Filtered Leapfrog Time Integration with Enhanced Stability Properties

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1 Journal of Applied Mathematic and Phyic, 6, 4, 4-7 Publihed Online July 6 in SciRe. Filtered Leapfrog Time Integration with Enhanced Stability Propertie Ari Aluthge, Scott A. Sarra, Roger Etep Department of Mathematic, Marhall Univerity, Huntington, WV, USA Received 7 May 6; accepted July 6; publihed 8 July 6 Copyright 6 by author and Scientific Reearch Publihing Inc. Thi work i licened under the Creative Common Attribution International Licene (CC BY). Abtract The Leapfrog method for the olution of Ordinary Differential Equation initial value problem ha been hitorically popular for everal reaon. The method ha econd order accuracy, require only one function evaluation per time tep, and i non-diipative. Depite the mentioned attractive propertie, the method ha ome unfavorable tability propertie. The abolute tability region of the method i only an interval located on the imaginary axi, rather than a region in the complex plane. The method i only weakly table and thu exhibit computational intability in long time integration over interval of finite length. In thi work, the ue of filter i examined for the purpoe of both controlling the weak intability and alo enlarging the ize of the abolute tability region of the method. Keyword Leapfrog Method, Computational Mode, Weak Stability, Method of Line, Numerical Weather Prediction. Introduction In thi work the tability propertie of a well known linear multitep method (LMM) are examined. The method i baed on a three point polynomial finite difference approximation of the firt derivative. The Leapfrog method (alo known a the explicit midpoint rule or Nytröm method) for the firt order Ordinary Differential Equation (ODE) initial value problem (IVP) y = F t, y t, y = y () i a imple three-level cheme with econd order accuracy. The leapfrog cheme ha a long hitory in application. The application include general oceanic circulation model and atmopheric model that go back at leat How to cite thi paper: Aluthge, A., Sarra, S.A. and Etep, R. (6) Filtered Leapfrog Time Integration with Enhanced Stability Propertie. Journal of Applied Mathematic and Phyic, 4,

2 a far a the pioneering work of Richardon [] (p. ) in 9 where it wa called the tep-over method. In thi context, the time integration method i ued to advance in time a emi-dicrete ytem in which the pace derivative of a partial differential equation have been dicretized by an appropriate method. Such an application i called the method of line. The leapfrog method remain relevant today a it i a component of many numerical weather forecating ytem and mot current geophyical fluid dynamic (GFD) code ue a form of leapfrog time differencing. Reference [] lit maor GFD code currently uing thi approach. The recent textbook [] (p. 9) decribe the leapfrog method a an important method for numerical weather forecating. The Leapfrog method i efficient a it only require one function evaluation per time tep, with a function evaluation referring to the function F in Equation (). The leapfrog cheme ha the deirable property of being non-diipative, but it uffer from phae error. The mot eriou drawback of the cheme i it well documented [4] weak tability property which reult in computational intability when it i ued in long time integration. Method that have been ued to control the weak intability of the leapfrog method involve a proce called filtering. A Filtering proce replace a time level with an average of that time level and neighboring time level. The ue of filtering in numerical ODE method originated in [] with a two-level averaging. Filtering method that have been ued to control the weak intability of the leapfrog method include the following. The Aelin time filter [6] add a mall artificial vicoity to the leapfrog method and i applied each time tep. Gragg [7] ued a three-point ymmetric filter to replace a time level of the leapfrog olution after every N tep. Bulirch and Stoer [8] extended Gragg method to build an extrapolation cheme. In [9] filtering wa hown not to be neceary in extrapolation cheme involving the leapfrog method ince the Euler tarting tep and the extrapolation proce alone have been able to control the weak intability. Five point ymmetric and aymmetric filter were conidered by Iri [] who recommended the ue of an aymmetric filter. Stabilizing the Leapfrog method continue to be an active reearch area. Example of recent work in the area can be found in [] and []. In the next ection LMM and their tability propertie are reviewed. Then three and five point filter are developed and it i hown that a five-point ymmetric filter can be effectively ued to control the weak intability that i inherent to the leapfrog method. After the five-point ymmetric i hown to be effective, way to incorporate the filter into a leapfrog time differencing cheme are examined. The application of the filter lead to new LMM a well a variou filter and retart algorithm. The ideal application of the filter will control the weak intability a well a retain the favorable feature of the method: efficiency, econd order accuracy, an abolute tability region with imaginary axi converge, and will be non-diipative. Finally, the concluion are illutrated with numerical example.. Linear Multitep Method A general -tep linear multitep method for the numerical olution of the ODE IVP () i of the form n+ m n+ m n+ m αm βm m= m= () y = k F t, y, n =,, where α m and β m are given contant and k i the ize of the time tep. It i conventional to normalize () by etting α =. When β = the method i explicit. Otherwie it i implicit. In order to tart the method, the firt time level need to be calculated by a one-tep method uch a a Runge-Kutta method. The tability of ODE method i a relatively new concept in numerical method and wa rarely if ever ued before the 9 []. The firt decription of numerical intability of numerical ODE method may have been in 9 in [4]. It wa the advent of high peed electronic computer that brought the concept of tability to the forefront. Varying definition of and type of tability for ODE method have been propoed over the lat 6 year. Indeed, when urveying the literature and textbook that have been publihed over that time period it i not traightforward to grap what i meant by the tability of a LMM due to the multitude of definition and the ame concept often being given more than one name. The tability propertie of LMM are examined by applying the method to the ODE y = λ y, y = y () where λ i a complex number with R( λ ). Equation () i commonly known a the Dahlquit tet equation. The firt concept of tability reult from conidering the imple cae of λ = in () for which the olution i the contant y. In thi cae the application of the LMM () to the tet equation reult in a difference equation

3 with a general olution of the form where the ζ i are the zero of the firt characteritic polynomial n n n y = cζ + + c ζ (4) ρζ m = αζ m () m= n of the LMM (). For a conitent method one of the term, ay ζ i, approache the olution of () a n while the other root correpond to extraneou olution. The following tability definition quantify to what extent the computed olution of (4) remain bounded: A method i table (alo called zero table) if ρ ha a imple root equal to one and if all the other root of ρ are le than one in abolute value. The criteria that the root mut atify for the method to be table are called a root condition. A weakly table method ha all root of ρ atifying ζ and ha at leat one root of abolute value ζ =. The olution may remain bounded for a fixed interval of time [,T ] one other than the imple root but will eventually become unbounded a t. A method for which a root of ρ i greater in magnitude than one i untable. In thi cae the olution will rapidly become unbounded. A tronger definition of tability can be made by conidering non-zero λ in Equation (). In thi cae the exact olution of () i y( t) = y e λt which approache zero a t ince the real part of λ i negative. A n n method that produce numerical olution with the ame aymptotic behavior, that i y a t, for a fixed value of k i aid to be abolutely table. Thi type of tability ha alo been called aymptotic tability, or eigenvalue tability. Application of the LMM () to the tet equation again reult in a difference equation with a general olution of the form (4), but thi time ζ i the zero of the tability polynomial where z i zσ ( ζ ) ρ ζ = (6) = kλ, ρ i the firt characteritic polynomial (), and σ i the econd characteritic polynomial σ ζ = βζ m (7) m= of the LMM (). The method i abolutely table for a particular value of z if the zero of (6) atify the root condition. The et of all number z in the complex plane for which the zero of (6) atify the root condition i called the abolute tability region,, of the method. The boundary of the abolute tability region i found by plotting the root locu cure iθ ( e ) iθ ( e ) ρ C = θ π (8) σ that i the ratio of the characteritic polynomial of the method. For z in the tability region, the numerical olution exhibit the ame aymptotic behavior a doe the exact olution of (). For a linear ytem of ODE, a neceary condition for abolute tability i that all of the caled (by k) eigenvalue of the coefficient matrix of the ytem mut lie in the tability region. For nonlinear ytem, the ytem mut be linearized and the caled n eigenvalue of the Jacobian matrix of the ytem mut be analyzed at each t. Standard reference on numerical ODE method [4] []-[9] can be conulted for more detail.. The Leapfrog Method The leapfrog method for the ytem () i n+ n n n y = y + kf t, y. (9) The method i contructed by replacing the time derivative in () by the econd order accurate central difference approximation n+ n n y y y ( t ) () k 6

4 of the firt derivative. The method need two time level to tart, y which i provided by the initial condition and y. Typically, the time level y i obtained by an explicit Euler tep The characteritic polynomial of the leapfrog method are y = y + kf t, y. () z σ ( ζ ) ρ ζ = and = z. () Obviouly the root of ρ are ζ = ± and the method i only weakly table. The boundary of the abolute tability region i iθ ρ ( e ) ( iθ) ( iθ) e e C = = = iin iθ σ ( e ) and the abolute tability region only conit of the interval = [ i, i] 4. Filter Contruction θ on the imaginary axi. It i hown in [] that the olution of the IVP () obtained by the leapfrog method uing tarting value only aumed to have an aymptotic expanion of the very general form ha an aymptotic expanion of the form The phyical mode y p n m 4 ηmn ηn k = y + k + k, n =, () m= 4 ( p) ( c) n y = phyical mode y + computational y + k. (4) λ λ yp = y + yt+ + k + y + + k 6 6 conit of the exact olution tational mode λt λt λt e ( η η ) e λη η η e y e λt and a ( k ) n n λ [ ] () truncation error that goe to zero a k. The compu- y c η η e k = y λη η η e k 6 λt λt contribute a non-phyical ocillatory factor to the olution which grow in ize with increaing t. Reference [7] and [] can be conulted for the detail of the derivation of the expreion for y p and y c. In order to damp the computational mode while retaining the econd order accurate approximation of the phyical mode, a filter of the form P E (6) r = ae (7) = r, and E i the forward hift operator defined by Ey( t) y( t k) i conidered where a, r, r + n n continuou notation and Ey = y + in dicrete notation. Let = λk and then note that λt λ( t+ k) λt λk λt = = =. Similarly, e λ t ( e λ t ) ( e λ t e λ E E E E ) e t ( e ) Ee e e e e e t t E λ λ and e = e e. Now = = = and in follow that e λ r r r t t t t t = e λ = e λ ( e ) = e λ ( e ) = e λ ( e ) P E ae a a P = r = r = r = + in 7

5 r λt λt = = ( + ) P E te a E te a t k e = r = r λ( t+ k ) a k r λt λt λt a t = r = e e e + e e λt d = e tp ( e ) + k P ( e ). d Thu, applying the filter (7) to the phyical mode () reult in λ 6 r [ η η] λt λt λt yˆ p = y e P e + y e tp e + k P e + + e P e λ 6 + y λη + η + η and applying the filter to the computational mode give λt e P( e ). n ( ) [ ] λt ( η η ) yˆ c = + e P e k n λ λt + y + λη + η η e P( e ) k. 6 Equation (8) ugget that in order for the phyical mode to be kept unchanged up to the order that P mut atify N (8) (9) N k term P e = + +. () In a imilar manner Equation (9) reveal that in order for the computational mode to be reduced by a factor M of k that P mut atify 4.. Three Point Filter M ( ) P e =. () Firt, filter of length three are conidered. A three point ymmetric filter i of the form. P E = a E + a + ae () The filter coefficient can be found by equating the coefficient of () with the coefficient from Equation () and Equation (). Thi can be done by etting N = and M = for which the firt term in the expanion of () and () repectively become and Additionally P ( e ) ( ) = + () ( ) P e =. (4) ( e ) = ( e ) + + ( e ) = a ( ) + a + a( + ) + ( ) P a a a where on the econd line e ha been replaced by it linear Taylor erie expanion. Equating the contant term () and () reult in the equation () a a a + + = (6) 8

6 and equating the coefficient of give Similarly a + = (7) a. ( e ) = ( e ) + + ( e ) P a a a = a + a a + where e ha been replaced by it Taylor erie truncated after the contant term. Equating the contant term (4) and (8) reult in the equation. (8) a + a a = (9) The linear ytem coniting of Equation (6), (7), and (9) ha the olution a =, a =, and a =. 4 4 Thu, the three-point ymmetric filter i or equivalently P E E 4 = ( + + ) n n n n+ yˆ = ( y + y + y ). () 4 Three point filter alter the accurate phyical mode by a factor of k and alo damp the computational mode by a factor of k. The coefficient of non-ymmetric filter P = ae and P = ae can be found in a imilar manner. Their coefficient are lited in Table. 4.. Five-Point Filter = The conideration of a five point filter preent two more degree of freedom that can be ued to develop a filter that alter the phyical mode by one le power of k and that damp the computational mode by one more power of k in comparion to a three point filter. That i, in the aymptotic expanion of the phyical mode N = and in the aymptotic expanion of the computational mode M =. A five point ymmetric filter i of the form and =. P E = a E + a E + a + ae + ae () In thi cae the firt few term in the expanion of () and () repectively are Additionally Table. Three point filter coefficient. P ( e ) ( ) = + () ( ) P e =. () 4a 4a 4a 4a 4a P - - P - - P - - 9

7 ( e ) = ( e ) + ( e ) + + ( e ) + ( e ) P a a a a a ( ) ( ) = a + + a + + a where thi time e ha been replaced by it Taylor erie expanion. Equating the contant term () and (4) reult in the equation and then equating the coefficient of give and finally equating the coefficient of Similarly ( ) (4) + a a ( ) () a + a + a + a + a = (6) a a + a + a = (7) give a a. + a a + = (8) ( e ) = ( e ) + ( e ) + + ( e ) + ( e ) = a ( + ) + a ( + ) + a + a( ) + a( ) + ( ) P a a a a a where e ha been replaced by it Taylor erie expanion. Equating the contant and linear term in () and (9) reult in the equation and (9) a a + a a + a = (4) a a + a a =. (4) The linear ytem coniting of Equation (6)-(8), (4), and (4) ha the olution a =, a =, 6 4 a =, a =, and a =. Thu, the five-point ymmetric filter i or The coefficient of non-ymmetric filter P P E E E E 6 = ( ) n n n n n+ n+ yˆ = ( y + 4y + y + 4 y y ). (4) 6 ( ) P = ae, = 4 ( ) P = ae, = 4 = ae can be found in a imilar manner. Their coefficient are lited in Table. = P = ae, and = The advantage of the five point filter when compared to the three point filter are now clear. The five point filter retain the econd order accuracy of the leapfrog method and damp the computational mode by a factor of k. In contrat, the three point filter reduce the accuracy of the method to firt order and only damp the computational mode by a factor of k. It remain to find the bet way to apply the filter. In the following ection the filter i applied to derive new LMM and variou filter and retart algorithm are analyzed. 6

8 Table. Five point filter coefficient. 6a 6a 4 6a 6a 6a 6a 6a 6a 6a 4 P P P P P New LMM Incorporating the filter in each leapfrog tep by averaging or y = y + kf t, y n+ n n n n y with = = P reult in the new LMM ( n y n 4y n y n 4y n+ y ) n n kf ( t, y ) ( n y n 4y n y n 4y n y n n kf ( t, y ) ) n n kf ( t, y ) n+ n 9 n n n n n y = y + y + y y + kf ( t, y ) (4) Thi approach ha the advantage that the reult may be analyzed within the LMM framework. The characteritic 4 9 polynomial of the method are ρ ( z) = z z z z+ and σ ( z) = z. The firt characteritic polynomial ha ζ = a a root and three other root le than one in magnitude and thu the method i zerotable. The abolute tability region of the method i hown in Figure. The maximum tability ordinate along the imaginary axi i approximately.87 and along the real axi.. The new method i econd order accurate. n Averaging y with P in each leapfrog tep reult in the new LMM n+ n n n n n y = y + y + y + hf ( t, y ). (44) 4 4 The root of the firt characteritic polynomial of the method are ζ = and two other root with magnitude one-half. The method i zero-table but only firt order accurate. The maximum tability ordinate along the imaginary axi i approximately /4 and along the real axi /. A trong point of the leapfrog method, epecially in the method of line integration of wave propagation type PDE, i that it i non-diipative. The phae and amplitude error of the two new LMM are compared with the phae and amplitude error of the leapfrog method in Figure. The P LMM ha phae error are imilar to thoe of the leapfrog method and i by far the leat diipative of the two filtered method. The P LMM uffer from larger phae error than doe the leapfrog method and i very diipative. The effect of the diperion and diipation error are illutrated in example P Filter Application In thi ection, four filter and retart algorithm are explored. For each option the effect on accuracy and tability propertie are examined. The option include: Method (M). Filter after every tep. Method (M). Filter followed by Euler retart every N tep. Method (M). Alternative tarting/retarting-algorithm. Method 4 (M4). Alternative tarting/retarting-algorithm. 6

9 Figure. Stability region for the (olid line) P LLM (4) and (dahed line) P LLM (44). Figure. Amplitude and phae error of the leapfrog method (9), the P LMM (4), and the P LMM (44). To allow for the abolute tability region to be examined, all four algorithm have a tart up procedure following by multiple tep of leapfrog and/or filtering. Then after a fixed number, N, of tep the algorithm i retarted with the tart up procedure. In each of the four algorithm N = or N = ha been ued. In mot cae, increaing N reult in the method having an abolute tability region with le coverage of the left half plane and with more imaginary axi coverage. The deired filtering algorithm will maintain econd order accuracy, retain a much of the tability interval on the imaginary axi a poible, and extend the abolute tability region into the left half plane. 6.. Filter after Every Step (M) The firt approach ue an explicit Euler tep () to calculate the firt time level. Each ubequent tep take one leapfrog tep to the time level n plu two additional leapfrog tep to calculate time level n + and n + that are needed by the filter. Then the P filter i applied and time level n i replaced with the average. Time level n + and n + are diregarded and then the proce i repeated until time level N i reached. At that time the method i retarted uing time level N a the initial condition. The M approach retain econd order accuracy but require three function evaluation per time tep. 6

10 The abolute tability region of the M approach with N = i hown in Figure. In the left image of the figure the tability region i een to cover ignificant area in the left half plane and appear to retain coverage of the portion of the imaginary axi over the interval [ ii, ]. However, the right image of the figure zoom in on the imaginary axi and reveal that a large portion of the interval i not included in the tability region. The abolute tability region of method M, M, M, and M4 are found in the following manner. For example, the tability region of M with N = i found by applying the method to the tet problem (). Then R z of the method i found by looking at the amplification factor and y = y + kλ y = + z y y = y + kλ y = + z+ z y y = y + kλ y = + z+ 4z + 4z y y = y + kλ y = + 4z+ 8z + 8z + 8z y 4 4 y = y + kλ y = + z+ z + z + 6z + z y 4 4 y = ( y + 4y + y + 4y y ) y = + z+ z + z + z z y 4 4 = R z y. 4 The abolute tability region contain all the z for which R( z). 6.. Filter, Euler Retart Every N Step (M) Method M approximate the firt time level with Euler method () and then advance in time with N + N leapfrog tep. Then the P filter i applied and the value of y i aigned to be the filtered average. Then N the method i retarted with the initial condition y. Thu the method i y = y + kf t, y n n n n y = u + kf t, y, n =,,, N + (4) N N N N N+ N+ y = P y, y, y, y, y. (a) Figure. (a): Abolute tability region of method M. (b): cloe up of the tability region along the imaginary axi. (b) 6

11 + N function evaluation per time tep. In the given example an average of. function evaluation per time tep are needed. The main drawback of approach M i that the tability region (Figure 4) lack any coverage along the imaginary axi. Method M i untable for all value of z > for which leapfrog method i abolutely table. The M approach remain econd order accurate and require ( N ) 6.. Alternative Start/Retart-Algorithm (M) The abolute tability region of Euler method i a circle of radiu one that i centered at z = in the complex plane. The region doe not include any purely complex number. Calculating the firt tep of method M with Euler method i cauing the method tability region to lack imaginary axi coverage. The following tarting procedure k y = y + F t, y M n n k n y = y + F t km, y, n,, M M + = (46) M y = y i uggeted for minimizing thi effect. To advance from the initial condition to time level one, M maller ubtep are taken coniting of one Euler tep followed by M leapfrog tep. After the firt tep, algorithm M i identical to algorithm M. The abolute tability region of method M with N = and M = 4 i hown in Figure. The propoed tarting procedure ha improved the imaginary axi coverage of the tability region to approximately the interval [. i,.i], or about half the ize of the leapfrog method tability interval. Method M require ( N + M + ) N function evaluation per time tep, which in the given example i equal to.. Method M4 in the next ection make one more modification to further increae the coverage of the tability region along the imaginary axi Alternative Start/Retart Algorithm (M4) Algorithm M4 ue the ame tarting procedure (46) a doe M but it doe not terminate and retart after filtering at time tep N. Intead time level N and N are filtered and then N more leapfrog tep are taken. The filtering and continuing proce i done C time before retarting. At the end of each continuation tep the final time level i filtered. The region of abolute tability for the M4 algorithm with M = 4, C =, and N = 7 (a total of time tep in order to approximate the time tep ued in the M, M, and M example) i in Figure 6. The imaginary axi coverage of the M4 tability region i nearly equal to that of the leapfrog method, (a) Figure 4. (a): Abolute tability region of method M. (b): cloe up of the tability region along the imaginary axi. (b) 64

12 (a) Figure. (a): Abolute tability region of method M. (b): cloe up of the tability region along the imaginary axi. (b) (a) Figure 6. (a): Abolute tability region of method M4. (b): cloe up of the tability region along the imaginary axi. covering approximately the interval [.9 i,.9i] ( N M C N) (( C ) N). The M4 algorithm require an average of function evaluation per time tep which i approximately.4 in the example given. Matlab ource code that implement method M4 i available on the webite of the econd author at 7. Numerical Reult 7.. Example The purpoe of the firt example i to demontrate the ability of the propoed method to control the computational mode that i introduced by the leapfrog weak intability. The example conider the nonlinear IVP which ha the exact olution y( t) tanh ( t) y = y, y =, (47) =. A time tep ize of k =. i ued with all method. Figure 7 (b) 6

13 illutrated behavior typical of a weakly table method. Over the finite time interval [,8 ] the leapfrog method give a good approximation to the true olution. Table report the olution having five digit of accuracy at t =. However, after approximately t = 8, non-phyical ocillation appear in the olution and eventually the olution become unbounded. Table record the reult of applying the two new LMM from ection and a well a uing the four filter and retart algorithm. Additionally, the five point aymmetric filter P that wa recommend in [] i ued. By t = and later, all even of the filtered method have either zero or near zero error. The numerical olution nearly exactly approximate the exact olution which become the contant y = a t. In thi example, all even method have effectively damped the computational mode and prevented the numerical olution from becoming infinite. In five of the method, graphically non-phyical ocillation are not oberved. However, in method P and M light ocillation (a in Figure 8) appear in the numerical olution prior to the computational mode being damped ufficiently. 7.. Example Thi example numerically calculate the order of convergence of the leapfrog method and the even filtered method. The tet Equation () i ued with λ = and with the initial condition y =. Each method i repeated on the time interval [, ] uing the equence of number of time tep N =, 4,8,6,, 64,8, 6,. The reulting convergence plot are hown in the left and right image of Figure 9. On a log-log plot of the abolute value of the error veru the time tep ize the plot will be a traight line with a lope that i the order of convergence of the method. The LMM, Aym -M, M, and P Figure 7. Exact (dahed) and leapfrog (olid) olution of problem (47) with k =.. Table. Error from problem (47). t = t = t = leapfrog 4.e 6. P LMM.9e P LMM 4.e 6.e 6.e 6 M.e 6.e 6.e 6 M 7.8e M.7e 7 M4.6e 6 P M 4.e.e 6.e 6 66

14 Figure 8. Ocillation appear in the aymmetric damped. P and M method olution prior to the computational mode being (a) (b) Figure 9. Convergence plot. (a): LF, LMM, LMM, nearly identical. P -M. (b): M, M, M, M4. The M and M4 plot are viually M method are only firt order accurate while the LMM, M, and M4 method retain the econd order accuracy of the leapfrog method. In thi example, the three econd order accurate filtered method are about a decimal place more accurate for a given value of k than i the leapfrog method. 7.. Advection The purpoe of the third example i to give an illutration of the effect of the diperion and diipation propertie of the two LMM decribed in Section and to illutrate the effectivene of the M4 algorithm a a method of line integrator of hyperbolic PDE. A model hyperbolic PDE problem i the advection equation u u =. (48) t x 67

15 , and periodic boundary condition are enforced. The initial = +. The pace derivative of (48) i dicretized with the Fourier Peudopectral method [] []. A relatively large number of point, N = 79, are ued to dicretize the pace interval o that the pace error will be negligible. After dicretizing in pace the linear ytem of ODE The domain for the problem i the interval [ ] 4 condition i taken from the exact olution u( xt, ) in ( π ( x t) ) ut = Du (49) remain to be advanced in time where D i the N N Fourier differentiation matrix []. The eigenvalue of D are purely imaginary requiring an ODE method with a tability region having imaginary axi coverage. The four method in thi work that are applicable are the leapfrog method, the P LMM, the P LMM, and method M and M4. However, the imaginary axi coverage of the tability region of the M i mall compared to that of M4 o the M method i not applied. Run with two different time tep ize are made. The firt with k =.4 λ and the econd with k =.7 λ, where λ i the magnitude of the larget eigenvalue of D. With the firt time tep the P LMM in practically non-diipative while the econd i near the tability limit for the P LMM. A analyzed in ection 6.4, the M4 method i abolutely table for time tep ize a large a k =.9 max ( ew) for the advection equation. With the maximum table time tep ize the error at t = i.7e. The reult at t = are hown in Figure and additional reult are in Table 4 and Table Advection-Diffuion Dicretized advection-diffuion operator do not have purely imaginary eigenvalue and thu the leapfrog method i not an appropriate method of line integrator for thi type of problem. Advection-diffuion problem in which the advection term i dominant will have differentiation matrice with eigenvalue that have negative real part that are of mall to moderate ize. Algorithm M4 can be effectively applied in uch a problem. Figure. Solution of problem (48) at P LMM; (b): P LMM. Table 4. Error from problem (48), k.7 max ( ew) (a) (b) t = (exact dahed, numerical olid) with k.7 max ( ew) =. = : (a): t = t = t = leapfrog.e.e.e P LMM.7e.e 7.e P LMM.4e.e.e M4.8e.8e.e 68

16 A an example, the advection diffuion equation u + u = ν u () t x x where ν i conidered. The vicoity coefficient i taken a ν =. and periodic boundary condition are applied on the interval interval [,]. The pace derivative are dicretized with the Fourier Peudopectral method with 99 evenly paced point. The caled pectrum of the dicretized pace derivative operator and the tability region of method M4 i hown in the left image of Figure. An initial condition of (,) u x x = otherwie i ued and the problem i advanced in time to t = uing a time tep of k =.7 λ, where λ i the maximum magnitude of the imaginary part of the eigenvalue of the dicretized operator. In the right image of Figure the numerical olution at t = i compared with the exact olution. The maximum error i.e. The reult illutrate the ability of method M4 to accurately advance advection dominated problem over long time interval. 8. Concluion Filter of length three and five have been examined for the purpoe of controlling the weak intability of the leapfrog method. Filtering the leapfrog method with a three point filter reduce the accuracy of the overall cheme to firt order. If a five point filter i ued, it i poible for the method to retain the econd order accuracy a well a damp the computational mode by a factor of k. Several way of incorporating the filter n into the leapfrog method have been examined. Applying the five-point ymmetric filter to the time level Table. Error from problem (48), k.4 max ( ew) =. u t = t = t = leapfrog.e 7.9e.e P LMM.9e.e 6.7e P LMM.e.e 8.e M4.e.e.e (a) (b) Figure. (a): Stability region of method M4 and the caled pectrum of the dicretized problem (); (b): Numerical olution (olid) v. exact (dahed) at t =. 69

17 of the leapfrog method (9) reult in a new econd order accurate LMM (4) which i zero table and that ha an abolute tability region that include both coverage of a portion of the imaginary axi a well a a region in the left half plane. Of the four filter and retart algorithm decribed, algorithm M4 i the mot attractive. The M4 method, which retart after every time tep, ha an abolute tability region that include nearly a much of the imaginary axi a doe the leapfrog method a well a include a region in the left half plane. Thi allow the method to be an effective method of line integrator for pure advection and advection dominated problem a i illutrated in the example. In addition to the econd order accuracy, the method alo retain the computational efficiency of the leapfrog method. Method M4 i a potential replacement for exiting leapfrog type time differencing cheme that are ued in geophyical fluid dynamic code. Reference [] Richardon, L.F. (9) Weather Prediction by Numerical Proce. Cambridge Univerity Pre, Cambridge. [] William, P. (9) A Propoed Modification to the Robert-Aelin Time Filter. Monthly Weather Review, 7, [] Coiffier, J. () Fundamental of Numerical Weather Prediction. Cambridge Univerity Pre, Cambridge. [4] Butcher, J. () Numerical Method for Ordinary Differential Equation. Wiley. [] Milne, W. and Reynold, R. (99) Stability of a Numerical Solution of Differential Equation. Journal of the ACM, 6, [6] Aelin, R. (97) Frequency Filter for Time Integration. Monthly Weather Review,, [7] Gragg, W.B. (96) Repeated Extrapolation to the Limit in the Numerical Solution of Ordinary Differential Equation. PhD Thei, UCLA. [8] Bulirch, R. and Stoer, J. (966) Numerical Treatment of Ordinary Differential Equation by Extrapolation Method. Numeriche Mathematik, 8, -. [9] Shampine, L. and Baca, L. (98) Smoothing the Extrapolated Midpoint Rule. Numeriche Mathematik, 4, 6-7. [] Iri, M. (96) A Stabilizing Device for Untable Numerical Solution of Ordinary Differential Equation Deign Principle and Application of a Filter. Journal of the Information Proceing Society of Japan, 4, [] Li, Y. and Trenchea, C. (4) A Higher-Order Robert-Aelin Type Time Filter. Journal of Computational Phyic, 9, -. [] Norton, T. and Hill, A. () An Iterative Starting Method to Control Paraitim for the Leapfrog Method. Applied Numerical Mathematic, 87, [] Dahlquit, G. (98) Year of Numerical Intability, Part I. BIT, [4] Todd, J. (9) Solution of Differential Equation by Recurrence Relation. Mathematical Table and Other Aid to Computation, 4, [] Hairer, E., Norett, S. and Wanner, G. () Solving Ordinary Differential Equation I: Nontiff Problem. Springer. [6] Hairer, E. and Wanner, G. () Solving Ordinary Differential Equation II: Stiff and Differential Algebraic Problem. nd Edition, Springer. [7] Henrici, P. (96) Dicrete Variable Method in Ordinary Differential Equation. Wiley. [8] Ierle, A. (97) A Firt Coure in the Analyi of Differential Equation. Wiley. [9] Lambert, J. (97) Computational Method in Ordinary Differential Equation. Wiley. [] Aluthge, A. (98) Filtering and Extrapolation Technique in the Numerical Solution of Ordinary Differential Equation. Mater Thei, Univerity of Ottawa, Ottawa. [] Boyd, J.P. () Chebyhev and Fourier Spectral Method. nd Edition, Dover Publication, Inc., New York. [] Canuto, C., Huaini, M., Quarteroni, A. and Zang, T. (988) Spectral Method in Fluid Dynamic. Springer. [] Weideman, J.A.C. and Reddy, S. () A MATLAB Differentiation Matrix Suite. ACM Tranaction on Mathematical Software, 6,

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