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A linear focusing mechanism for dispersive and non-dispersive wave problems

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A linear focusing mechanism for dispersive and non-dispersive wave problems
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  A linear focusing mechanism for dispersive and non-dispersivewave problems Yogesh G. Bhumkar a , Manoj K. Rajpoot b , Tapan K. Sengupta a, ⇑ a Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India b Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India a r t i c l e i n f o  Article history: Received 1 August 2010Received in revised form 3 November 2010Accepted 17 November 2010Available online 27 November 2010 Keywords: Fourier–Laplace spectral theoryFocusing phenomenonDiscretization errorDispersion errorSignal and error propagation dynamicsLinearized rotating shallow water equation(LRSWE) a b s t r a c t A linear focusing mechanism for a wave-packet propagation in a non-periodic domain isexplained here using Fourier–Laplace spectral theory. The global analysis method is usedhere to obtain numerical properties at each node of the full domain. In this work, we showthe spectacular growth of an error-packet at a particular node and wavenumber, a phe-nomenon termed as focusing which depends on: (a) CFL number,  N  c   and (b) numericalmethod used for discretization. In this study, we have shown the focusing phenomenonfor the numerical solution of the 1D convection equation and linearized rotating shallowwater equation.   2010 Elsevier Inc. All rights reserved. 1. Introduction In many disciplines of engineering and applied mathematical physics, numerical solution of wave problems are oftensought. Thus, it is necessary to obtain numerical properties of the numerical methods used for discretization in simulatingthe wave phenomena. The wave problems are generic and can arise as solution of hyperbolic partial differential equation(PDE)orasthedispersivesolutionsofanyothertypesofPDEs.Infact,therecanbewavesgeneratedforagoverningequationwhichdoesnothavetimeappearingexplicitly,whilethedispersionarisesduetotime-dependentboundaryconditions,suchasthecaseofsurfacegravitywavesgovernedbyLaplaceequation.Theseaspectsofdispersiveandnon-dispersivewavesolu-tions have been variously described and discussed in [1–5].ClassicalmethodoferroranalysisduetovonNeumannhasbeenfollowedin[1,6,7]andotherreferences,whiledescribingnumerical methods for wave problems. The central assumption in this analysis is that the error and the signal follow thesame discrete linear equation. Propagation of the energy of the system is governed by group velocity which is variously de-scribedin[8,9].UseofgroupvelocitytoaccountforthedispersionerrorincomputationwasintroducedbyTrefethen[4]and Vichnevetsky and Bowles [5] with the help of Fourier–Laplace transform of the discretized equation. However despite theadoption of von Neumann’s theory in computing alongside Fourier analysis, there were many situations which remainedunexplainedthatpromptedZingg[10]tocommentthat‘‘throughFourieranalysis,onecanevaluatethephaseandamplitudeerror of a given method as a function of the wavenumber. However, this information can be difficult to interpret’’. 0021-9991/$ - see front matter   2010 Elsevier Inc. All rights reserved.doi:10.1016/j.jcp.2010.11.026 ⇑ Corresponding author. Tel.: +91 512 2597945; fax: +91 512 2597561. E-mail address:  tksen@iitk.ac.in (T.K. Sengupta). Journal of Computational Physics 230 (2011) 1652–1675 Contents lists available at ScienceDirect  Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp  A new global spectral analysis tool was proposed in [11–13] to explain the simultaneous effects of one-sided boundaryand near-boundary nodes. This allowedfull domainanalysis without any assumptions regarding boundarynodes whichhasbeen the drawback of previous analysis technique, apart from the shortcomings of the von Neumann analysis [11]. In theseworks,theauthorsobtainedthecorrectnumerical dispersionrelationwiththewavenumber( k )astheindependentvariable.For the analysis of 1D convection equation @  u @  t   þ  c  @  u @   x  ¼  0 ;  c   >  0 ;  ð 1 Þ the numerical dispersion relationis given by  x N   ¼  c  N  k , as opposed to the physical dispersion relation x  ¼  kc  . Here,  c  N   is thewavenumber-dependent numerical phase speed to be calculated from the discrete equation, the details of which aredescribed in detail in [11,13]. The error propagation equation given in [11] clearly identifies the sources of error that con- taminatesthenumericalsolutionof Eq. (1). Theveryfact that onecannotevensolveEq. (1)keepingthephasespeed( c  ) con-stant, shows dispersion as a major source of numerical error.Variation of numerical properties near the boundaries are clearly indicated for high accuracy, explicit and implicit meth-ods by the global matrix analysis of  [12]. Problems associated with the boundary closure have attracted attention of manyresearchers. For example, Trefethen [4] tried to explain numerical instabilities near the boundary in terms of GKS stabilitytheory, as described in the same reference. However, this did not explain the problemunambiguously, as often instability isnoted not exactly at the boundary nodes – but in its vicinity.Additionally,wheninstabilityisnotedneartheboundaries,itisseentobeviolentlyunstableascomparedtoeventsintheinteriorofthedomain.Thisobservationledtootherstudiesthattriedtoexplainthisintermsofnonlinearmechanisms,asin[14–17].Briggsetal.[14]proposedanonlinearmechanismbywhichtheerrorgetsfocusedatonepointinthecomputational domain with respect to a nonlinear partial differential equation. This was for a periodic nonlinear problem that was quasi-linearized and a three time level leap-frog method was used for time advancement, along with second order central differ-encescheme.Duetotheuseofthreetimelevelmethodforthedifferentialequationwithfirstderivativeoftime,oneobtainsa spurious computational mode, in addition to the physical mode. In these results, role of spurious mode was found to becentral for this nonlinear instability. In using the leap-frog method of time advancing, Sloan & Mitchell [17] highlightedthe Fourier side-bandinstability in the context of envelope modulation. The same time discretization method was also usedin [15], in discussing a nonlinear instability problem.In the present work, a detailed investigation is made with the help of full-domain analysis [12] to investigate the nodalproperties of the points and relate the noted violent instability with a linear mechanism. In view of this, we will provide agraphicdemonstrationoferrorgettingfocusedataparticularnodeandaparticularwavenumberforanon-periodicproblem.This focusingmechanismis shownfor solutionof 1Dconvectionequationandlinearizedrotatingshallowwater wave equa-tion. Moreover, only two-time level method is adopted in numerical computations, so that no computational mode comesinto play, unlike the case described in the previous paragraph.This paper is formatted in the following manner. In the next section, we describe the full-domain analysis methodbriefly, which helps in obtaining the major properties of a numerical method for different nodes. In Section 3, we analyzethe numerical results of Eq. (1) for the propagation of wave-packets. This analysis exactly pinpoints the focusing of errorthat can only be explained with the full-domain analysis method [12]. In addition to it, the focusing mechanism is alsoshown for two-dimensional linearized rotating shallow water wave equation in Section 4 along with the numerical prop-erties of the methods used to obtain numerical solution. In Section 5, we summarize the results and provide someconclusions. 2. Fourier–Laplace spectral theory for non-periodic problems In many physical problems, one must maintain symmetry of the numerical schemes. All compact schemes, includingoptimal upwind compact scheme  ð OUCS  3 Þ  and other high accuracy schemes [12,13] display strong directionality of algo-rithms. We will confine our discussion on  OUCS  3 scheme, as it has been found to be one of the better optimal high accu-racy compact scheme which has been used in many applications for solving Navier–Stokes equation. However, theunphysical bias of compact schemes was found to be a major impediment in solving Navier–Stokes equation in a channelflow [18]. It was noted that even obtaining the undisturbed flow was impossible for the channel due to the bias. This biaswas removed using the symmetrization approach in [18] and the symmetrized optimal upwind compact scheme  ð SOUCS  3 Þ was shown to be adequate for the problem. Such symmetrization can be performed by the general methodology explainedin [18] and such schemes are always preferable as compared to general schemes. This has been used in the presentanalysis.Basic numerical properties of the high accuracy  SOUCS  3 of  [18] to solve Eq. (1) are explained again for the ease of under- standingtheanalysis forthe studiedfocusingphenomena. The stencil usedfor  SOUCS  3is thesame as usedin OUCS  3schemewith the directional bias present in obtaining first derivative in the latter scheme removed by symmetrization.Consider the following initial condition for the numerical solution of Eq. (1) u 0 m  ¼  u ð  x m ; t   ¼  0 Þ ¼ Z   A 0 ð k Þ e ikx m dk :  ð 2 Þ Y.G. Bhumkar et al./Journal of Computational Physics 230 (2011) 1652–1675  1653  Substituting Eq. (2) in Eq. (1), one obtains the general numerical solution as u nm  ¼  u ð  x m ; t  n Þ ¼ Z   A 0 ð k Þð G 2 r   þ  G 2 i  Þ n = 2 e i ð kx m  n b Þ dk ;  ð 3 Þ wherethenumericalamplificationfactoris G ð k Þ ¼  G r   þ  iG i  andtan b  ¼  G i = G r  .Fromtheknowledgeofthesignsfor G r   and G i ,one can exactly identify the quadrant for calculating  b . The solution is also represented by  u nm  ¼ R   U  ð k ; t  Þ e ikx m dk  and theamplification factor is therefore defined as,  G ð k Þ ¼  U  ð k ; t  n þ 1 Þ U  ð k ; t  n Þ  .FromEqs. (2) and (3), it is apparent that the numerical dispersionrelation fixes numerical phase speed which is differentfrom the physical phase speed. In the above,  b  is a function of   k  and is related to the numerical phase speed (as shown in[13,19]) c  N  ð k Þ c   ¼  b x D t  :  ð 4 Þ The numerical group velocity, using numerical dispersion relation, is given by ([13,19]): V   gN  ð k Þ c   ¼  1 N  c  D  xd b dk ;  ð 5 Þ where  N  c   denotes the CFL number and  D  x  is mesh spacing.If   e  ¼  u    u N  , denotes the error, then the equation for error dynamics [11] is given by: @  e @  t   þ  c  @  e @   x  ¼  c   1   c  N  c  h i @  u N  @   x   Z   V   gN     c  N  k   Z   ik 0  A 0 ½j G j t  = D t  e ik 0 ð  x  c  N  t  Þ dk 0   dk   Z   Ln j G j D t  A 0 ½j G j t  = D t  e ik ð  x  c  N  t  Þ dk :  ð 6 Þ Equation(6)iscentraltothedevelopedDRPschemein[20]forthe1Dconvectionequationandalsopresentingtheresults for the numerical properties here. Present equation assembles the error based on generic properties of any numericalscheme. The first term on the right hand side of this equation is due to phase mismatch given by  ð 1   c  N  = c  Þ  and this errorbecomes higher for solutions with sharp discontinuities, as in the propagation of fronts or compressible flows with shocks.Thesecondtermontherighthandsideisduetonumericaldispersion,measuredbytheratio V   gN  = c  ,whilethelasttermisdueto numerical attenuation or amplification and will be absent for a neutrally stable method. Ideally for a chosen numericalmethod,onetherefore,requiresneutralstability ðj G j ¼  1 Þ  andzerophaseanddispersionerror[11]. Thenumericalproperties j G ð k Þj ; c  N  ð k Þ  and  V   gN  ð k Þ  are adequate to describe any numerical method, as has been explained in [11,19].In [3,12,13], a matrix spectral analysis has been introduced to account for nodal properties in non-periodic problems.Keeping in view of the analysis of Eq. (1), the first derivative  u 0 ¼  @  u @   x  is written as an explicit relation f u 0 g ¼  1 D  x ½ C  f u g ;  ð 7 Þ where  ½ C    denotes a matrix, with  f u g  denoting a column vector.The numerical first derivative at  x  j  is written in the spectral plane as,  u 0 ð  x  j ; t  Þ ¼ R  k max k min ik eq U  ð k ; t  Þ e ikx  j dk , so that ik eq ð  x  j Þ ¼  1 D  x X N l ¼ 1 C  lj e ik ð  x l   x  j Þ ;  ð 8 Þ where  N   represents number of grid points.It is important torealizethat for Fourier spectral methodsthemaximumandminimumresolvedwavenumbers aregivenby the Nyquist limit as,  k max  ¼  k min  ¼  p = D  x  – as explained in [3]. For a particular choice of grid spacing, this limit tells oneabout the maximumrange of wavenumbers that could be resolved by a Fourier spectral method. Computing in the physicalspace,  k eq  is a complex quantity, with the real part signifying the phase representation and the negative imaginary part pro-vidingtheaddednumericaldissipation. Anumericalscheme,that hasapositiveimaginarypartof   k eq  becomeslocallyunsta-ble, because that is equivalent to adding  anti-diffusion , as shown in [12] for some well-known compact schemes for theirproperties near the boundaries. This observation is crucial for the present study.For the  OUCS  3 scheme, following stencil is used for the evaluation of first derivative for interior points a  j  1 u 0  j  1  þ a  j u 0  j  þ a  j þ 1 u 0  j þ 1  ¼  1 D  x X k ¼ 2 k ¼ 2 q k u  j þ k :  ð 9 Þ In developing the  OUCS  3 scheme for non-periodic problems, one is forced to use one-sided boundary stencils at  j  ¼  1 ; 2 and N    1 ; N  . This type of boundary closure can violate the physical principle of information propagation. In developing highorder compact schemes, various researchers have used implicit boundary closure schemes, as referred to in [3,12]. Suchimplicitschemesduetotheirinherentbias,actuallyaccentuatetheabovementionedproblemofviolatingphysicalprinciple.Thiswasclearlydemonstratedin[3,12] formanyearliercompactschemeswhichleadstonumericalinstability, byusingthenewly developed spectral analysis tool. In these references, having diagnosed the problem of numerical instability for thevarious proposed schemes due to implicit boundary closure, a new set of upwind schemes were introduced with explicit 1654  Y.G. Bhumkar et al./Journal of Computational Physics 230 (2011) 1652–1675  boundaryclosureschemes.Onesuchschemewasthe OUCS  3schemewhichusedthefollowingboundaryclosureschemesfor  j  ¼  1 and  j  ¼  2 u 0 1  ¼  12 D  x ð 3 u 1  þ 4 u 2    u 3 Þ ;  ð 10 Þ u 0 2  ¼  1 D  x 2 c 2 3   13   u 1    8 c 2 3  þ 12   u 2  þ ð 4 c 2  þ 1 Þ u 3    8 c 2 3  þ 16   u 4  þ 2 c 2 3  u 5   ;  ð 11 Þ where  c 2  is an user specified parameter used to achieve better numerical properties of neutral stability, lesser phase anddispersion errors. A value of   c 2  ¼  0 : 025 was found to be an optimum for the solution of convection dominated problems.Similarly, one can write down boundary closure schemes for  j  ¼  N   and  j  ¼  N    1 using another parameter,  c N   1  ¼  0 : 09, u 0 N   ¼  12 D  x ð 3 u N    4 u N   1  þ  u N   2 Þ ;  ð 12 Þ u 0 N   1  ¼  1 D  x 2 c N   1 3   13   u N     8 c N   1 3  þ 12   u N   1  þ ð 4 c N   1  þ 1 Þ u N   2    8 c N   1 3  þ 16   u N   3  þ 2 c N   1 3  u N   4    ð 13 Þ Fig. 1.  Comparison of   ðj G jÞ ; ð V   gN  = c  Þ  and  ð 1   c  N  = c  Þ  contours for the near-boundary nodes  j  ¼  2 ; 100 (left column) and central node  j  ¼  51 (right column),using  RK  4    SOUCS  3 scheme for the solution of Eq. (1). The line at  N  c   ¼  2 corresponds to the cases computed and shown in Fig. 4. Y.G. Bhumkar et al./Journal of Computational Physics 230 (2011) 1652–1675  1655  The coefficients in Eq. (9) are upwinded to achieve numerical stability and are given in [3,12];  a  j  1  ¼  0 : 3793894912  g = 60 ; a  j  ¼  1 ; q  2  ¼  0 : 183205192 = 4 þ g = 300 ; q  1  ¼  1 : 57557379 = 2 þ g = 30 and  q 0  ¼  11 g = 150, where  g  ¼  2 : 0 is the upwindparameter for the interior stencil.Estimate of scale-wise amplification factor at the  j th node,  j G  j ð k Þj  is obtained, once the time integration method is fixed.For example, with Euler time integration, the amplification factor at the  j th node is obtained using Eq. (8) as G  j ð k D  x ; N  c  Þ ¼  1   N  c  X N l ¼ 1 C  lj  cos k ð  x l    x  j Þ þ  i sin k ð  x l    x  j Þ  " # :  ð 14 Þ Thisisanunconditionallyunstablemethod.Forthisreason,wewillusefour-stageRunge–Kutta ð RK  4 Þ  timeintegrationstrat-egy. Amplification factor for the  j th node for this combination is given by [11,18]: G  j ð k D  x ; N  c  Þ ¼  1   A  j  þ  A 2  j 2    A 3  j 6  þ  A 4  j 24 ;  ð 15 Þ where,  A  j  ¼  N  c  X N l ¼ 1 C   jl e ik ð  x l   x  j Þ : Fig. 2.  Comparison of   ðj G jÞ ; ð V   gN  = c  Þ  and  ð 1   c  N  = c  Þ  contours for the near-boundary nodes  j  ¼  4 ; 98 (left column) and nodes  j  ¼  6 ; 96 (right column), using RK  4    SOUCS  3 scheme for the solution of Eq. (1). The line at  N  c   ¼  2 corresponds to the cases computed and shown in Fig. 4.1656  Y.G. Bhumkar et al./Journal of Computational Physics 230 (2011) 1652–1675
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