Accuracy of Symmetric Partitioned Runge-Kutta Method for Differential Equation on Lie-Group WUB / 11-23 BUW-IMACM 11-19, Michael Günther, Franceco Knechtli and Michèle Wandelt Bergiche Univerität Wuppertal, Faculty of Mathematic and Natural Science, Germany E-mail: {triebel, guenther, wandelt}@math.uni-wuppertal.de, knechtli@phyik.uni-wuppertal.de Computer imulation in QCD are baed on the dicretization of the theory on a Euclidean lattice. To compute the mean value of an obervable, uually the Hybrid Monte Carlo method i applied. Here equation of motion, derived from an Hamiltonian, have to be olved numerically. Commonly the Leapfrog (Stoermer-Verlet) method or plitting method with multiple timecale à la Sexton-Weingarten are ued to integrate the dynamical ytem, defined on a Lie group. Here we formulate time-reverible higher order integrator baed on implicit partitioned Runge- Kutta cheme and how that they allow for larger tep-ize than the Leapfrog method. Since thee method are baed on an infinite erie of exponential function, we concentrate on the truncation of thi erie with repect to the global error and accuracy. Finally, we ee that the global error of a SPRK cheme i alway even uch that a convergence order of one i gained for method with odd convergence order. The XXIX International Sympoium on Lattice Field Theory - Lattice 211 July 1-16, 211 Squaw Valley, Lake Tahoe, California Speaker. Thi work wa upported by the Deutche Forchunggemeinchaft through the Collaborative Reearch Centre SFB-TR 55 Hadron phyic from Lattice QCD. Furthermore, acknowledge the Marie Curie Initial Training Network STRONGnet Strong Interaction Supercomputing Training Network for travel upport. c Copyright owned by the author() under the term of the Creative Common Attribution-NonCommercial-ShareAlike Licence. http://po.ia.it/
1. Introduction and Motivation In the molecular dynamic tep of the Hybrid Monte Carlo method [1], Hamiltonian equation of motion have to be olved. Thee equation form coupled ytem of matrix differential equation of the form H ([U],[A]) U ν = = A ν U ν, A ν (1.1a) H ([U],[A]) Ȧ ν = = g(u ν ), U ν for ν = 1,...,n. (1.1b) In thi notation, U ν i an element of a matrix Lie group G and A ν an element of it aociated Lie algebra g. Thu, [U] can be imagined a a vector of n matrix Lie group element U 1,U 2,...,U n, and [A] a a vector of n Lie algebra element A 1,A 2,...,A n. In pure lattice gauge theory, the element U ν can be een a the link U x,µ between the lattice ite x and x + a ˆµ. Thu, A µ i it aociated momentum P x,µ time the complex i. In thi context, the vector of matrice [U] and [A] are the whole configuration of the link and it momenta. The equation of motion have to be olved in a Lie group, repectively in a Lie algebra with a timereverible and area-preerving cheme. In a recent paper [2], we have invetigated the potentialitie of higher order partitioned Runge-Kutta cheme for olving the equation of motion uch that the deired propertie are met. We found out that ymmetric partitioned Runge-Kutta method baed on the Magnu and Munthe-Kaa approach can be time-reverible. So far, area-preervation i not fulfilled and mut be corrected in the acceptance tep. Furthermore, the global error of thi cheme i alway even and invetigated in detail in thi paper. In doing o, we tart with a hort derivation of ymmetric partitioned Runge-Kutta cheme baed on the idea of Magnu and Munthe-Kaa. Afterward, we focu on the global error and accuracy of the method and how ome numerical reult. 2. Numerical Integration The differential equation (1.1) become an initial value problem (IVP) by precribed initial value: U ν () := U ν, and A ν () := A ν, for ν = 1,...,n. Thereby, the initial value U ν () have to be in the Lie group and the element A ν () in the Lie algebra. Conidering the tructure of equation (1.1), (1.1a) i a differential equation in a Lie group uch that it ha to be olved with a numerical cheme that guarantee a olution in the Lie group a decribed in paragraph 2.1 For the econd equation (1.1b), no pecial treatment ha to be applied. It i an equation in the Lie algebra g, which i a linear pace. Thu, thi equation can be olved with any time-reverible and areapreerving numerical cheme. For convenience, we leave out the index ν from now on. Thi mean we invetigate jut one coupled differential equation for a pecial but arbitrary index ν: U = A U and Ȧ = g(u). The reult can then be extended traightforward to the whole vector [U] and [A]. 2.1 Differential Equation in Lie Group Concerning equation (1.1a), we follow the idea of Magnu and Munthe-Kaa. Magnu [3] tated that the differential equation (1.1a) in the Lie group can be replaced through a differential equation in the Lie algebra. Thi new differential equation can be olved directly due to the linearity 2
of the Lie algebra. The trategy i a follow: Identify U(t) with exp(ω(t)) uch that the variable change from U to Ω. Ω i the olution of the differential equation Ω = d exp 1 Ω (A), (2.1) with Ω(t) g and initial value Ω() :=. The derivative of the invere exponential map (2.1) i given by an infinite erie a d expω 1 (A) = k k! adk Ω (A). In thi erie, the variable are the k-th Bernoulli number and the adjoint operator adω k i a mapping in the Lie algebra g given by ad Ω (A) := [Ω,A] = ΩA AΩ. It follow the convention ad Ω (A) = A and adk Ω (A) = [Ω,adk 1(A)]. Thi mean, Ω i the olution of the differential equation Ω Ω = k k! adk Ω (A). Knowing Ω, the olution U of (1.1a) can be attained via U = exp(ω)u. In total, we record that the initial value problem (1.1) i equivalent to Ω = k= k! adk Ω (A), Ȧ = g(u) with U = exp(ω)u (2.2) and U() := U G, A() := A g and Ω() := g. Thi tranformed problem can now be olved directly by a Runge-Kutta method without detroying the Lie group tructure: A the Lie algebra g i a vector pace, the analytic olution (Ω(t),A(t)) a well a it approximation (Ω 1,A 1 ) attained by a numerical integration cheme both are element of the Lie algebra g. Furthermore, a for any a g the matrix exponential exp(a) i in the aociated matrix Lie group G, alo U i in G. 2.2 Symmetric Partitioned Runge-Kutta cheme The problem in olving (2.2) i that Ω i given a infinite erie which ha to be uitably truncated after q + 1 term. Thi mean, the truncation index q of Ω ha to be choen properly uch that a numerical integration cheme meet a precribed convergence order p. Thereby, the convergence order of a numerical integration method i p if the deviation between the exact olution and it numerical approximation after one tep i of order p + 1 in a uitable norm. Here, the idea of Munthe-Kaa come into play. He tate in [4] that the truncation index q of Ω ha to be choen a a value larger than the deired convergence order p minu one. Conequently, for a Runge-Kutta cheme of convergence order p, Ω i et a a function depending on the truncation q p 2 of the aforementioned infinite erie, i. e. Ω = q k= k! adk Ω (A) =: f q(ω,a). (2.3) All in all, the exact olution of (2.2) i approximated through an integration cheme of order p of the truncated model Ω = q=p 2 k= k! adk Ω(Â), Â = g(û) with Û = exp( Ω)Û, (2.4) 3
Û() := U G, Â() := A g and Ω() := g. Thereby, thi model can be olved with higher order time-reverible ymmetric partitioned Runge-Kutta (SPRK) cheme derived in [2] a follow: Compute the approximation Ω 1 = h i=1 b i K i, A 1 = A + h i=1 bi L i, (2.5) with increment K i = f q ( Ω i,ā i ) and L i = g(ū i ) for i = 1,...,. In the coure of thi, the internal tage are defined a Ω i = h j=1 α i j K j, Ā i = A + h j=1 α i j L j, Ū i = exp( X i )exp ( 1 2 Ω 1) U, X i = h j=1 γ i j K j. At the end, the olution U 1 i attained via U 1 = exp(ω 1 )U. In thi cheme, the coefficient b i, b i,α i j, α i j and γ i j for i, j = 1,..., can be determined to guarantee time-reveribility (and ymmetry). Their value for convergence order p = 3 can be found in [2]. 3. Global Error and Accuracy of the SPRK Method For the local error, the olution of the integration method after one tep ha to be compared with the exact olution U(t + h),a(t + h) of the differential equation (1.1). The SPRK method (2.5) i of convergence order p if U(t + h) U 1 = O(h p+1 ) and A(t + h) A 1 = O(h p+1 ) (3.1) hold. Since the approximation to U 1 i computed from evaluating the matrix exponential (we aume here that we can evaluate thi exactly), which i Lipchitz on every cloed interval, it uffice to demand Ω(t + h) Ω 1 = O(h p+1 ) and A(t + h) A 1 = O(h p+1 ) with exact olution Ω(t + h),a(t + h). According to Munthe-Kaa, the approximation Ω 1 and A 1 of the exact olution of the uitably truncated problem (2.4) can alo be interpreted a approximation to the original problem (2.2). With the ame argument, we can even formulate a tronger tatement on the local accuracy (3.1). A the method i ymmetric, theorem 3.2 in [6, II.3] applie, which tate that the maximal convergence order p of a ymmetric method i even, which mean that the local error i alway odd. Hence, the SPRK method developed a a method of an odd convergence order p i of order p + 1. The global error of a numerical integrated cheme i computed a the um of the local error. Thi mean, for the computation of a trajectory with length τ, an integration method with fixed tep ize h i applied N = τ/h time. Hence, the global error i of the order local error minu one. Thu, the SPRK method of convergence order p ha at leat a global error of order p. Again, in cae of an odd p, the global error i of order p + 1. The accuracy depend on the truncation k = q = p 2 of the erie given in (2.2) with p being the convergence order of the Runge-Kutta method that i ued to olve the problem numerically. We recall the baic tep of the proof given in [6] for a deeper undertanding of the choice of the truncation parameter q. For thi purpoe, we retrict to an uncoupled Lie algebra problem Ω = F (Ω) := k= k! adk Ω (A), with Ω() = g, (3.2) 4
which arie from the Magnu approach mentioned in paragraph 2.1. The truncation of the erie in (3.2) at k = q yield the truncated problem Ω = F q ( Ω), with Ω() =, (3.3) uch that F (Ω(t)) F q (Ω(t)) = k=q+1 k! adk Ω(t) (A). For ufficiently mooth A we recognize ad k Ω(t) (A) = O(tk+1 ), which i due to the neted tructure of the ad-operator [6]. Hence, we have F q ( Ω(t)) = F ( Ω(t)) +C(t) with C(t) = c 1 t q+2 + c 2 t q+3 + and contant value c 1,c 2,... g. For fixed h > and t [,h] the exact olution Ω(t) of (3.2) and Ω(t) of (3.3) atify Ω(t) Ω(t) = Ω() + F (Ω(τ))dτ ( = F (Ω(τ))dτ F (Ω(τ)) F ( Ω(τ)) dτ + ( Ω() + ) F q ( Ω(τ))dτ F ( Ω(τ)) +C(τ)dτ) C(τ) dτ. The function F i Lipchitz continuou on every cloed interval for ufficiently mooth A. We aume that for an interval where both Ω(t) and Ω(t) reide in for t [,h], the Lipchitz contant i L R, i. e., Furthermore, for t [,h] we ee that Hence from (3.4) it follow that F (Ω(t)) F ( Ω(t)) L Ω(t) Ω(t). (3.4) C(t) c 1 h q+2 + c 2 h q+3 +... := c(h) R. (3.5) Ω(t) Ω(t) c(h) t + L Ω(τ) Ω(τ) dτ, uch that the requirement of the "Gronwall lemma" [5] are atified by which Ω(t) Ω(t) c(h) ( ) e L h 1 L = 1 ( ( (1 c1 h q+2 + c 2 h q+3 + ) + L h + 1 L 2! (L h) 2 + ) 1) = ( ( c 1 h q+2 + c 2 h q+3 + ) h + 1 2 L h 2 + 1 ) 3! L2 h 3 +. 5
Thu, the difference between the exact olution of the full problem (3.2) and the truncated problem (3.3) i Ω(h) Ω(h) = O(h q+3 ) (3.6) after one time tep h. Applying a one tep method of convergence order p on the truncated problem (3.3) mean to calculate an approximation Ω 1 to the exact value Ω(h) uch that Ω(h) Ω 1 = O(h p+1 ). (3.7) Finally, we interpret Ω 1 a an approximation to the exact olution of the original problem (3.2). The quality of thi approximation i determined by the deviation (3.6) introduced by the modeling and the dicretization error (3.7): Ω(h) Ω 1 Ω(h) Ω(h) + Ω(h) Ω 1 = O(h q+3 ) + O(h p+1 ). Thi clearly indicate that Ω 1 i a numerical approximation to Ω(h) of convergence order p, i. e., 4. Numerical Tet Ω(h) Ω 1 = O(h p+1 ) if q + 3 p + 1, i. e., q p 2. We conider a pure lattice gauge theory in SU(2,C) with Wilon action and compare the SPRK method decribed in (2.5) with the Leapfrog method. For thi purpoe, we invetigate a ymmetric partitioned Runge-Kutta cheme of convergence order p = 4 which contain the truncated function Ω = f q (Ω,A) given in (2.3). Becaue of the ymmetry, the method ha an even convergence order uch that the choice p = 3,q = 1 already lead to a local error of order 5. Thi mean, we ue the equation Ω = f 1 (Ω,A) = A 1 [Ω,A] (4.1) 2 according to (2.3) and perform imulation on a 2-dimenional lattice with lattice ize L = T = 32. There are 2 reult hown in figure 1: On the left ide, the convergence order of the different method can be een. For thi purpoe, we conider the energy change H of two ucceive configuration after a whole trajectory of length 1 and take the mean of 5 configuration. The 1 1.15 Leapfrog SPRK h 2 1.1 H 1 5 h 4 Determinant 1.5 1 Leapfrog SPRK.995 1/32 1/16 1/8 1/4 1/3 Step ize 1/32 1/16 1/8 1/4 1/3 Step ize Figure 1: Left: Convergence order. Right: Area-preervation. 6
tatitical error are o mall that they are not viible in the plot. Since the energy change H deviate from zero jut becaue of the numerical error of the integration cheme, the violation of the energy preervation give the global error. We ee that for a given H the SPRK allow for larger tep ize. On the right ide of figure 1, we ee the violation of the area-preervation in dependence of the tep ize choen in the numerical method. Area-preervation (up to roundoff error) i given if the determinant of (Ω 1,A 1 )/ (Ω,A ) ha exactly the value 1. Here, the determinant i numerically approximated by firt order difference quotient. 5. Concluion We invetigated the accuracy of the time-reverible ymmetric partitioned Runge-Kutta cheme (2.5). The order of accuracy conit of two component: On the one hand, the convergence order depend of coure on the order p of the method itelf. A the method i ymmetric, the local error i alway odd, i. e., the cheme ha a local error of order p+1 for an even convergence order p. On the other hand, the SPRK cheme contain one truncated erie (2.3). The truncation index q ha to be larger than or equal to p 2 to meet the precribed convergence order. All in all, chooing an SPRK method with an even local error hould be preferred ince a convergence order of one i gained by the ymmetry. We performed imulation for an SPRK cheme of convergence order 4 and ee that the global error given in the numerical reult ha order 4 a theoretically expected. In the development of the SPRK method, area-preervation ha not been conidered. Thu, it i not urpriing, that area-preervation i not met applying thi cheme. Thi property ha to be invetigated in future work. Reference [1] S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Hybrid Monte Carlo, Phyic Letter B, 216 222, 195, 1987. [2] M. Wandelt, M. Günther, F. Knechtli, M. Striebel, Symmetric Partitioned Runge-Kutta Method, 211, arxiv:119.33 [hep-lat]. [3] Wilhelm Magnu, On the exponential olution of differential equation for a linear operator, Communication on Pure and Applied Mathematic, 4, 649 673, doi:1.12/cpa.316744. [4] Han Munthe-Kaa, High order Runge-Kutta method on manifold, Appl. Numer. Math, 29, 115 127, 1999. [5] Hairer, E and Nørett, S. P. and Wanner, G., Solving Ordinary Differential Equation I Nontiff Problem, Springer, econd revied, 2. [6] E.Hairer, C.Lubich and G.Wanner, Geometric Numerical Integration Structure-Preerving Algorithm for Ordinary Differential Equation, Springer Ser. Comput. Math. 31, 2nd ed., Springer, 26. [7] W. Kamleh and M. Peardon, Polynomial Filtered HMC an algorithm for lattice QCD with dynamical quark, arxiv:116.5625 [hep-lat]. [8] M. A. Clark, B. Joó, A. D. Kennedy, P. J. Silva, Improving dynamical lattice QCD imulation through integrator tuning uing Poion bracket and a force-gradient integrator, arxiv:118.1828 [hep-lat]. 7