A-posteriori Diffusion Analysis of Numerical Schemes in Wavenumber Domain

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1 2th Annual CFD Symposium, August 9-1, 218, Bangalore A-posteriori Diffusion Analysis of Numerical Schemes in Wavenumber Domain S. M. Joshi & A. Chatterjee Department of Aerospace Engineering Indian Institute of Technology, Bombay Mumbai 47, India July 12, 218 Abstract: A new technique for a posteriori diffusion analysis of numerical schemes is proposed. In this technique, the modal and total energy of a broadband signal in wavenumber domain is analyzed to understand diffusion characteristics of a candidate numerical scheme. The analysis reveals information such as the effect of spatial and temporal orders, reconstruction method and time step size on numerical diffusion. In addition to this, an onset of numerical instability can be recognized at an early stage through this analysis. The analysis is simple and inexpensive to perform and can be used to analyze linear and nonlinear including space-time coupled numerical schemes. Several popular numerical schemes are analyzed using this technique. The technique is particularly useful for numerical schemes used for simulation of transient linear waves. Keywords: FFT, PSD, Higher-order, Signal energy, Numerical diffusion 1 Introduction Higher-order accurate numerical schemes are routinely used for applications involving traveling linear waves such as computational aeroacoustics (CAA) and computational electromagnetics (CEM). In this context, higher-order indicates order of accuracy greater than two. Such schemes are known to introduce less dispersion and dissipation errors in the solution in addition to having a more relaxed points-per-wavelength (PPW) requirement than conventional first or second-order accurate schemes [1, 2]. However, formal order of accuracy may not be a sufficient metric for judging quality of a numerical scheme. A more thorough understanding of the error characteristics of a numerical scheme is possible through error analysis in wavenumber domain. For example, optimized schemes often have a lower formal order of accuracy in space compared to conventional schemes using a similar stencil, yet introduce a lower amount of dispersion error in the solution [2]. Similarly, nonlinear numerical schemes such as t he weighted essentially non oscillatory (WENO) scheme may show differential order of accuracy for different wavenumber components [4]. Thus, a thorough analysis of errors in wavenumber space is important in addition to the conventional order of accuracy analysis for modern numerical schemes used for simulation of traveling linear waves. For simple linear schemes, such an analysis showing diffusion and dispersion error characteristics is easily possible in a priori manner. However, an smjoshi@aero.iitb.ac.in 1

2 a priori estimate of dispersion and diffusion errors for nonlinear schemes or space-time coupled schemes may not be possible. Nonlinear in this context indicates a data-driven numerical scheme. In this paper, we propose a novel technique for diffusion analysis of numerical schemes in wavenumber domain in a posteriori fashion. Wavenumber representation of the signal is obtained using the discrete Fourier transform (DFT) technique. Diffusion characteristics are found out from modal and total energy of the signal in the wavenumber domain. In addition to this, variation of the maximum resolved wavenumber over simulation time is found out. The technique described in this paper is useful for analysis of linear as well as nonlinear including space-time coupled numerical schemes. The technique also helps identify onset of a numerical instability at an early stage. Several modern higher-order accurate numerical schemes are analyzed using this technique. 2 A-posteriori Diffusion Analysis: Detailed Procedure For this analysis, consider the 1D scalar advection equation, u t + au x =, a > (1) over domain x [x l,x u ] with periodic boundary conditions and u(x,) = u (x) as the initial condition. Let the solution u(x,t) be sampled at N discrete equidistant points (x,x 1,...,x N 1 ) such that at time level t n, the value at j th point is given as u n j. The diffusion analysis is performed as follows 1. Grid spacing in spatial and wavenumber domain is arbitrary selected. 2. Based on this, the minimum resolved wavenumber k min, total number of cells N and the domain boundaries x l and x u are found out. 3. On a 1D domain x [x l,x u ], define broadband initial conditions given by a Gaussian signal u(x) = e ( 8x αl ) 2 (2) The parameter α controls the spread of the Gaussian curve thereby controlling energy distribution in the higher Fourier modes as shown in Figure 1a. 4. Solve equation 1 numerically for the final time T using a candidate numerical scheme. 5. Take DFT of both the analytical and the numerical results using fast Fourier transform. The DFT of function u(x) is defined as, û (m) = 1 N 1 N u(x m )e ik mm (3) m=. Obtain power spectral density (PSD) graph of the solution along with variation of total energy of the signal over time. The total energy of a discrete signal u(x) : x {x,x 1,...,x N 1 } is defined as N 1 E = (u(x i )) 2 = 1 N 1 i= N k= (û(k)) 2 (4) 7. Similarly, obtain maximum wavenumber corresponding to a well resolved mode and its variation with time 2

3 1. Gaussian Pulse α =1 α = Power Spectrum.8 α = 1 4 α = u ˆf(k) 2 ˆf(k) 2 α =1/ α =1 α = α = 1 4 α = x (a) Space domain (b) Normalized in frequency domain Figure 1: Gaussian signal with different values of α in space and frequency domains It is well known that the zeroth Fourier mode of a signal indicates average value of the signal. Thus, if the zeroth Fourier mode of the signal varies over time, it can be concluded that the candidate numerical scheme is not conservative in nature. On the other hand, if energy corresponding to the zeroth Fourier mode remains unchanged, then the candidate numerical scheme proves to be conservative in nature. Higher gradients in the space domain are characterized by presence of more energy in higher wavenumber modes. A diffusive numerical scheme results in smoothing of the signal, which essentially means reduction in energy corresponding to higher wavenumber modes. Thus, the PSD curve corresponding to the numerical solution found out using a diffusive numerical scheme would agree with the analytical solution only in the lower wavenumber range and will likely deviate at some point indicating addition of numerical diffusion in modes corresponding to higher wavenumbers. If the scheme is conservative in nature, the zeroth wavenumber continues to posses the same amount of energy as that corresponding to the initial conditions. Fig. 1b shows all spectra normalized with respect to the spectrum for α = 1. An important observation from fig.1b is that, smoother signals (larger values of α) have more energy in the lower wavenumbers whereas signals with a sharper slope have energy distributed uniformly throughout the wavenumber range. Signs of numerical instability can be recognized at an early point through the Fourier signature of the numerical scheme. The modal as well as total energy of the signal goes on increasing over time if the candidate numerical scheme is unstable. However, an increase in energy of higher wavenumber modes doesn t always indicate an instability. For example, nonlinear schemes such as the essentially non oscillatory (ENO) and WENO schemes are known to show spurious modes as well as numerical turbulence [, 7]. Thus, time evolution of total energy of the signal is studied to identify the numerical instability. Similarly, numerical artifacts such as the Gibb s phenomenon does not indicate presence of a numerical instability. This is also confirmed by tests conducted on a discontinuous signal. In the subsequent sections, we study effect of spatial order of accuracy on numerical diffusion, effect of spatial order of accuracy on total energy of smooth as well as discontinuous signals, identification of a numerical instability and variation of maximum resolved wavenumber over simulation time. For this analysis, we consider the Arbitrary DERivatives (ADER), essentially non oscillatory (ENO), weighted essentially non oscillatory (WENO), fixed stencil (FS) and discontinuous Galerkin (DG) schemes. 3

4 3 Results 3.1 Effect of Spatial Order on Diffusion O(1) O(2) O(3) O(4) O(1) O(2) DG, RK3, RK3 O(4) DG, RK k x (a) Fixed-Stencil scheme (b) Discontinuous Galerkin method Figure 2: Normalized PSD for FS and discontinuous Galerkin methods O(2) FIXED RK2 O(2) ADER-FIXED O(3) FIXED RK3 O(3) ADER-FIXED O(4) FIXED RK4 O(4) ADER-FIXED O(2) ENO RK2 O(2) ADER-ENO O(3) ENO RK3 O(3) ADER-ENO O(4) ENO RK4 O(4) ADER-ENO 5 (a) Fixed-stencil reconstruction (b) ENO reconstruction Figure 3: Comparison of ADER schemes with semidiscrete schemes Fig. 2a shows comparison of power spectra using fixed-stencil finite volume schemes of different spatial orders. Fig.2 shows results for the discontinuous Galerkin scheme. In both the cases, it is observed that energy associated with the zeroth Fourier mode remains unchanged when spatial order of accuracy is changed. As explained earlier, 4

5 the zeroth mode indicates average value of the signal. This confirms that both FS and DG schemes are conservative in nature. Higher-order reconstruction shows more energy in the higher Fourier modes, whereas lower-order variants show addition of numerical diffusion in the higher modes. Figs.3a and 3b show PSD graphs for ADER-FS, ADER-ENO and equivalent semidiscrete numerical schemes. ENO scheme, which is a nonlinear scheme, shows presence of spurious modes in the wavenumber domain. It is noted that the time step is kept smaller than that dictated by the Courant Friedrich Lewy (CFL) condition, thus ensuring stability for all numerical schemes. ENO scheme uses adaptive stencils for data reconstruction. In the case of both FS and ENO schemes, the ADER counterparts are found to be less diffusive. Addition of energy in higher wavenumbers in case of semidiscrete ENO and ADER-ENO schemes does not necessarily indicate a numerical instability. This is explained from evolution of total energy of the signal over simulation time. 3.2 Effect of Spatial Order on Total Energy % Energy 4 % Energy 4 2 O(2) ADER O(2) DG O(2) ENO O(2) FS Simulation time (%) (a) Second-order 2 O(3) ADER O(3) ENO O(3) FS O(3) WENO Simulation time (%) (b) Third-order Figure 4: Effect of spatial order on total energy of the signal Figs.4a and 4b show signal energy over time for various numerical schemes. Identical Courant number ensuring stability as well as identical number of degrees of freedom (DOFs) is used for all numerical schemes. It is seen that even if the numerical schemes have same spatial order of accuracy, they show differences in ability to preserve signal energy over simulation time. Among all the candidate numerical schemes, DG scheme is found to best preserve the signal energy over time. Nonlinear schemes such as the ENO and WENO schemes result in maximum loss of total energy over the course of simulation. This also indicates that, although spurious modes are present in the high wavenumber range for nonlinear numerical schemes, they are in fact diffusive in nature. 4 Wave Resolution It is found that the numerical PSD curve deviates from the analytical curve at a particular point. This wavenumber (called as the threshold wavenumber) changes over time. Wavenumbers lower than the threshold wavenumber can be considered to be well resolved. Diffusion first starts affecting higher wavenumbers and progressively shifts towards the lower wavenumber range. The threshold wavenumber is defined as the wavenumber at which modal energy of the 5

6 numerical solution is lower than 95% of that of the analytical solution as shown in Fig.5. The threshold wavenumber is different for different numerical schemes even though they show identical order of spatial accuracy. 1 2, RK Energy numerical Energy analytical <.95 Threshold wavenumber Figure 5: Maximum resolved wavenumber for a third-order DG scheme 5 O(3) ADER O(3) ENO O(3) FS O(3) WENO 5 O(4) ADER O(4) DG O(4) ENO O(4) FS Simulation time (%) (a) Third-order Simulation time (%) (b) Fourth-order Figure : Threshold wavenumber () over simulation time for different numerical schemes Fig.a and Fig.b show threshold wavenumber as a function of time for different numerical schemes. It is seen that the DG schemes show highest threshold wavenumber for the length of the simulation. Nonlinear schemes such

7 as ENO and WENO schemes introduce dissipation in lower wavenumbers earlier and thus show a lower threshold wavenumber than other schemes. This is analogous to the plot of variation of total energy over simulation time. ADER schemes show similar threshold wavenumbers as their semidiscrete counterparts and therefore may not have significant advantages over them. 5 Onset of Instability CFL=.99 CFL=1. CFL= CFL=1. CFL=1.1 CFL=.99 1 % Energy Simulation time (%) (a) Power spectral density (b) Evolution of total energy Figure 7: First-order upwind FV scheme Fig.7a shows energy spectrum and Fig.7b shows time-evolution of total signal energy for a first-order upwind scheme with forward Euler time-stepping at different Courant numbers. The instability is characterized by increase in the modal as well as the total energy of the signal over time. For Courant number 1, there is no dissipation added to the solution as well as no presence of an instability detected. For Courant number less than 1, numerical dissipation is seen to get added to higher wavenumbers. Discontinuous Signal The diffusion analysis technique is verified with discontinuous initial conditions. A square pulse centered at zero is taken as the initial conditions. The initial conditions are defined as, u(x,) = { 1 if x <.4 if x >.4 ; x [ 1,1] (5) Analysis as described earlier is performed on this signal. Effects of spatial orders and reconstruction method on advection of the square pulse are observed. The results are shown in Fig.8. Figures 8a, 8c and 8e show effect of spatial order on the signal in space domain, frequency domain and total energy respectively. Figures 8b, 8d and 8f show effect of reconstruction method on the square pulse in time domain, frequency domain and total energy respectively. Higher-order schemes show less diffusion added in the Fourier modes. In addition, higher-order schemes show better preservation of signal energy over time. 7

8 1.8 Square pulse, T=2 units O(2) DG O(4) DG 1.8 Square pulse, T=2 units O(3) FIXED O(3) ENO O(3) WENO O(3) ADER.. u u x x 1 4 (a) Effect of order, space domain 1 4 (b) Effect of reconstruction, space domain O(2) DG O(4) DG O(3) FIXED O(3) ENO O(3) WENO O(3) ADER (c) Effect of order, frequency domain O(2) DG O(4) DG (d) Effect of reconstruction, frequency domain 1 % Energy 99 % Energy Simulation time (S) (e) Effect of order, energy of the signal 9 O(3) FIXED O(3) ENO O(3) WENO O(3) ADER 1 2 Simulation time (S) (f) Effect of reconstruction, energy of the signal Figure 8: Effect of spatial order and reconstruction method on advection of a discontinuous signal 8

9 Linear schemes such as the DG, ADER and FS schemes show Gibbs oscillations in case of the discontinuous signal as expected. These oscillations do not indicate a numerical instability. This is also confirmed from the energy plot. The total energy of the signal reduces over time indicating addition of the diffusion in the signal. Nonlinear schemes such as the ENO and WENO schemes do not show oscillations. Moreover, these schemes show higher reduction in the total energy of the signal over time, indicating addition of higher amount of diffusion. The fixed stencil (FS) reconstruction scheme uses an upwind biased reconstruction stencil. The ADER scheme also uses FS reconstruction method. Both of these schemes show large amount of diffusion in the higher modes of the Fourier spectrum but show less numerical diffusion in energy-rich lower wavenumbers. 7 Conclusion A new technique for qualitative analysis of diffusion characteristics of numerical schemes is presented. The frequency representation of the numerical solution is obtained using the DFT technique. Numerical schemes are compared based on modal energy content and time evolution of total energy of the broadband signal. The analysis is applicable to semidiscrete as well as space-time coupled linear and non linear schemes used for simulating propagating linear waves. The analysis yields important information on properties of a numerical scheme like conservation, ability to preserve signal energy and overall diffusion characteristics. Among candidate numerical schemes, the DG scheme is found to best preserve signal energy over simulation time. Nonlinear schemes such as ENO and WENO schemes result in maximum amount of numerical dissipation in the solution. ADER scheme is found to be slightly less dissipative than its semidiscrete counterpart with same order of accuracy. References [1] C. K. W. Tam, Computational Aeroacoustics-Issues and Methods, AIAA J., vol. 33, no. 1, pp , 212. [2] D. W. Zingg, Comparison of High-Accuracy Finite-Difference Methods for Linear Wave Propagation, SIAM J. Sci. Comput., vol. 22, no. 2, pp , 2. [3] C. K. W. Tam and J. C. Webb, Dispersion-Relation-Preserving Finite Difference Schemes for Computational Acoustics, J. Comput. Phys., vol. 17, pp , [4] D. Fauconnier and E. Dick, On the spectral and conservation properties of nonlinear discretization operators, J. Comput. Phys., vol. 23, no. 12, pp , 211. [5] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics, Cambridge, New York: Cambridge University Press, 22. [] F. Jia, Z. Gao, and W. S. Don, A Spectral Study on the Dissipation and Dispersion of the WENO Schemes, J. Sci. Comput., vol. 3, no. 1, pp , 215. [7] F. Ladeinde, X. Cai, M. R. Visbal, and D. V. Gaitonde, Turbulence spectra characteristics of high order schemes for direct and large eddy simulation, Appl. Numer. Math., vol. 3, no. 4, pp , 21. [8] C. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., vol. 77, pp , [9] G. Jiang and C. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., vol. 12, no. 1, pp , [1] B. Cockburn and C. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., vol. 35, no., pp , [11] B. Cockburn, S. Lin, and C. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, J. Comput. Phys., vol. 84, pp , sep

10 [12] B. Cockburn and C. Shu, TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws II: General Framework, Math. Comp., vol. 52, p. 411, apr [13] V. A. Titarev and E. F. Toro, ADER: Arbitrary High Order Godunov Approach, J. Sci. Comput., vol. 17, no. 1-4, pp. 9 18, 22. [14] T. Schwartzkopff, C. D. Munz, and E. F. Toro, ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D, J. Sci. Comput., vol. 17, no. 1-4, pp , 22. 1

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