NUMERICAL METHODS FOR SECOND-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS

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1 SIAM J SCI COMPUT Vol 9, No, pp c 7 Society for Industrial and Applied Mathematics NUMERICAL METHODS FOR SECOND-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE Abstract We seek numerical methods for second-order stochastic differential equations that reproduce the stationary density accurately for all values of damping A complete analysis is possible for scalar linear second-order equations damped harmonic oscillators with additive noise, where the statistics are Gaussian and can be calculated exactly in the continuous-time and discrete-time cases A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep The only Runge Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case Key words damped harmonic oscillators with noise, stationary distribution, stochastic Runge Kutta methods, implicit midpoint rule, multiplicative noise AMS subject classifications 6-8, 65C3 DOI 37/56463 Introduction Newton s second law of motion relates force to acceleration Consequently, second-order differential equations are common in scientific applications, in the guise of Langevin, Monte Carlo, molecular, or dissipative particle dynamics [,, 3], and the study of methods for second-order ordinary differential equations is one of the most mature branches of numerical analysis [4] The most exciting advances in recent decades have been the development of symplectic methods, capable of exactly preserving an energy-like quantity over very long times [5], and their extension to stochastic systems [6] In the stochastic setting, the long-time dynamics of a typical physical system is governed by fluctuation-dissipation, so that the amount of time spent in different regions of phase space can be calculated from the stationary density [7] This density can have a relatively simple explicit expression even when the dynamics is highly nonlinear [8] Numerical methods replace continuous-time with discrete-time dynamics, generating values at times t,t, Usually t n+ t n is a fixed number Δt The criterion for a good numerical method that will be examined in this work is that its discrete-time dynamics has a stationary density as close as possible to that of the continuous-time system The differential equations describing second-order systems contain a parameter known as damping The stationary density is independent of damping, but dynamical quantities, and the usefulness of numerical algorithms, are strongly dependent on it In the infinite-damping limit, the system becomes first order The limit of zero damping, on the other hand, corresponds to Hamiltonian systems, where symplectic methods can be applied The aim in this paper is to devise methods capable of accurately reproducing the stationary density for all positive values of damping Received by the editors November 4, 5; accepted for publication in revised form August, 6; published electronically February, 7 Advanced Computational Modelling Centre, Department of Mathematics, University of Queensland, Queensland 47, Australia kb@mathsuqeduau, ijl@mathsuqeduau Department of Applied Mathematics, University of Leeds, Leeds LS 9JT, England grant@ mathsleedsacuk 45

2 46 KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE We shall consider equations of the following form: ẍ = fx ηs xẋ + ɛsxξt, where ξtξt = δt t and the damping parameter is denoted η Angled brackets denote mean over realizations The second-order stochastic differential equation SDE describes the position of a particle subject to deterministic forcing fx and random forcing ξt The deterministic forcing is related to the potential function V x via fx = V x The amplitude of the random forcing, ɛ, is related to the temperature T and damping coefficient η by the fluctuation-dissipation relation [8] ɛ =ηkt We can write as a pair of first-order equations for X t and V t, the position and velocity variables: dx t = V t dt, dv t = ηs X t V t dt + fx t dt + ɛsx t dw t, where W t is a Wiener process satisfying W t W s = mint, s If sx is not a constant, the noise amplitude is a function of position and the equation is commonly said to have multiplicative noise However, because the coefficient of dw t in the SDE for V t is a function of X t only, there is no difference between the Ito and Stratonovich see Gardiner [8] forms of The probability density at time t is P x, v; t, where P x, v; t = d d 3 dx dv Prob X t <x,v t <v The stationary density, P x, v, defined as 4 P x, v = lim t P x, v; t, has the following analytical form, independent of η and sx as long as η> and suitable conditions on V x are satisfied [8]: 5 P x, v =N exp v /KT V x/kt Thus the late-time statistics of the velocity are Gaussian and uncorrelated with the position It is notable that the stationary density has a closed tractable form for many nonlinear functions fx when analytical study of the full evolution is not possible In this paper we examine how faithfully the stationary density is reproduced by standard timestepping methods for SDEs These methods produce approximate values for position X n and velocity V n at discrete times t,t,, where t n+ t n =Δt We consider the evolution of X n and V n, and their statistical properties as t n, and compare it with the exact form 4 In particular, we consider P y, u; t n, the discrete-time analogue of 3, and compare the limit P y, u = lim P y, u; t n, t n where it exists, with P y, u In this work we base our analysis on linear secondorder equations, where the statistics are Gaussian and completely characterized by

3 NUMERICAL METHODS FOR SECOND-ORDER SDES 47 three quantities: the mean squares of the position and velocity variables, and the correlation between the position and velocity The Euler and Heun methods simply treat as a pair of SDEs, without attempting to take advantage of their special structure due to their origin in a single second-order differential equation We shall show that they perform reasonably well at intermediate values of damping but fail at high damping, when the equations are stiff, and low damping, when it is important to conserve energy-like quantities over long times The Verlet algorithm [, 9, ] produces numerical solutions of second-order differential equations by updating the position variable without reference to the velocity variable Extensions of this idea to Langevin equations [,,, 3] have proved successful, and convergence properties may be analyzed using second-order difference equations [, 4, 5] The implicit midpoint rule cannot be written as a difference equation but can be written as a matrix equation [5] Schurz [6, 7] studied multidimensional Ornstein Uhlenbeck processes and showed that the implicit midpoint rule possesses the correct stationary distribution Time-dependent properties of numerical algorithms for the class of linear equations considered in this paper can be considered by means of a modified frequency [5] or by the method of modified equations [8] Recent analysis by Mannella [9, ] of the stationary distribution resulting from numerical timestepping methods, based on an expansion of the exponent in 5, has led to a proposed modification of the leapfrog method We shall take this method as one of our examples and find that it provides a notable improvement on the standard leapfrog method A slightly different system has been studied by Strømmen Melbø and Higham [] They considered a second-order system, but without a damping term Rather than approaching a stationary density, the sum of the mean squared velocity and position grows proportional to time Partitioned methods are superior to the Euler method for this system In section, we consider the case fx= gx, where g> is a constant, and sx = That is, we consider linear second-order equations describing harmonic oscillators with additive white noise and damping The statistics for linear systems are Gaussian, described by a covariance matrix, and can be calculated exactly in the continuous-time and discrete-time cases A matrix equation is derived for the stationary variances and correlations resulting from a large class of numerical methods We then calculate the stationary variances produced by some well-known methods All methods we shall consider use one realization per timestep of a Gaussian distribution with mean zero and variance Δt This framework includes many methods with multiple intermediate steps and implicit methods which are in fact explicit for linear equations In section 3 we seek measure-exact methods for linear second-order SDEs, that is, methods that give the correct late-time mean square of the position and velocity and absence of correlation between them In section 4, we restrict our consideration to Runge Kutta methods for systems of additive noise SDEs and show that the unique measure-exact method is the implicit midpoint rule In section 5 we report on numerical experiments carried out with fx =x x 3 and sx = ; in section 6 we report on numerical experiments carried out with fx =x x 3 and sx =x In both cases, of additive and of multiplicative noise in the double-well system, the implicit midpoint rule is the most satisfactory, with no error in two of the three quantities of

4 48 KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE interest over the full range of values of damping However, leapfrog methods have the benefit of being fully explicit Linear equation and matrix notation If fx = gx, then the stationary density 4 is P x, v =N exp gx /KT v /KT The distributions of the position and velocity variables are Gaussian with lim Xt = KT, lim V t g t = KT, and lim Xt V t = t t If fx = gx and sx =, then the linear second-order SDE can be written in matrix notation as Xt Xt d = Q dt + ɛ dw t, V t V t where 3 Q = g η We consider numerical updates for the linear system that can be summarized as 4 Xn+ V n+ = R Xn V n + ɛrδw n, where r r R =, r = r r r r and ΔW n is sampled from a Gaussian distribution with mean zero and variance Δt, independently of ΔW m for n m Since the numerical update 4 is a linear transformation, P x, v is Gaussian Let the correlation matrix be 5 where σ x = σ Σ= x μ μ lim X n, σ v = lim V n, and μ = lim Xn V n t n t n t n σ v, Then P x, v = π Σ exp x x, vσ v The stationary density P x, v has the property of invariance under the transformation 4 Now, if X n V n is Gaussian with mean zero and correlation matrix Σ, then R X n V n is Gaussian with mean zero and correlation matrix RΣR T Thus the correlation

5 NUMERICAL METHODS FOR SECOND-ORDER SDES 49 matrix that results from a method of the form 4 satisfies Σ = RΣR T + ɛ rr T Δt, or 6 RΣR T =Σ ɛ rr T Δt In the remainder of this section, we shall use 6 to calculate the σ x, μ, and σ v as a function of η and Δt under six numerical methods for the case sx = Wecan rewrite 6 in a form suitable for inversion: r r r r r r r r + r r r r r r r r σ x μ σ v = ɛ Δt r r r The stability boundaries of the methods are functions of the parameters ηδt and gδt Of the methods discussed below, the implicit midpoint rule and the leapfrog methods have the virtue of reducing to a symplectic method if η = [5] Further explicit methods with this property have been devised [6] Forward Euler method Under the Euler method, the position and velocity variables are updated as follows: X n+ = X n + V n Δt, V n+ = V n ηv n Δt + fx n Δt + ɛδw n With the notation of 4, R = R E and r = r E, where Δt 7 R E =+ΔtQ =, gδt ηδt r E = and the solution of 6 yields 8 Σ E = KT g η Δt ηδt + gδt g ηδt + gδt Δt Δt In Figure we display the differences between the stationary variances, obtained by applying the Euler method to the linear equation, and the exact values The forward Euler method works best at intermediate values of η It is unstable if ηδt <gδt or ηδt >+ gδt In Figure, we plot the correlation between velocity and position, which is zero in the exact solution but is proportional to Δt under the Euler method The Heun method Under the Heun method, intermediate values are obtained via the Euler method: ˆX = X n + V n Δt, The update is ˆV = V n ηv n Δt + fx n Δt + ɛ ΔW n X n+ = X n + V n + ˆV Δt, r V n+ = V n ηv n + ˆV Δt + fx n+f ˆX Δt + ɛ ΔW n

6 5 KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE error in mean square σ x KT/g σ v KT numerical η Fig Mean squares versus damping for the forward Euler method The differences between the late-time mean squares and the exact values are plotted against η for fixed Δt =, g =, and KT = The lines use 8 Dots with error bars are obtained from the numerical solution of the linear SDE With the notation of 4, R = R H and r = r H, where R H = I +ΔtQ + Δt Q = gδt Δt ηδt gδt + ηgδt ηδt + g + η Δt and r H = Δt ηδt Let k = ηδt and k = gδt Then, for the stochastic Heun method, 9 σx = KT g σ v = KT 4 k + 8 k 3 k k + k k k 3 k k + k + 4 k k 8 k k + 6 k k k + 4 k k 8 k k + 6 k μ = KTΔt k k k k + 4 k k 8 k k + 6 k k + k 4 k /k 8 k3, k + k 4 k /k, k + k 4 k /k Compared with the Euler method, the upper limit on η for stability is little changed The lower limit is, however, proportional to Δt 3 As Δt, the Heun method is stable if gδt <ηδt <, and the correlation between position and velocity is proportional to Δt See Figure 3

7 NUMERICAL METHODS FOR SECOND-ORDER SDES 5 - correlation η Fig Position-velocity correlation versus damping for the forward Euler method The latetime correlation, μ, obtained using the forward Euler method, is plotted against η for fixed Δt =, g =, and KT = The line uses 8 Dots with error bars are obtained from the numerical solution of the linear SDE 3 The leapfrog method Under leapfrog methods, velocity and position are updated successively rather than together The simplest possible method is ˆX = X n + V nδt, V n+ = V n ηv n Δt + f ˆXΔt + ɛ ΔW n, X n+ = ˆX + V n+δt Under this method, with the notation of 4, R = R le and r = r le, where R le = gδt Δt ηδt 4 gδt3 gδt ηδt gδt and r le = Δt The leapfrog method maintains the independence of position and velocity and produces the exact stationary variance of the position variable However, the error in σ v is an increasing function of η: Σ le = KT g ηδt 4 gδt

8 5 KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE error in mean square σx KT/g σv KT numerical η Fig 3 Mean squares versus damping for the Heun method The differences between the late-time mean squares and the exact values are plotted against η for fixed Δt =, g =, and KT = The lines use 9 Dots with error bars are obtained from the numerical solution of the linear SDE Mannella [9, ] has proposed the following modification of the leapfrog method: ˆX = X n + V nδt, V n+ = c c V n + f ˆXΔt + ɛ ΔW n, X n+ = ˆX + V n+δt, where c = ηδt and c =+ ηδt The corresponding quantities are c R M = gδt Δt +c c c Δt c gδt c c c gδt, r M = c Δt, and Σ M = KT g 4 gδt The only error, in σ v, is independent of η 4 The BBK method Under the Brünger Brooks Karplus BBK method [], the position variable is updated without reference to the velocity variable The

9 NUMERICAL METHODS FOR SECOND-ORDER SDES 53 two most recent values of the position variable need to be retained, and the update is X n+ = X n + ηδt Δt + ηδtx n X n + + ηδt fx n+ɛδw n With the definition of velocity V n Δt = X n X n, proposed in [], the BBK method can be written in the form 4 with [] and R B = r B = gδt + ηδt Δt ηδt + ηδt gδt + ηδt ηδt + ηδt + ηδt Δt Whatever the definition of velocity, we find [, 5, 8] 3 σx = KT g 4 gδt However, σv and μ do depend on the definition of velocity Note that, with the velocity definition V n Δt = X n+ X n, the BBK update cannot be put into the form 4 5 The implicit midpoint method Under the implicit midpoint method, intermediate values are obtained via the implicit procedure ˆX = X n + ˆV Δt, The update is ˆV = V n η ˆV Δt + f ˆXΔt + ɛ ΔW n X n+ = X n + ˆV Δt, V n+ = V n η ˆV Δt + f ˆXΔt + ɛ ΔW n With the notation of 4, R = R im and r = r I, where R im = I I ΔtQ + ΔtQ = κ gδt κ Δt κ gδt κ, and r im = κ Δt, where κ =+ ηδt + 4 gδt Thus g 4 Σ im = KT Since R im and r im satisfy 6, the implicit midpoint method gives the exact stationary variances for all values of η Only an algebraic error in Appendix A of [5] prevented this observation from being made by Mishra and Schlick The behavior of the implicit midpoint method is explored further in section 4

10 54 KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE 3 Measure-exact algorithms We shall call a method measure-exact if its correlation matrix is the exact one for : g 3 Σ e = KT The R and r matrices for a measure-exact method satisfy 6 with Σ = Σ e, that is, g R R T g 3 = ηrr T Δt We shall assume Σ is positive definite and let Δt be chosen small enough to ensure G =Σ ɛ rr T Δt is also positive definite Matrices Σ and G can be constructed to satisfy G = G G T and Σ = Σ Σ T Let Σ and G be the inverses of Σ and G Condition 6 can then be rewritten as PP T = I, where P = G RΣ and I is the identity matrix Any orthogonal matrix P generates a solution of 3 as 33 R = G P Σ A real orthogonal matrix can be written as a P = / a a a / for some a This form is convenient because we can take a Δt, so that P reduces to the identity matrix when a = With Σ given by 3, we can choose Σ =KT / g /, So G =Σ rr T KT/g r Δt = Δt r r Δt r r Δt KT rδt, and KT/g r G Δt = r r Δt KT rδt + rrδt KT/g r Δt KT/g r Δt Any measure-exact method, using one Gaussian random variable per timestep, can be obtained from 33 with some choice of a, r, and r For example, we may obtain a measure-exact Euler-like method by choosing r = That is, G / =KT / g / ηδt / and a / ag / R = G / P Σ / = ag / ηδt / a / ηδt /

11 NUMERICAL METHODS FOR SECOND-ORDER SDES 55 The Euler method is obtained as Δt ifa = g / Δt Thus gδt / Δt 34 R = gδt ηδt / gδt / ηδt / Notice that the difference between 34 and 7 is proportional to Δt This modified Euler method reproduces the exact stationary density as long as ηδt < As a next step, we may impose agreement of R with expδtq to order Δt : R = I +ΔtQ + Δt Q + Then a = g Δt g ηδt, r = Δt +, and r = η Δt + Methods constructed in this way will have a stationary density differing from the exact density by some power of Δt Having established that measure-exact numerical methods do exist, in the next section we return to the question of solving condition 3 exactly, this time in the context of Runge Kutta methods 4 Runge Kutta methods for additive noise From the point of view of general Runge Kutta methods, the system of two SDEs with sx = isa special case of the m-dimensional additive noise SDE that can be written [3] dy t = fy t dt + ɛhdb t, where Y t and B t are m column vectors, the entries of B t are independent Wiener processes, and H is an m m matrix with constant entries Let the numerically generated approximations be denoted by column vectors y n Under an s-stage Runge Kutta method, y n+ is obtained from y n as a weighted sum of s evaluations of the function f at intermediate values Y i : y n+ = y n + s b j fy j Δt + ɛhδb n, j= where j b j = The m entries of ΔB n are drawn independently from a Gaussian distribution with mean zero and variance Δt The column vectors of intermediate values satisfy 4 s Y i = y n + a ij fy j Δt + ɛc i HΔB n j= Runge Kutta methods for the case sx also exist [3] We apply this general formalism to the linear equation with m = We have y n = Xn V v,fy =Qy, and H = Let Y =Y,Y,,Y s T, c =c,c,,c s T, e =,,, T and A be the s s matrix whose entries are the a ij in 4 Then 4 can be written Y i = e y n + A QY i Δt + ɛc ΔW n

12 56 KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE If b =b,b,,b s T, we can write y n+ = y n + b T Y QΔt + ɛ ΔW n = y n + b T QΔt I s I m A QΔt e y n + ɛc ΔW n + ɛ ΔW n = I m + b T QΔt I s I m A QΔt e y n + ɛ I m + b T QΔt I s I m A QΔt c ΔW n Thus y n+ = RΔtQy n + R ΔtQ ɛδw n, where, for scalar z, Rz =+b T zi s Az e and R z =+b T zi s Az c In the notation of 4, r = R For example, the Euler method has R =ΔtQ and r T = Let us now examine the condition for an exact stationary measure We shall take g =, which can always be achieved by rescaling time, so The equation to be satisfied is now 4 and Let Σ e = KT RR T I +ηδtr R T = RΔtQ =I + R ΔtQ =I + α i ΔtQ i i= β i ΔtQ i i=

13 NUMERICAL METHODS FOR SECOND-ORDER SDES 57 Then 4 can be expanded in powers of Δt as Δt α Q + Q T +η +Δt α QQ T + α Q +Q T +ηβ Q +Δt 3 α 3 Q 3 +Q T 3 +α α Q Q T + QQ T + Q T +η β Q +β Q T +β Q Q T + = If this equation is to be satisfied for all Δt, then each coefficient in the expansion must be zero It is easily shown that this can hold if and only if α i = i and β i = i Thus the family of Runge Kutta methods that gives the exact stationary density is characterized by RΔtQ = I I ΔtQ + 43 ΔtQ and Explicitly, with Q given by 3, r = R R ΔtQ =ΔtQ RΔtQ I =Δt r ηr + r While Runge Kutta methods with more than one stage can be constructed that have the stability function given by 43, they all have a singular tableau matrix The unique Runge Kutta method, with a nonsingular tableau matrix, extended with a single Gaussian random variable per timestep, that preserves the exact stationary density of the linear equation for all values of damping, is thus the implicit midpoint method The method also has the virtue of being symplectic in the limit η [5] For linear equations, the method is explicit In the case of, the implicit midpoint rule is implemented as follows We first generate ˆX and ˆV : ˆV = κ V n gx n Δt + 44 ɛ ΔW n 45 ˆX = X n + ˆV Δt, where κ =+ ηδt + 4 gδt Then X n+ = X n + ˆV Δt, V n+ = V n η ˆV Δt g ˆX Δt + ɛδw n Steps are carried out in the order given

14 58 KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE 5 Double-well system: Additive noise The good properties of the implicit midpoint and leapfrog methods applied to linear second-order SDEs prompt us to investigate their accuracy for nonlinear equations In this section we take as our example system, with the double-well potential 5 V x = x + 4 x4 and with sx = The stationary density is given explicitly by 5 The statistics of the position variable are non-gaussian, but the mean is zero and σx can be evaluated to arbitrary accuracy by a numerical evaluation of an integral In general, using an implicit method on a nonlinear equation requires an iterative procedure at each timestep However, the structure of second-order systems makes a very simple iteration possible First, the intermediate value ˆX is generated by fixedpoint iteration with the starting value ˆX = X n This amounts to repeated evaluation of ˆX = X n + + ηδt Δt V n + Δtf ˆX+ 5 ɛδw n The rest of the algorithm is explicit: ˆV = V n + f ˆXΔt + / ɛ ΔW n + ηδt, X n+ = X n + ˆV Δt, V n+ = V n η ˆV Δt + f ˆXΔt + ɛδw n In our numerical experiments with the double-well system we used six iterations of 5, which was not a large overhead, comparable to an explicit predictor-corrector approach However, in larger systems, where evaluation of fx is computationally expensive, the number of iterations needs to be considered along with Δt In Figure 4 we display results obtained at KT =, using the Heun, leapfrog, and implicit midpoint methods, as a function of η for Δt = The Heun method is seen to be poor both at large and small values of damping, while the leapfrog method is accurate for small damping but inaccurate in σv at large values of damping The implicit midpoint method produces the exact values of σv and μ at all values of damping but has a nonzero error in σx Results for Mannella s modification of the leapfrog method are not shown in Figure 4 but are summarized as follows: the errors show little or no dependence on η, the error in σx is comparable to that of the implicit midpoint method, while the error in σv is similar to that of the standard leapfrog method as η In Figure 5 we display results obtained using the Heun and implicit midpoint methods, the standard leapfrog method, and Mannella s modified leapfrog method as a function of Δt, with η = and KT = Both leapfrog methods and the implicit midpoint method maintain the independence of position and velocity for all values of Δt The implicit midpoint method also gives the exact value of the late-time mean square of the velocity variable Mannella s modification of the leapfrog method leads to a reduced error in σv It is appropriate to compare numerical methods in a way that reflects their demand for computer resources Figure 6 plots the errors in σx and σv as a function of

15 NUMERICAL METHODS FOR SECOND-ORDER SDES σ x He le im 95 5 σ v - μ η Fig 4 Mean squares and correlation versus damping: double-well system Results obtained at KT = using the Heun circles, leapfrog larger filled circles, and implicit midpoint small filled circles methods with Δt = The top graph shows the late-time mean square of the position variable; the exact value is shown as a solid line The middle graph shows the late-time mean square of the velocity variable; the exact value is The lower graph shows the correlation between the position and velocity variables; the exact value is the number of evaluations of fx per time unit In the cases of the leapfrog methods, the horizontal axis is simply Δt The Heun method requires two evaluations of fx per timestep; the horizontal axis in this case is Δt The implicit midpoint rule requires a total of eight such evaluations, six of them in the iterative step; the horizontal axis is thus 8Δt All four methods have errors in σ x proportional to Δt ; the extra computer time required for the iteration makes the implicit midpoint rule more expensive On the other hand, the implicit midpoint rule reproduces the exact value of σ v Mannella s modification reduces the error in σ v associated with the leapfrog method 6 Double-well system: Multiplicative noise It is of interest to see how the methods analyzed and tested in this paper perform when extended to multiplicative noise systems, ie, sx We first describe in the following steps how we have implemented them in our numerical experiments i Under the Heun method, intermediate values are obtained via the Euler

16 6 KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE σ v 4 im le Ma He μ - Fig 5 Mean-squared velocity and correlation versus Δt: double-well system Results obtained at KT = using the Heun method circles, leapfrog method larger filled circles, Mannella s modification squares, and the implicit midpoint method small filled circles with η = The top graph shows the late-time mean square of the velocity variable; the exact value is The lower graph shows the correlation between the position and velocity variables Δt error in σ x error in σv im le Ma He function evaluations per time unit function evaluations per time unit Fig 6 Error in mean squares: double-well system Results obtained at KT = and η = using the Heun method circles, leapfrog method larger filled circles, Mannella s modification squares, and the implicit midpoint method small filled circles with η = The horizontal coordinate is Δt multiplied by the number of evaluations of fx per timestep

17 NUMERICAL METHODS FOR SECOND-ORDER SDES 6 method: ˆX = X n + V n Δt, ˆV = V n ηs X n V n Δt + fx n Δt + sx n ɛ ΔW n The update is X n+ = X n + V n + ˆV Δt, V n+ = V n ηs X n V n + s ˆX ˆV Δt + fx n ii The standard leapfrog method is now + f ˆX Δt + ɛsx n+s ˆXΔW n ˆX = X n + V nδt, V n+ = V n ηs ˆXV n Δt + f ˆXΔt + ɛs ˆXΔW n, X n+ = ˆX + V n+δt iii Mannella s leapfrog method is now ˆX = X n + V nδt, V n+ = + s ˆXηΔt s ˆXηΔt V n + f ˆXΔt + ɛs ˆXΔW n, X n+ = ˆX + V n+δt iv When using the implicit midpoint method, intermediate values ˆX are generated by repeated evaluation of ˆX = X n + + s ˆX ηδt V Δt n + Δtf ˆX+ ɛs ˆXΔW n, with the starting value ˆX = X n The rest of the algorithm is explicit: ˆV = V n + Δt ˆX, X n+ = X n + ˆV Δt, V n+ = V n ηs X n+ ˆV Δt + f ˆXΔt + sx n+ ɛδw n We have performed numerical experiments with sx = x In Figure 7 we display results obtained at KT =, using the Heun method, Mannella s modification of the leapfrog method, and the implicit midpoint method, as a function of η for Δt =

18 6 KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE error in σ x He le Ma im σ v 4 98 μ 5-5 η Fig 7 Mean squares and correlation versus damping: double-well system with multiplicative noise Results obtained at KT = and Δt = using the Heun method circles, the leapfrog method larger filled circles, Mannella s modification of the leapfrog method squares, and the implicit midpoint method small filled circles The top graph shows the error in the late-time mean square of the position variable log scale The middle graph shows the late-time mean square of the velocity variable; the exact value is The lower graph shows the correlation between the position and velocity variables In terms of the error in σ x, Mannella s method performs best The implicit midpoint method, however, is the only one that appears to give the exact value of σ v at all values of η In Figure 8 we display numerical results with KT = and η = as a function of Δt Mannella s method is the only one that has second-order convergence in σ x, but the implicit midpoint method is more accurate in σ v Note that, for systems of SDEs where there is a difference between Ito and Stratonovich forms, the implicit midpoint rule will converge to the Stratonovich form However, it will only have strong order 5 for noncommutative SDEs with more than one noise term, in which case other approaches are needed to obtain a strong order of [3, 4] 7 Conclusion Exact calculations for linear equations are of interest in their own right and because the qualitative form of the error as a function of Δt and η carries over to the nonlinear equations used as examples here The implicit midpoint rule is the only Runge Kutta method with a nonsingular tableau matrix that reproduces

19 NUMERICAL METHODS FOR SECOND-ORDER SDES 63 e-4 error in σ x He le Ma im 5 σ v -5e-5 5e-5 μ Δt Fig 8 Mean squares and correlation versus Δt: double-well system with multiplicative noise Results obtained at KT = and η = using the Heun method circles, the leapfrog method larger filled circles, Mannella s modification of the leapfrog method squares, and the implicit midpoint method small filled circles The top graph shows the error in the late-time mean square of the position variable log scale; the dotted lines are Δt, 6Δt, and 8Δt The middle graph shows the late-time mean square of the velocity variable; the exact value is The lower graph shows the correlation between the position and velocity variables the exact stationary distribution for all values of damping It also performs well when applied to second-order nonlinear and multiplicative noise equations, with no apparent error in the stationary mean square of the velocity variable even for the nonlinear double-well system with multiplicative noise However, leapfrog-type methods have the advantage of being fully explicit even for nonlinear equations In particular, they require only one evaluation of the deterministic force per timestep Applied to linear equations, the simplest leapfrog method gives the exact stationary variance of the position variable and maintains the independence of position and velocity Mannella s modification of the leapfrog method maintains this virtue while reducing the magnitude of the error in the variance of the velocity variable We found similar qualitative behavior in our numerical experiments on the nonlinear double-well system: the implicit midpoint method and Mannella s method perform best, the former being superior in the error in σ v and the latter in σ x In the deterministic case when solving separable Hamiltonian problems, explicit partitioned Runge Kutta methods can be constructed that are symplectic A chal-

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