Uniformly best wavenumber approximations by spatial central difference operators: An initial investigation

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1 Uniformly best wavenumber aroximations by satial central difference oerators: An initial investigation Vitor Linders and Jan Nordström Abstract A characterisation theorem for best uniform wavenumber aroximations by central difference schemes is resented. A central difference stencil is derived based on the theorem and is comared with disersion relation reserving schemes and with classical central differences for a relevant test roblem. 1 Introduction Modelling wave roagation over sizeable intervals using finite differences is a common roblem in fields ranging from aeroacoustics to seismology. For high frequency roblems the numerical error may over time be dominated by inaccurate aroximations of the disersion relation, leading to errors in hase and grou velocity, unless the satial increment, x is very small. A remedy, resented in [1] for central differences, is to erturb the classical schemes by an extra arameter but decreasing the formal accuracy. The new arameter is used to minimise the disersion error in the L 2 [,π/2] norm. Such schemes are nown as Disersion Relation Preserving (DRP). For other aroaches based on similar ideas, see e.g. [2, 3, 4]. The DRP aroach is disadvantageous in that it rovides no means of obtaining wavenumber-secific error bounds. For roblems involving a range of wavenumbers it is more convenient to minimise the disersion error in the L -norm. In this aer we resent a characterisation theorem for best uniform wavenumber aroxima- Vitor Linders Linöing University, Deartment of Mathematics, Comutational Mathematics, SE Linöing, Sweden, vitor.linders@liu.se Jan Nordström Linöing University, Deartment of Mathematics, Comutational Mathematics, SE Linöing, Sweden, jan.nordstrom@liu.se 1

2 2 Vitor Linders and Jan Nordström tions. New central difference schemes are derived and comared with their classical and DRP counterarts. 2 Central difference schemes We begin by demonstrating why classical central differences are subotimal for wavenumber aroximation. Consider a central difference stencil of order 2, (u x ) j = 1 x =1 (u j+ u j ) + O( x 2 ). The numerical wavenumber of this scheme is (see e.g. [1]) ξ c = 2 =1 sin(ξ ) (1) where ξ = xκ and κ is the exact wavenumber of the roagating solution. Here we let ξ [,ξ max ] [,π]. A Taylor exansion reveals that to obtain desired accuracy, must satisfy The following observation is useful. For the roof, see [5]. Lemma 1. Consider the function f (x) = 1 2 =1 2 =.. (2) T (x) where satisfies (2) and T (x) is the th order Chebyshev olynomial of the first ind, uniquely defined through the relation T (cos(φ)) = cos(φ). Then f (x) = d (1 x) for some d that deends exclusively on. Theorem 1. Classical central difference stencils underestimate the seed of roagating solutions. Proof. Let E c (ξ ) ξ ξ c = ξ 2 =1 sin(ξ )

3 Title Suressed Due to Excessive Length 3 be the error function associated with a classical central difference stencil of order O( x 2 ). Note from the definition of f and the Chebyshev olynomials that de c dξ = f (cos(ξ )). It follows that de c dξ = only when cos(ξ ) = 1. It is thus clear that E c(ξ ) has an inflection oint at ξ = and no other extrema in the domain of interest. Consequently E c (ξ ) is monotonic and since E c () = and E c (π) = π it is also increasing. We thus have ξ c ξ with equality only at ξ =. Therefore the classical central difference stencils underestimate the analytic wavenumber. It follows now that the relative error in the hase seed is v v v and so the hase seed is underestimated. = ξ c ξ 1 Let us now, lie for DRP schemes, erturb the stencil by adding an additional coefficient, a without increasing the accuracy. The coefficients of the new stencil must solve the system (2) for each a, though this system is now underdetermined. Calling the coefficients of the new system a, = 1,..., + 1, we must have a linear deendence of the first coefficients on the added arameter, a. We write a = + c a, = 1,...,. In view of (3) the numerical wavenumber of the erturbed stencil is ξ = 2 =1 Let us define the error function of this scheme as a sin(ξ ), ξ ξ max π. (3) E(ξ ) ξ ξ = ξ 2 a sin(ξ ). (4) Our goal is to choose a such as to minimise the magnitude of any extrema of E(ξ ). In order to do so we will extend Lemma 1 to the erturbed stencil. For a detailed roof, see [5]. Lemma 2. Let =1 g (x) = 1 2 a T (x) where a = + c a are the coefficients of the erturbed central difference stencil as defined reviously, and T (x) is the th order Chebushev olynomial. Then [ ( )] g (x) = (1 x) (1 x) d c a + d 1 a c =1

4 4 Vitor Linders and Jan Nordström where d and are defined as before. Corollary 1. E(ξ ) has at most one extremum in the oen interval (,ξ max ] and it is located at ξ = ξ r = arccos ( 1 d d ]) [1 c. a For good aroximations ξ r is a minimum. This occurs only when a and c have the same sign and a c, where equality holds only for classical stencils. Again, for the roof we refer to [5]. From the above corollary we conclude that we can have E(ξ ) = E only at two ossible oints, namely at ξ r or ξ max, i.e. E = max{ E(ξ max ), E(ξ r ) }. (5) Of course E(ξ r ) and E(ξ max ) deend on how we choose a and in view of (5) it is of interest to investigate this deendency. Our goal is to find the choice of a that minimises (5). In fact we have Theorem 2. Consider a oint central difference scheme of order O( x 2 ) with numerical wavenumber ξ defined as in (3), and a corresonding error function E(ξ ) = ξ ξ. The stencil that uniformly minimises the error function, i.e. min ξ E = min a R ξ ξ, is uniquely characterised by the roerty E(ξ r ) + E(ξ max ) =. (6) The roof is found in [5] where it is also demonstrated how this result generalises to an arbitrary number of free arameters, a,...,a +n. 3 A numerical examle For a given ξ max solving (6) is a simle matter. Even for general ξ max good estimates may be found by relacing E(ξ r ) by a suitable olynomial aroximation. For the case = 1 this is shown in Fig. 1 for ξ max [,π/2]. Here e = min ξ E. To illustrate the strength of Theorem 2 we consider a rofoundly olychromatic solution to the advection equation over a eriodic domain: u t + u x =, x 3, t u(x,) = ex( 32(x 1/2) 2 ). This ulse is narrow and its Fourier transform is wide resulting in a significant contribution from a broad range of wavenumbers. The dominating wavenumber is

5 Title Suressed Due to Excessive Length Ξ max Fig. 1: Error of best uniform wavenumber aroximation for given ξ max [,π/2] (orange dots) and general solution using third degree olynomial aroximation (blac line). κ =. Contributions from larger wavenumbers decay exonentially but slowly. It maes sense to use a scheme that accurately aroximates the disersion relation near ξ = κ x = and for some suitably chosen region of larger wavenumbers. From Fig. 1 we see that if we choose ξ max = π/4.785 we will have an error in the disersion relation of around 1 3. Solving (6) gives the scheme a 1 = , a 2 = In Fig. 2 the disersion error of the new scheme is lotted as a function of ξ. The aroximation starts to deviate from the exact result for ξ ξ max. The errors of a fourth order classical central difference scheme and of a five-oint DRP scheme [1] are also included. As exected from Theorem 1 the classical stencil underestimates the disersion relation, seen by the ositive sign of the error. For the shown range of ξ it seems that the DRP scheme overestimates the disersion relation whereas our new scheme stays within tight error bounds. We set x = 1/12 and integrate in time using the classical fourth order Runge- Kutta scheme with time ste t = 1 3 so the contribution from the temoral discretisation is small. The exact and numerical solutions are shown in Fig. 3 (to) together with the error as a function of time (bottom). All numerical solutions quicly diserse into a train of ulses of decaying amlitude trailing behind the main ea. As exected from Fig. 2, the DRP scheme overestimates the seed of some ulses. Our new scheme does this as well but to a much reduced extent. The smaller ulse train behind the DRP solution with resect to the new scheme may be attributed to a better aroximation for very high wavenumbers. However, since the contribution of these wavenumbers are comarably small, the resulting error remains larger for the DRP scheme as comared with the classical and the new stencil.

6 6 Vitor Linders and Jan Nordström.1.5 New Classical DRP E = e 4 E(ξ) ξ Fig. 2: Disersion error for the new scheme, the classical 4 th order stencil, and a five-oint DRP scheme. 4 Extension to multile dimensions Extending the new stencil to multile dimensions is in rincile straight forward. As an examle, for the advection roblem in 2D we may, after discretising in sace, write v t + (D x I y )v + (I x D y )v = where v is a grid vector aroximating the true solution, D x,y are eriodic oerators containing the central difference stencil and oerating on a given cartesian grid in the x and y direction resectively. Here I x,y are identity matrices of aroriate dimensions and denotes the Kronecer roduct. For this situation, the solution roagates at an angle θ with resect to the x- axis. It should be noted that the stencils resented here are otimal for the onedimensional roblem, that is for the cases when θ = nπ/4, n =,1,2,3. For any other angle the stencils will be subotimal since the numerical disersion relation deends on the direction of roagation. In other words, these stencils may be sensitive to numerical anisotroy. For a comrehensive overview of methods that handles this issue, see e.g. [6]. At resent we shall not consider this henomenon further.

7 Title Suressed Due to Excessive Length New DRP Classical Exact 3 u(x, t) x.35.3 New DRP Classical Error develoment in time.25 Error Time Fig. 3: (To) Exact solution and numerical aroximations after 2 time stes. (Bottom) Corresonding errors. 5 Conclusion We have roved a characterisation theorem for best uniform wavenumber aroximations by central difference stencils with one free arameter. The best aroximation is unique and may be easily obtained numerically for a given range of wavenumbers. This allows for accurate aroximations of roblems of high frequency waves, or multi-frequency solutions, with a relatively coarse satial mesh.

8 8 Vitor Linders and Jan Nordström References 1. C.K.W. Tam, J.C. Web, Disersion-Relation-Preserving Finite Difference Schemes for Comutational Acoustics, J. Comut. Phys. 17, (1993) 2. D.W. Zing, H. Lomax, H. Jurgens, High-accuracy finite-difference schemes for linear wave roagation, SIAM J. Sci. Comute. 17, (1996) 3. D.W. Zingg, H. Lomax, H. Jurgens, An otimized finite-difference scheme for wave roagation roblems, AIAA aer 93(459) (1993) 4. C. Bogey, C. Bailly, A family of low disersive and low dissiative exlicit schemes for flow and noise comutations, J. Comute. Phys. 194, (24) 5. V. Linders, J. Nordström, Uniformly best wavenumber aroximations by satial central difference oerators, LiTH-MAT-R, 214:17, 214, Deartment of Mathematics, Linöing University 6. A. Sescu, Numerical anisotroy in finite differencing, Advances in Difference Equations 215:9 (215), Mississii State University