INSTABILITY OF CRANK-NICOLSON LEAP-FROG FOR NONAUTONOMOUS SYSTEMS

Transcription

1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 5, Number 3, Pages c 2014 Institute for Scientific Computing and Information INSTABILITY OF CRANK-NICOLSON LEAP-FROG FOR NONAUTONOMOUS SYSTEMS WILLIAM LAYTON, AZIZ TAKHIROV AND MYRON SUSSMAN Abstract The implicit-explicit combination of Crank-Nicolson and Leap-Frog methods is widely used for atmosphere, ocean and climate simulations Its stability under a CFL condition in the autonomous case was proven by Fourier methods in 1962 and by energy methods for autonomous systems in 2012 We provide an energy estimate showing that solution energy can grow with time in the nonautonomous case, with worst case rate proportional to time step size We present two constructions showing that this worst case growth rate is attained for a sequence of timesteps t 0 The construction exhibiting this growth for leapfrog is for a problem with a periodic coefficient Key words partitioned methods, energy stability 1 Introduction Stability of CNLF, the Crank-Nicolson Leap-Frog method CNLF) below, is considered for systems with nonautonomous At), Λt): du 11) dt + At)u + Λt)u = 0, for t > 0, and u0) = u 0 Here At), Λt) are d d matrices and ut) is a d vector At) is positive semi-definite symmetric and Λt) is skew symmetric Let 2 denote the euclidean norm The CNLF discretization of 11) is expressed as follows Let t n = n; given u 0, u 1 find u n X for n 2 satisfying u n+1 u n 1 CNLF) + At n ) un+1 + u n 1 + Λt n )u n = 0, 2 2 with approximations to appropriate accuracy, 23, at the first two time steps CNLF is the implicit-explicit IMEX) method used for the dynamic core of most current atmosphere, ocean and climate codes, eg, 6, 15, 22, 13 and other geophysics problems, 16 Stability was shown for the scalar, autonomous case under the timestep condition 12) Λ 2 α < 1, in 1963 by Johansson and Kreiss 14 and for non-commuting) autonomous systems in , see also 23, 6 for background We prove herein weak instability in the nonautonomous case Remark 1 CNLF is often used in geophysical fluid dynamics codes together with a time filter The general strategy used is to split the nonlinear) equations of motion into terms corresponding to high speed/low energy waves and low speed/high energy waves The respective terms are discretized by CN and LF with time filters Williams 24, for example, lists 20 atmosphere codes, 15 ocean codes and 24 coupled / geophysics codes based on this approach The precise realization varies with Received by the editors February 6, Mathematics Subject Classification 65M12, 65J08 The research of the first two authors described herein was partially supported by NSF grant DMS and AFOSR grant FA

2 290 W LAYTON, A TAKHIROV AND M SUSSMAN the effects included and the implementation For one detailed development of one such splitting see Section 3 in 9) Linearization of the split system leads to nonautonomous systems of the form 11) above Time filters are used with CNLF, eg, 3, 19, 24, 13, with the usual explanation that the latter controls the unstable mode of CNLF However, the so-called) unstable mode or computational mode) of CNLF has recently been proven in 12 to be asymptotically stable under 12) We do not study time filters herein However, the instability result does suggest that one positive contribution of time filters may be to control the weak instability identified herein that arises from nonautonomous and periodic Λt) The extension of stability theory for linear multistep methods from autonomous to nonautonomous with test problem y = λt)y) has a rich history Dahlquist 5 proved that an A-stable method is similarly stable for y = λt)y when Re λt)) 0, further developed in 18 For the corresponding AN-stability theory for Runge- Kutta methods, see Hundsdorfer and Stetter 10 For non-a-stable multi-step methods, nonautonomous stability theory was recently developed in 4 with both stability and instability conditions for y = λt)y Given a linear multistep method for y = λt)y, let ρz), σz) be the complex polynomials associated with the method in a standard way and form ρz) A := Re σz) z=i Even if is small enough to be in the stability region of the method, if A < 0 then there exists a λt) < 0 for which the method is unstable 4 While the theory in 4 does not apply to IMEX methods like CNLF, it can be applied to the special case At) = 0 The leapfrog method At) = 0) is an important wedge example Indeed, for leapfrog, we calculate ρz) = 1 2 z2 1 2, and σz) = z Thus A = 0 and the theory of 4 is inconclusive Many interesting behaviors are possible between exponential asymptotic stability and exponential instability One hint is that there is a rich catalog eg, 1,20,21,25 ) of exotic behavior of leapfrog for Burgers equation starting to our knowledge) with Fornberg s 1973 paper 8 The results are clearest for the case that Λt) is Lipschitz, 13) Λt n ) Λt n 1 ) 2 a 0 We prove in Theorem 1 that any instability is, at worst, a weak one: 14) u N u N 2 2 Cα, u 0, u 1 a 0 ) exp tn The rate constant a 0 /) 0 as 0 but exp a0 1 α tn as t N However, the true solution of 11) is uniformly bounded and if At) > 0, ut) 0 as t Section 3 presents give two constructions that show that 14) is best possible for the leapfrog case At) = 0) Remark 2 Theorem 25 in 4 shows that for A < 0, then there is an alternating pair of states that together lead to growth The first construction in Section 3 shows similar behavior for A = 0 In Section 3 we give two constructions that show that LF and thus CNLF) is exponentially unstable for arbitrarily small timesteps when Λt) is a bounded function that changes sign periodically A numerical study is also given in Section 3 The numerical data suggest that the instability occurs for a sparse set of

3 INSTABILITY OF CRANK-NICOLSON LEAP-FROG METHODS 291 timesteps The two constructions in Section 3 show conclusively that the upper estimate in Theorem 1 is attained and that it persists for arbitrarily small timesteps Naturally, upper estimates in Theorem 1) follow more readily than non-generic lower estimates in Section 3 that prove instability Thus, the proof of instability in Construction 2 in Section 3 is necessarily technical 2 CNLF when Λ = Λt) We prove the claimed stability bound for the CNLF method In the Lipschitz case, we then show that the upper estimate implies one slightly sharper than the bound claimed in the introduction Theorem 1 Assume for every t n that 12) holds Then CNLF satisfies: for every N 2 21) u N u N 2 2 Cα, u 0, u 1 N ) exp Λt n+1 ) Λt n ) 2, where Cα, u 0, u 1 ) = u u u 1 ) T Λt 0 )u 0 /) Proof Define E n+1/2 := u n u n 2 Take the dot product of CNLF) with u n+1 + u n 1) and add and subtract u n 2 2 This gives E n+1/2 E n 1/2 + u n+1 + u n 1) T At n ) u n+1 + u n 1) + +2 u n+1 + u n 1) T Λt n )u n = 0 Since A is positive semi-definite u n+1 + u n 1) T At n ) u n+1 + u n 1) 0 and can be dropped From skew symmetry of Λt n ) we can write E n+1/2 E n 1/2 + 2 u n+1 ) T Λt n )u n u n ) T Λt n )u n 1 0 Rewrite the Λ terms as C n+1/2 C n 1/2 Q n with Q n the extra term that now occurs due to time dependence u n+1 ) T Λt n )u n u n ) T Λt n )u n 1 = C n+1/2 C n 1/2 Q n, C n+1/2 := u n+1) T Λt n )u n, C n 1/2 := u n ) T Λt n 1 )u n 1, Q n := u n ) T Λt n ) Λt n 1 ) u n 1 Note that by the usual inequalities 2Q n Λt n ) Λt n 1 ) 2 E n 1/2, giving, for n 1, E n+1/2 + 2C n+1/2 E n 1/2 + 2C n 1/2 + Λt n ) Λt n 1 ) 2 E n 1/2 The timestep condition 12) implies E n+1/2 + 2C n+1/2 )E n+1/2 Summing, we have E N+1/2 1 E 1/2 + C 1/2 + N Λt n ) Λt n 1 ) 2 E n 1/2 E N+1/2 1 Thus by the discrete Gronwall lemma we have, for every N 1 E 1/2 + C 1/2 + E 1/2 + C 1/2 ) 2 N Λt n ) Λt n 1 ) 2 M N,

4 292 W LAYTON, A TAKHIROV AND M SUSSMAN where M N := n<j N In particular, as a special case α Λtj ) Λt j 1 ) 2 ) E N+1/2 1 E 1/2 + C 1/2 exp N Λt n ) Λt n 1 ) 2 This completes the proof under the assumed timestep restriction The predicted growth rate for Lipschitz Λt) Consider the growth term on the RHS of 21) If Λt) is Lipschitz continuous 13) then N exp Λt n ) Λt n 1 ) 2 exp a 0t N Returning to the proof, the growth rate arises from 1 + a 0 2 ) N 1 + a a0 2) ) 2 N ) )2! + exp a 0N, in which the first step obviously is not sharp If we rescale by s = a0 1 α n)2 = ) n a 0 1 α tn ) 2, m = n 2, we have 1 + a0 1 α 2 = 1 + s m) m Sharp double asymptotic limits m and s can be obtained using calculus giving a slight improvement for large timesteps: 1 + a ) N 0 2 exp exp 3 Exponential Instabilities a 0t N 1 α 0807a 0t N 1 α ), for < ) 06117a 1 0 ), for > 06117a 1 0 ) In this section, we present one construction exhibiting a disastrous exponential instability This is followed by a numerical study which suggests strongly that the leapfrog may be generically stable and that the set of t for which instability occurs is sparse The numerical data can be misinterpreted to suggest that for small enough t leapfrog may even be stable This is not the case Our second construction shows that for a periodic Λt) there is a sequence of t 0 for which exponential growth is exhibited Construction 1: Exponential instability under 12) Take At) = 0 so CNLF reduces to LF), = 1/2, u = u 1, u 2 ) t, y 0 = 1, 1) t, y 1 = 1, 1) t and choose lt) = cos2πt) Consider LF for the 2 2 system: 31) u Λt)u = 0, Λt) = lt) +1 0 For this choice of and Λt), Λt n ) 2 = 1 2 < 1, inside the stability region for autonomous systems LF is given by: u 0, u 1 given, u n+1 = u n 1 2Λt n )u n, where Λt n ) = 1) n It is easily proven by induction that u n is given by u n = F n 1, 1) n+1 ) T where {F n } is the Fibonacci sequence Thus u n, as n The instability in this construction is so dramatic that it is surprising that in the next numerical investigation stability seemed incorrectly) to be the generic behavior

5 INSTABILITY OF CRANK-NICOLSON LEAP-FROG METHODS 293 Numerical Study of Stability: For smaller instability occurs on a sparse set We computed the growth rates of CNLF for smaller time steps in the previous example with lt) = cos2πt), and smaller) timesteps = P/K for integers K, P K, K 1400 We applied LF, CNLF) with At) = 0 The average growth rate ρ was computed by constructing the 2 2 matrix given in 37) below and computing its eigenvalues λ j The value σ = max{ λ j } is the largest possible net growth after K steps of LF Thus, for some starting values, the n th LF step satisfies u n u 0 σ n/k = u 0 e ρn, with ρ = 1 K log σ When ρ = 0, we shall take 10 8 as numerically zero) the iteration is stable while if ρ > 0 the LF iterates grow exponentially Figure 1 Average growth rate ρ vs for lt) = cos2πt) Figure 1 plots the largest growth rate, ρ, of LF vs The vertical axis is ρ scaled logarithmically and labelled with ρ values and the horizontal axis is Values of ρ smaller than 10 8 can be regarded as numerically zero because it would take 10 6 steps before u n would increase 1% in size Notice that the values of which give ρ = 0 to numerical precision) are overwhelmingly more common than values of for which ρ is of significant size Thus, a randomly-chosen along with randomly-chosen initial values would be unlikely to produce an instability Notice also that as becomes smaller, values of which produce an instability become even less common It even appears incorrectly) that there are no values of which produce an instability for < 005 We shall show that this is not true for a different lt) in the construction below) While Figure 1 suggests considerable structure, especially for larger, the authors have been unable to observe a pattern in ρ for small It is clear that values of that give rise to significant growth rates are not easy for the authors to guess

6 294 W LAYTON, A TAKHIROV AND M SUSSMAN Construction 2: Exponential instability for t 0 We give next a construction of a 2 2 system 11) that is equivalent to a complex scalar case and for which LF is unstable for arbitrarily small timesteps The coefficient lt) is chosen as a square wave with unit amplitude and unit period see 38) below), so that 0 1 Λt) = lt) 1 0 satisfies Λt) 2 1 and Λt n ) Λt n 1 ) 2 takes the value 0 except twice per period, when it is 2 A sequence k 0 is exhibited for which the solution to CNLF) with At) = 0 grows as e k ct This example shows that the RHS of 21) has asymptotic behavior that cannot be improved and thus LF is unconditionally unstable when At) = 0 The construction is performed in several steps: 1) Define a sequence of complex z n derived from the original sequence v n, thus reducing the original 2 2 system to a complex scalar system 2) Rewrite the nonautonomous recursion CNLF) as an autonomous recursion A 1 Z m = A 2 Z m 1, where the complex vectors Z m are defined by grouping the iterates z n according to the period of lt) 3) Write an explicit recursive expression for Z m = A 1 1 A 2Z m 1 4) Construct a 2 2 matrix B 2 2 whose eigenvalues agree with the nontrivial eigenvalues of A 1 1 A 2 5) Based on the choice of lt), show that the eigenvalues of B 2 2 are real and one of them has magnitude O 2 ) > 1 Step 1 Equivalent complex recursion Rewrite the vector u 1, u 2 ) as a complex scalar z = u 1 + iu 2 Since A = 0, CNLF) becomes 32) z k+1 = z k 1 2i lt k )z k Step 2 Autonomous recursion Assume that lk) is periodic with period K Set a k = 2 lk) Periodicity implies that a k+k = a k Define a complex vector Z m with components Zk m, for k = 1,, K as 33) Z m k = z mk+k Substituting m 1 for m gives Z m 1 k = z mk+k K Writing 32) in terms of Zk m gives ia ia ia ) ia K ia K Z m 1 Z m 2 Z m 3 Z m 4 Z m K Z m 1 1 Z m 1 2 Z m 1 3 Z m 1 4 Z m 1 K =

7 INSTABILITY OF CRANK-NICOLSON LEAP-FROG METHODS 295 Denoting the two matrices in 34) as A 1 and A 2, 35) A 1 Z m = A 2 Z m 1 where Z m is the vector of Zk m and A 1 and A 2 are the above matrices The recursion 35) is autonomous Step 3 Explicit recursion expression The only nontrivial columns of the matrix A 2 are the final two, so the the only nontrivial columns in the product B = A 1 1 A 2 are its final two columns Furthermore, the matrix A 1 is lower triangular with only three nontrivial diagonals, so B can be written in recursive form as 36) B 1,K 1 = 1 B 2,K 1 = ia K B 1,K = ia 1 B 1,K 1 B 2,K = 1 ia 1 B 2,K 1 B k,j = B k 2,j ia k 1 B k 1,j for k = 3,, K and j = K 1, K Step 4 Reduction to a 2 2 matrix Since the recursion 35) is autonomous, its stability is determined by the spectral radius of the matrix B It turns out that the spectral radius of B is equal to the spectral radius of a derived 2 2 matrix B 2 2 We will choose a function lt) for which the spectral radius of B 2 2 is larger than one for a sequence of values K Because of its structure, there are necessarily K 2 null eigenvalues of B If a vector Z is an eigenvector of B with eigenvalue λ 0, then the following 2 2 system must be satisfied 37) B 2 2 ZK 1 Z K BK 1,K 1 B = K 1,K ZK 1 B K,K 1 B K,K Z K ZK 1 = λ Z K Each eigenvector of B 2 2 can be expanded into an eigenvector of B, by choosing its two components as the two initial values in 32), so the non-null eigenvalues of B and B 2 2 agree Step 5 Spectral radius Choose lt) to be the periodic function of period 1 38) lt) = +1 0 t < 1/4 1 1/4 t < 3/4 +1 3/4 t < 1 periodic otherwise For an even 1 integer k 0, let K = 22k 0 + 1), and = 1/K With this choice of K, the values t k = k never exactly equal a point of discontinuity of lt), and can be chosen arbitrarily small A straightforward but tedious induction shows that 39) B 2 2 = The eigenvalues of B 2 2 are 1 2i 2i Thus, the spectral radius is larger than 1 + O 3 ) λ = 1 ± O 3 ) σ = sprb 2 2 ) = spra 1 1 A 2) = O 2 ) > 1 Choose an initial vector Z 0 as the dominant eigenvector of B, Z k = σz k 1 Since each vector Z k represents K = 1/ LF timesteps, the complex iterates satisfy 1 Odd integers work similarly and with the same spectral radius, but the formulæ are slightly different

8 296 W LAYTON, A TAKHIROV AND M SUSSMAN z Kn = z 0 e Kn) log σ + O 2 ) On average, then, with t n = n, z n z 0 e 2/K)tn + O 2 ) Denote the average growth rate per timestep as ρ = 2 + O 2 ) For each even integer k 0 = 2, 4,, letting K = 22k 0 + 1), and = 1/K 0 results in an exponentially divergent LF iteration The average rate of growth of the iterates is ρ = 2 + O 2 ) There is no limiting size of below which the iteration does not diverge This choice of Λt) is not Lipschitz, but Theorem 1 does apply The exponential factor in the theorem is exp N Λt n+1 ) Λt n ) 2 For this choice of Λ, the difference Λt n+1 ) Λt n ) 2 is nonzero and then equal to 2) only twice per period Replacing N with nk, the exponential factor becomes exp 2nK2/K) = exp 4t since nk = t Thus the average growth rate in Theorem 1 is ρ = 4/), and α can be taken to be small Remark 3 It is interesting to note that 39) contains no explicit dependence on the value k 0 This is a consequence of the average of lt) over a period being zero Just after the point t = k 0 /K, where lt) jumps from +1 to -1, the analogous matrix is ik 0 1 k 0 k 0 + 2)/2 2 1 k0k0 + 2)/2 2 ik 0 + 2) Remark 4 The function lt) was chosen because it has a Fourier series with terms related to the function in the first example, lt) = 4/π) n=0 cos22n + 1)πt)/2n + 1) For K = 22k 0 + 1), and = 1/K, as above, the term n = k 0 takes the values ±1, so that growth appears just as in the first example Remark 5 The leapfrog method for the differential equation du dt ilt)u = 0 is precisely 32), and the solution of this equation, starting with u0) = u o is 310) ut) = u 0 e it 0 t < 1/4 u o e it 1/2) 1/4 t < 3/4 u 0 e it 1) 3/4 t < 1 periodic t > 1 This function is bounded, so the growth of the leapfrog approximation is due to the numerical approximation 4 Conclusions In contrast to the autonomous case, in the nonautonomous case we have shown through two constructions that CNLF and, as a special case, leapfrog) computed solutions can grow even when the system itself has bounded or decaying solutions In Theorem 1 we prove that the growth rate, however, is at worst proportional to when Λt) is Lipschitz, a rate attained in the examples constructed One important open problem is related to Remark 1 Time filters are an important and proven method for GFD simulations However, their computational

9 INSTABILITY OF CRANK-NICOLSON LEAP-FROG METHODS 297 practice is far ahead of their analytic foundations Understanding the impact of time filters on the nonautonomous instability delineated herein is an important open problem References 1 A Aoyagi, Nonlinear leapfrog instability for Fornberg s pattern, Journal of Computational Physics, ) U Asher, S Ruuth and B Wetton, Implicit-Explicit methods for time dependent partial differential equations, SINUM, ) RA Asselin, Frequency filter for time integration, Mon Weather Review, ) BR Boutelje and AT Hill, Nonautonomous stability of linear multistep methods, IMA J Numer Anal, ) G Dahlquist, G stability is equivalent to A stability, BIT, ) DR Durran, Numerical methods for fluid dynamics with applications to geophysics, Second Edition, Springer, Berlin, J Frank, W Hundsdorfer and J Verwer, On the stability of Implicit-Explicit linear multistep methods, Appl Num Math, ) B Fornberg, On the instability of the Leap-Frog and Crank-Nicolson approximation of a nonlinear partial differential equation, Math Comp, ) FX Giraldo, JF Kelly and EM Constantinescu, Implicit-Explicit formulations of a three dimensional nonhydrostatic unified model of the atmosphere NUMA), SIAM J Scientific Computing ) B1162- B WH Hundsdorfer and MN Spijker, A note on B- stability of Runge-Kutta methods, Numer Math, ) WH Hundsdorfer and J Verwer, Numerical solution of time dependent advection diffusion reaction equations, Springer, Berlin, N Hurl, W Layton, Y Li and M Moraiti, The Unstable Mode in the Crank-Nicolson Leap-Frog Method is Stable, tech report C Jablonowski, and D Williamson, The pros and cons of diffusion, filters and fixers in atmospheric general circulation models In Numerical Techniques for Global Atmospheric Models, Lecture Notes in Computational Science and Engineering Eds PH Lauritzen, C Jablonowski, M Taylor, RD Nair), p , vol 80, Springer, Berlin, O Johansson, H-O Kreiss, Über das Verfahren der zentralen Differenzen zur Lösung des Cauchy problems für partielle Differentialgleichungen, Nordisk Tidskr Informations- Behandling, ) E Kalnay, Atmospheric Modeling, data assimilation and predictability, Cambridge Univ Press, Cambridge, M Kubacki, Uncoupling Evolutionary Groundwater-Surface Water Flows Using the Crank- Nicolson Leapfrog Method, Numerical Methods for Partial Differential Equations, ) , DOI: /num W Layton and C Trenchea, Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations, ANM, ) O Nevalinna and F Odeh, Multiplier techniques for linear multistep methods, Num Funct Anal and Opt, 31981) A Robert, The integration of a spectral model of the atmosphere by the implicit method, Proc WMO/IUGG Symposium on NWP, Japan Meteorological Soc, p 19-24, Tokyo, Japan, JM Sans-Serna, Studies in Numerical Nonlinear Instability I Why do Leapfrog Schemes Go Unstable?, SIAM J Sci and Stat Comput, 61985) I Sloan and AR Mitchell, On nonlinear instabilities in leap-frog finite difference schemes, Journal of Computational Physics, ) SJ Thomas, RD Loft, The NCAR spectral element climate dynamical core: semi-implicit Eulerian formulation, J Sci Comput, 251-2) 2005) J Verwer, Convergence and component splitting for the Crank-Nicolson Leap-Frog integration method, CWI report MAS-E0902, 2009

10 298 W LAYTON, A TAKHIROV AND M SUSSMAN 24 PD Williams, The RAW Filter: An Improvement to the Robert Asselin Filter in Semi- Implicit Integrations, Mon Weather Rev, ) W Zhou, An alternative Leap-Frog scheme for surface gravity wave equations, J Amer Meteorological Soc, ) Department of Mathematics, University of Pittsburgh Pittsburgh, PA 15260, USA wjl@lpittedu, azt7@pittedu and sussmanm@mathpittedu URL: URL: