinear ODEs. Hence, all tat remains concerning te local error is to ceck te difference between { ϕ A f Y 1 + } g j Y ξ j f Y 1 + m e A I g j Y ξ j and { f K 1 + } g j Y ξ j f K 1 As Y 1 = K 1 + O, we ave { ϕ A f Y 1 + } g j Y ξ j f Y 1 = were { f K 1 + r = 1 } g j Y ξ j f K 1 + 3/ + 3/ m A g j Y ξ j. + e A I m g j Y ξ j m A g j Y ξ j + 5/ r + O 3, { 1 4 A g j Y + 1 } 3 Af K 1 g j Y ξ j. Since E[r] = 0, te local error of 3. 4 is also of weak order tree. As a sufficient condition for 1 in Teorem 3.1, it is known tat te following two inequalities old for sufficiently small > 0: E [ yn+1 y n y n ] C 1 + yn, yn+1 y n Xn 1 + y n, were C is a positive constant and X n is a random variable wic as moments of all orders [5, p. 10]. From te definition of Y 1 in 3. 4 and te linear growt of g jg k, we ave 1 g j Y 1 + C 1 g j y n g k Y 1 ζ kj g j Y 1 g k y n ζ kj C 1 + y n g k Y 1 ζ kj 9

11 for constants C 1, C > 0. As g j y j = 0, 1,..., m are Lipscitz continuous, tey also satisfy te linear growt conditions of tem. From tese facts, we can see tat te two inequalities requested above old for Consequently, 3. 4 is of weak order two by Teorem 3.1. As anoter case, let us consider te family of SERK metods given by were y n+1 = Y 1 + γc + c 3 ϕ A { γf m Y 3 + b 1 g j Y ξ j m } +f Y 4 + b g j Y ξ j γ + 1fy n + m e A I g j Y ξ j + H, Y 3 = e c A y n + c ϕ 1 c Afy n, Y 4 = e c 3A y n + {c 3 ϕ 1 c 3 A ψa} fy n + ψafy and b 1 and b are parameters as well as c, c 3 and γ satisfying. 6. If te diffusion terms vanis, 3. 8 is equivalent to. 5. Teorem 3.3 Let g 0 y denote Ay + fy and suppose tat 3. 1 satisfies Assumption 3.1 for q =. Suppose also tat g jyg k y j, k = 1,,..., m satisfy Ten, 3. 8 is of weak order two if te parameters satisfy as well as. 6. γb 1 + b γc 1 + c = 1, γb 1 + b γc 1 + c = Proof. Te last term is te same in 3. 7 and If g j 0 for j = 1,,..., m, ten y n+1 ŷ n+1 = O 3 because 3. 7 and 3. 8 are of order two and tree for semilinear ODEs, respectively. Hence, all tat remains concerning te local error is to ceck te difference between { m ϕ A γf Y 3 + b 1 g γc + c j Y ξ j γf Y 3 3 m } +f Y 4 + b g j Y ξ j f Y 4 + m e A I g j Y ξ j and { f K 1 + } g j Y ξ j f K 1 + 3/ m A g j Y ξ j. 10

12 As Y 3 = K 1 + c 1g 0 y n + O and Y 4 = K 1 + c 3 1g 0 y n + O, using 3. 9 we ave { 1 m γf Y 3 + b 1 g γc + c j Y ξ j γf Y 3 3 m } +f Y 4 + b g j Y ξ j f Y 4 were = f K 1 g j Y ξ j + + γb3 + b 3 3 6γc + c 3 3/ r 1 = On te oter and, f = j,k,l=1 f K 1 [ g j Y, g k Y ] ξ j ξ k + 3/ r 1 j, f K 1 [ g j Y, g k Y, g l Y ] ξ j ξ k ξ l + O, γb c + b 3 c 3 m 1 f K 1 [ g γc + c 0 y n, g j Y ] ξ j. 3 K 1 + g j Y ξ j f K 1 f K 1 g j Y ξ j + + 3/ 6 j,k,l=1 f K 1 [ g j Y, g k Y ] ξ j ξ k j, f K 1 [ g j Y, g k Y, g l Y ] ξ j ξ k ξ l + O. By utilizing tese results and φ A = /I + /6A + O 3, tus, we obtain { m ϕ A γf Y 3 + b 1 g γc + c j Y ξ j γf Y 3 3 m } +f Y 4 + b g j Y ξ j f Y 4 = + m e A I { f K 1 + g j Y ξ j } g j Y ξ j f K 1 + 5/ r 1 + 5/ r + O 3, + 3/ m A g j Y ξ j 11

13 were r = 1 γb 3 + b 3 m 3 1 f K 1 [ g 1 γc + c j Y, g k Y, g l Y ] ξ j ξ k ξ l 3 j,k,l=1 + 1 { 1 4 A g j Y + 1 } 3 Af K 1 g j Y ξ j. Since E[r 1 ] = E[r ] = 0, te local error of 3. 8 is of weak order tree. In a similar way to te proof of Teorem 3., we can see tat 1 in Teorem 3.1 olds. Consequently, 3. 8 is of weak order two if te parameters satisfy. 6 and Remark 3.1 As a simple solution of. 6 and 3. 9, we can find c = 1, c 3 = 1, γ = 4, b 1 = 6 ± 6, b = double sign in order. For tis solution, te intermediate values Y 3 and Y 4 satisfy Y 3 = Y, Y 4 = Y 1 + ψa {f Y f y n }. 4 MS stability analysis for SERK metods Let us investigate te stability properties of our SERK metods. We consider te following scalar test SDE [13]: dyt = λytdt + σ j ytdw j t, t > 0, y0 = y 0, 4. 1 were y 0 0 wit probability one w. p. 1 and were λ and σ j 1 j m are complex numbers satisfying Rλ + σj < Because of 4., te solution of 4. 1 is MS stable lim t E[ yt ] = 0. Wen an SRK metod is applied to 4. 1, it is generally expressed by y n+1 = R, λ, {σ j } m, η y n, were η is a random vector wose elements are random variables appeared in te metod. Te metod is said to be MS-stable for particular, λ, σ j j = 1,,..., m if [ R ] E, λ, {σ j } m, η < 1, wic means tat E[ y n ] 0 as n for te given, λ, σ j j = 1,,..., m. Furter, te metod is said to be A-stable in te MS if it is MS-stable for any > 0 wenever 4. olds [13]. 1

14 Teorem 4.1 Te SERK metod 3. 4 is A-stable in te MS for te test equation Proof. If we apply 3. 4 to 4., ten, we ave y n+1 = R, λ, {σ j } m, {ξ j} m, {ζ jk} m j, y n, were R, λ, {σ j } m, {ξ j} m, {ζ jk} m j, = e {1 λ + σj ξ j + From tis, te MS stability function ˆR of 3. 4 is given by ˆRp r, q def = E [ R ] = e p r 1 + q + q, } σ j σ k ζ kj. def were p r = Rλ and q def = m σ j. As we can rewrite 4. by p r + q < 0, we ave ˆRp r, q < e p r 1 p r + p r. Te function in te rigt-and side is less tan 1 for any p r < 0. Tus, ˆRpr, q < 1 wenever p r + q < 0. Consequently, 3. 4 is A-stable in te MS.,k Teorem 4. Te SERK metod 3. 8 is A-stable in te MS for Proof. Te metod 3. 8 is equivalent to 3. 4 except te second term, and bot second terms disappear wen tey are applied to 4.. By Teorem 4.1, tus, 3. 8 is also A-stable in te MS As a comparison, let us look at stability properties of te SROCK metod. Wen m = 1, its MS stability function is given by ˆRp, q = Ap + Bp q + Cp q, were p def = λ and Ap, Bp, Cp are polynomial functions of p. For details, see [5]. Now, we can plot te MS stability domain, tat is, {p, q ˆRp, q < 1}. For te SROCK metod wit six stages, te MS stability domain and its profile are given in Figure 1. Te MS stability domain is indicated by te colored part in te left of te figure, and p i denotes Iλ. Te oter part enclosed by te mes indicates te domain in wic te solution of te test SDE is MS stable. In te rigt part of te figure, te colored area indicates te profile of te MS stability domain wen p i = 0. We can see tat te MS stability domain is large along te negative axis of p r, but it is tin in te axis of p i. On te oter and, we plot te MS stability domain of our SERK metods in Figure. 13

15 p r q q p i p r p i = 0 Figure 1: MS stability domain left and its profile rigt for te SROCK metod wit six stages p r p r q q p i p i Figure : MS stability domain for our SERK metods 5 Numerical Experiments In Section 3, we ave derived our SERK metods. For example, 3. 4 is of weak order two and deterministic order two. In wat follows, let us call tis te SERKWD metod. As we ave seen in Remark 3.1, 3. 8 wit c = 1/, c 3 = 1, γ = 4, b 1 = 6 + 6/10, b = 3 6/5 is of weak order two and deterministic order tree. Let us call tis te SERKWD3 metod. As an implementation of te SROCK metod, we do not directly use te Fortran codes from ttp://anmc.epfl.c/pdf/srock.zip, but ave implemented C codes by including rectp.f from te Fortran codes. Tus, te SROCK metod in our C codes as te same parameter values as tat in te Fortran codes. In order to confirm te performance of te metods, we investigate some statistics in numerical experiments. As first two examples, let us consider te following scalar, nonstiff, nonlinear SDEs [5, 9] for wic some functions of te exact solution are analytically 14

16 log relative error log log relative error log Example 5. 1 Example 5. Figure 3: Log-log plots of te relative error versus in te examples 5. 1 and 5. Solid: SERKWD, das-dotted: SERKWD3, das: SROCK, dotted: reference line wit slope obtained. One of te examples is 1 dyt = 4 yt + 1 yt + 1 yt + 1 dt + dw t, t > 0, y0 = 0 w.p.1. For te solution yt, E[arcsin yt ] = t /4 + t/. Te oter is 10 1 dyt = ytdt + yt + 1 dw j t, t > 0, b j y0 = 1 w.p.1, a j were a 1 = 10, a = a 8 = 15, a 3 = a 7 = a 9 = 0, a 4 = a 6 = a 10 = 5, a 5 = 40, b 1 = b 6 =, b = b 7 = 4, b 3 = b 8 = 5, b 4 = b 9 = 10, b 5 = b 10 = 0. For te solution yt, E [ yt ] = e t e t / In tese examples, using te Mersenne twister algoritm [3] we simulate independent trajectories for a given, and seek numerical approximations to E[arcsin y1 ] and E[y1 ] for 5. 1 and 5., respectively. Te results are indicated in Figures 3. Te solid, das-dotted and das lines denote te SERKWD metod, te SERKWD3 metod, and te SROCK metod wit 13 stages [5], respectively. Te dotted one is a reference line wit slope. Note tat te results of te SERKWD and SERKWD3 metods in 5. are te same because te drift term is linear. As a wole, we can observe tat all metods acieve teoretical convergence order weak order two, altoug te error of te SERKWD metod seems to be influenced by statistical errors wen = 4, 5, in te rigt plot. In order to deal wit stiff cases, let us consider te following SDE [ α 1 dyt = ω α y0 = [1 1] w.p.1 ] ytdt + [ σ 0 0 σ ] ytdw t, t > 0,

17 Table 1: Step size for Numerical stability in 5. 3 Metod Step size Absolute errors Case 1 SROCK 10 stages = 1/ stable SERKWD = 1/ stable Case SROCK 3 stages = 1/ stable SROCK all stages = 1/ 8 unstable SERKWD = 1/ stable Case 3 SROCK 3 stages = 1/ stable SROCK 5 stages = 1/ stable SROCK all stages = 1/ 5 unstable SERKWD = 1/ stable for α, ω, σ R. Since te eigenvalues of te matrix in te drift term are α ± iω, lim t E[ yt ] = 0 olds if α + σ < 0. We investigate tree cases: Case 1 α = 100, ω = 1, σ = 199, Case α = 1 4, ω = 30π, σ = 1 4, Case 3 α = 100, ω = 30π, σ = 199. In tis example, we simulate independent trajectories for a given until t = 10 and seek numerical solutions to E[ y10 ] by te SROCK and SERKWD metods. Note tat te SERKWD and SERKWD3 metods are equivalent for 5. 3 because te drift term is linear. For te solution yt in eac case, we ave Case 1 E [ y 1 10 ] = {1 + sin0}e 0, E [ y 10 ] = {1 sin0}e 0, Case E [ y 1 10 ] = E [ y 10 ] = e 35/8, Case 3 E [ y 1 10 ] = E [ y 10 ] = e 10. Table 1 gives numerical results, wic indicate ow small step size is necessary for eac metod to solve 5. 3 numerically stablely. In Case 1 te SROCK metod wit 10 stages can solve it for = 1/, but tose wit less tan 10 stages cannot. In Case te SROCK metod cannot solve te SDE for = 1/ 8 even if we make te stage number large. Tis is understandable because increasing stage number does not lead to making te MS stability domain large enoug in te axis of p i. Remember Fig. 1. In Case 3 te SROCK metod wit tree stages cannot solve te SDE for = 1/ 6, but tose wit five stages can. However, for = 1/ 5 te SROCK metod cannot solve by making te stage number large. On te oter and, te SERKWD metod can solve for = 1/ in all cases. Te fourt example comes from a stocastic Burgers equation wit wite noise in time only. Da Prato and Gatarek [8] ave proved te existence and uniqueness of te global solution of a scalar Burgers equation wit multiplicative noise driven by a scalar Wiener 16

18 log relative error log Figure 4: Log-log plot of te relative error of te variance versus Solid: SERKWD, das-dotted: SERKWD3, das: SROCK, dotted: reference line wit slope process. Now, we consider an extended version of teir equation: u dut, x = t, x + ut, x ut, x dt + k x 1 ut, xdw 1 t x + k 1 + ut, x dw t, t > 0, x [0, 1], 5. 4 ut, 0 = ut, 1 = 0 w.p.1, t > 0, u0, x = sinπx w.p.1, x [0, 1], were k 1, k R. If we discretize te space interval by N + equidistant points x i 0 i N + 1 and define a vector-valued function by yt def = [ut, x 1 ut, x ut, x N ], ten we obtain te following non-commutative SDE dyt = Ayt + fyt dt + k 1 ytdw 1 t + b yt dw t, t > 0, y0 = [ sinπx 1 sinπx sinπx N ] w. p by applying te central difference sceme to 5. 4, were A def = N , y 1 y y y 3 y 1 fy def = N + 1. y N 1 y N y N y N y N 1, by def = k 1 + y y. 1 + y N. For N = 17, k 1 = and k = 3/, we seek an approximation to te variance of eac element of yt at t = 1. As we do not know te exact solution of te SDE, we seek numerical approximations by te SROCK metod wit six stages for = 1 and use tem instead of te exact variance. In tis example, we simulate independent trajectories for a given. In order to solve te SDE numerically stably wit reasonable cost by te SROCK metod, we 17

19 Average of u1, x x Average of u1, x x 35 stages, = 7 6 stages, = 1 Figure 5: Approximations to E[ut, x] at t = 1 given by te SROCK metod Average of u1, x x Figure 6: Approximation to E[ut, x] at t = 1 given by te SERKWD metod for = 6 set te stage number of te metod at 49, 35, 4, 17, 1 and 8 corresponding to te step size 6, 7, 8, 9, 10 and 11, respectively. Te results are indicated in Figure 4. Te solid, das-dotted and das lines denote te SERKWD metod, te SERKWD3 metod and te SROCK metod, respectively. Te dotted one is a reference line wit slope. Te figure indicates tat te SERKWD and SERKWD3 metods ave almost te same error, wereas te SROCK metod seems to be inferior to tem. In Figures 5 and 6 plots for E[ut, x] at t = 1 are sown. Tese are obtained by te SROCK metod wit 35 stages for = 7, te SROCK metod wit 6 stages for = 1 and te SERKWD metod for = 6. Finally, Table indicates comparisons of computational cost for eac metod in one step and one trajectory. In te table, n e, n r and n m stand for te number of evaluations on te drift or diffusion coefficients, te number of generated pseudo random numbers and te number of te products of a matrix exponential function wit a vector, respectively. 6 Concluding remarks We ave derived explicit SERK metods wic acieve weak order two for non-commutative Itô SDEs wit a semilinear drift term, and simultaneously acieve order two or tree for ODEs. Using a scalar test SDE wit complex coefficients, we ave investigated te stability properties of te metods. As a result, we ave proved tat tey are A-stable in te MS for te test SDE. To our best knowledge, tere seems to be no weak second order metod for wic te A-stability in te MS is proven using te test SDE wit complex coefficients, except a drift-implicit metod of weak order two and deterministic order two 18

20 Table : comparisons of computational cost in one step and one trajectory Metod n e n r n m SROCK wit s stages s + 5m + m 0 SERKWD 6m + m 6 SERKWD3 6m + 4 m 7 in [4]. In addition, as one of explicit stabilized metods we ave picked up te SROCK metod, and ave plotted its MS stability domain. In order to ceck numerical performance of te metods as well as teir stability properties, we ave performed four numerical experiments. In te first two experiments, scalar, nonstiff, nonlinear SDEs ave been considered. Te experiments ave confirmed te teoretical convergence order, weak order two for our SERK metods and te SROCK metod. In te tird experiment, we ave dealt wit tree stiff cases. Te experiment indicates tat if te imaginary part of eigenvalues in te drift term is large, te SROCK metod needs a very small step size for stability, wereas te SERK metods do not need. In te last experiment, we ave considered a stocastic Burgers equation wit wite noise, and compared our SERK metods wit te SROCK metod wit several stages. Tis experiment as sown te superiority of te SERK metods to te SROCK metod in terms of computational accuracy for relatively large step size. Finally, we sould make te following remarks. As we ave seen, we can apply our metods to SDEs wit a semilinear drift term and tey ave very good performance if te stiffness of te problem is in te matrix A, not in te nonlinear function f. Te SROCK metod is applicable to more general SDEs witout suc restriction and tey can also cope wit stiff problems by increasing te stage number. Wen te dimension of a system of SDEs is not large and te stiffness is very strong, our metods will ave a significant advantage over te SROCK metod. Tis is because te metod as to increase te stage number significantly, wic leads to ig computational cost. On te oter and, wen te dimension of SDEs is very large, te SROCK metod can still cope wit ig dimensional stiff SDEs by just increasing te stage number, but our metods need tecniques in order to calculate matrix exponentials efficiently, suc as known metods. Altoug we ave not used suc approaces for matrix exponentials in tis paper, te application of tese tecniques will give considerably important impact on our metods to callenge very ig dimensional SDEs wit a semilinear drift term. Acknowledgments Tis work was partially supported by JSPS Grant-in-Aid for Scientific Researc No It was also partially supported by overseas study program of faculty at Kyusu Institute of Tecnology. 19

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