Numerical anisotropy in finite differencing
- Adrian Sescu^{1}Email author
https://doi.org/10.1186/s13662-014-0343-0
© Sescu; licensee Springer 2015
Received: 15 July 2014
Accepted: 25 December 2014
Published: 16 January 2015
Abstract
Numerical solutions to hyperbolic partial differential equations, involving wave propagations in one direction, are subject to several specific errors, such as numerical dispersion, dissipation or aliasing. In the multi-dimensional case, where the waves propagate in all directions, there is an additional specific error resulting from the discretization of spatial derivatives along the grid lines. Specifically, waves or wave packets in the multi-dimensional case propagate at different phase or group velocities, respectively, along different directions. A commonly used term for the aforementioned multi-dimensional discretization error is the numerical anisotropy or isotropy error. In this review, the numerical anisotropy is briefly described in the context of the wave equation in the multi-dimensional case. Then several important studies that were focused on optimizations of finite difference schemes with the objective of reducing the numerical anisotropy are discussed.
Keywords
1 Introduction
Numerical anisotropy is a discretization error that is specific to numerical approximations of multidimensional hyperbolic partial differential equations (PDE). This error is often neglected, and the focus is directed toward the reduction of other types of discretization errors, such as numerical dissipation, dispersion or aliasing (e.g., Lele [1], Tam and Webb [2], Kim and Lee [3], Zingg and Lomax [4], Mahesh [5], Hixon [6], Ashcroft and Zhang [7], Fauconnier et al. [8] or Laizet and Lamballais [9]), or toward improving the accuracy of various time marching schemes (e.g., Hu et al. [10], Stanescu and Habashi [11], Mead and Renaut [12], Bogey and Bailly [13] or Berland et al. [14]). There are several areas, however, where the numerical anisotropy can significantly affect the numerical solution based on finite difference or finite volume schemes (examples include computational acoustics, computational electromagnetics, elasticity or seismology). The numerical anisotropy can be reduced by using, for example, one-dimensional high-resolution discretization schemes, multi-dimensional optimized difference schemes, or sufficiently fine grids. However, by increasing the number of grid points the computational time may increase considerably, while one-dimensional high-resolution difference schemes may generate spurious waves at the boundaries of the domain. Oftentimes, optimizations of multi-dimensional difference schemes are more effective.
High-order finite difference schemes that are optimized in one dimension may not preserve their wave number resolution in multi-dimensional problems. These schemes may experience numerical anisotropy, because the dispersion characteristics along grid lines may not be the same as the dispersion characteristics associated with the diagonal directions. Over the years, several attempts to reduce the numerical anisotropy by various techniques were reported. A comprehensive analysis of the numerical anisotropy was performed in the book of Vichnevetsky and Bowles [15] where, among others, the two-dimensional wave equation was solved using two different finite difference schemes for the Laplacian operator. A considerable reduction of the numerical anisotropy was attained by weight averaging the two schemes. A slightly similar approach was previously used by Trefethen [16] who used the leap frog scheme to solve the wave equation in two dimensions. Zingg and Lomax [17] performed optimizations of finite difference schemes applied to regular triangular grids that give six neighbor points for a given node. They conducted comparisons between the newly derived schemes and conventional schemes that were discretized on square grids, and found that the numerical anisotropy can be significantly reduced by using triangular grids. Tam and Webb [18] proposed an anisotropy correction to the finite difference representation of the Helmholtz equation. They derived an anisotropy correction factor using asymptotic solutions to the continuous equation and its finite difference approximation.
Jo et al. [19], in the context of solving the acoustic wave equation, proposed a finite difference scheme over a stencil consisting of grid points from more than one direction, by linearly combining two discretizations of the second derivative operator. A notable reduction of the numerical anisotropy was obtained, but the numerical dispersion error was increased. Hustesdt et al. [20] proposed a two-staggered-grid finite difference schemes for the acoustic wave propagation in two dimensions, where the first derivative operator was discretized along the grid line and along the diagonal direction. Lin et al. [21] explored the dispersion-relation-preserving concept of Tam and Webb [2] in two dimensions to optimize the first-order spatial derivative terms of a model equation that resembles the incompressible Navier-Stokes momentum equation. They approximated the derivative using a nine-point grid stencil, resulting in nine unknown coefficients. Eight of them were determined by employing Taylor series expansions, while the ninth one was determined by requiring that the two-dimensional numerical dispersion relation is the same as the exact dispersion relation.
Kumar [22] derived isotropic finite difference schemes for the first and second derivatives in the context of symmetric dendritic solidification, and obtained a notable reduction of the numerical anisotropy. Patra and Karttunen [23] introduced several finite difference stencils for the Laplacian, Bilaplacian, and gradient of Laplacian, with the objective of improving the isotropic characteristics. Their stencils consisted of more grid points than the conventional schemes, but it was shown that the computational cost may decrease with more than 20% due to some gain in terms of stability. Stegeman et al. [24] applied spectral analysis to evaluate the error in numerical group velocity (both the magnitude and the direction) of vorticity, entropy, and acoustic waves, using the numerical solution to the linearized Euler equations in two dimensions. They showed that a different measure of the group velocity error must be used to account for the error in the propagation direction of the waves. They also stressed that the numerical group velocity is more important than the numerical phase velocity in analyzing the errors associated with wave propagation. In a series of papers [25–28], Sescu et al. proposed a technique to derive finite difference schemes in the multi-dimensional case with improved isotropy. The optimization performed in [25–28] improved the isotropy of the wave propagation and, moreover, the stability restrictions of the multi-dimensional schemes in combination with either Runge-Kutta or linear multistep time marching methods were found to be more effective. They found that the stability restrictions are more favorable when using multi-dimensional schemes, even if they involve more grid points in the stencils. However, this was advantageous for low order schemes, such as those of second or fourth order of accuracy, but it was also shown that favorable stability restrictions can be obtained for higher order of accuracy schemes (sixth or eight) by increasing the isotropy corrector factor. The approach was extended to prefactored compact schemes by Sescu and Hixon [29, 30]. Beside reducing the numerical anisotropy, the new multi-dimensional compact schemes are computationally cheaper than the corresponding explicit multi-dimensional scheme defined on the same stencil.
In computational electromagnetics, there were many attempts to reduce the numerical anisotropy, by applying various techniques. Berini and Wu [31] conducted a comprehensive analysis of the numerical dispersion and numerical anisotropy of finite difference schemes applied to transmission-line modeling (TLM) meshes. They found that, under certain circumstances, the time domain nodes introduce anisotropy into the dispersion characteristics of isotropic media, stressing the importance of developing schemes with improved isotropy. Gaitonde and Shang [32] proposed a class of high-order compact difference-based finite-volume schemes that minimizes the dispersion and isotropy error functions for the range of wave numbers of interest. Sun and Trueman [33] proposed an optimization of two-dimensional finite difference schemes, by considering additional nodes surrounding the point of differencing. They obtained a significant reduction in the numerical anisotropy, dispersion error and the accumulated phase errors over a broad bandwidth. Further optimizations of this scheme were performed in another paper of Sun and Trueman [34]. Koh et al. [35] derived a two-dimensional finite-difference time-domain method, discretizing the Maxwell equations, to eliminate the numerical dispersion and anisotropy. They showed that the new algorithm has isotropic dispersion and resembles the exact phase velocity, whose isotropic property is superior to that of other existing schemes. Shen and Cangellaris [36] introduced a new stencil for the spatial discretization of Maxwell’s equations. Compared to conventional second-order accurate FDTD scheme, their scheme experienced superior isotropy characteristics of the numerical phase velocity. They also showed that the Courant number cab be increased by using the newly derived schemes. Kim et al. [37] derived new three-dimensional isotropic dispersion-finite-difference time-domain schemes (ID-FDTD) based on a linear combination of the traditional central difference equation and a new difference equation using extra sampling points. Among all versions of the proposed finite-difference schemes, three of them showed improved isotropy of the wave propagation compared to the original scheme of the Yee [38]. Kong and Chu [39] introduced a new unconditionally stable finite-difference time-domain method with low numerical anisotropy in three dimensions. Compared with other finite-difference time-domain methods, the normalized numerical phase velocity of their proposed scheme was significantly improved, while the dispersion error and numerical anisotropy have been reduced.
This review will describe and discuss the numerical anisotropy in the framework of wave equation and will present some of the most important optimizations of finite difference schemes in the context of reducing the numerical anisotropy. In Section 2, the dispersion error and the numerical anisotropy existing in finite difference discretizations of the wave equation are introduced and discussed. In Section 3, several approaches to reduce the numerical anisotropy, which were developed over the years by various research groups, are reviewed and discussed. Concluding remarks are included in Section 4.
2 Dispersion error and numerical anisotropy
Weights of the selected spatial finite difference stencils
Stencil | \(\boldsymbol {\alpha_{1}}\) | \(\boldsymbol {\alpha_{2}}\) | \(\boldsymbol {a_{1}}\) | \(\boldsymbol {a_{2}}\) | \(\boldsymbol {a_{3}}\) |
---|---|---|---|---|---|
E2 | 0 | 0 | 1/2 | 0 | 0 |
E4 | 0 | 0 | 2/3 | −1/12 | 0 |
E6 | 0 | 0 | 3/4 | −3/20 | 1/60 |
DRP | 0 | 0 | 0.770882380 | −0.166705904 | 0.020843142 |
C4 | 1/4 | 0 | 3/4 | 0 | 0 |
Haras | 0.3534620 | 0 | 1.5669657/2 | 0.13995831/4 | 0 |
Lui | 0.5381301 | 0.0666331 | 1.36757772/2 | 0.823428170/4 | 0.0185207834/6 |
Lele | 0.5771439 | 0.0896406 | 1.3025166/2 | 0.99355/4 | 0.03750245/6 |
3 Reduction of the numerical anisotropy
In this section, several attempts to reduce the numerical anisotropy, performed by various research groups over the years, are briefly reviewed. The optimizations of the schemes are grouped according to the mathematical model: wave equation, Helmholtz equations, advection equation, Maxwell equation, and dendritic solidification equations.
3.1 Wave equation
Sescu et al. [28, 43] conducted a comprehensive stability analysis of the multi-dimensional schemes combined with either linear-multistep or multistage time marching schemes, and obtained several noteworthy results. For the Leap-Frog scheme applied to the advection equations, it was shown that the stability restriction corresponding to multi-dimensional schemes differs from the corresponding stability restriction via conventional schemes by the factor \((2\beta+2)/(\beta+2)\), where β is the isotropy corrector factor. The conclusion was that the stability restrictions corresponding to multi-dimensional schemes are more convenient compared to the conventional schemes. For an arbitrary direction of the convection velocity with \(|c_{x}| \geq|c_{y}|\), the stability restriction for conventional stencils was given by \(\sigma_{x}+\sigma_{y} \leq CFL\), where \(\sigma_{x}=k|c_{x}|/h\) and \(\sigma_{y}=k|c_{y}|/h\). For multi-dimensional stencils the stability restriction was given by \((1+\beta)\sigma_{x}+\sigma_{y} \leq CFL(1+\beta)\) (where, for example, \(CFL\) is 1, 0.72874 or 0.63052 corresponding to E2, E4 or E6 scheme, respectively). Adams-Bashforth and Runge-Kutta time marching schemes in combination with conventional and multi-dimensional schemes were also analyzed, and it was found that the multi-dimensional schemes provide less restrictive stability limits.
3.2 Helmholtz equation
3.3 Advection equation
3.4 Maxwell equations
3.5 Dendritic solidification
4 Concluding remarks
The numerical anisotropy in finite difference discretizations of partial differential equations was discussed and reviewed. In some instances, the numerical anisotropy can be neglected, and the focus is directed toward other types of one-dimensional errors, such as numerical dispersion, dissipation or aliasing. These errors can be analyzed in the context of one-dimensional difference equations, while the extension to multi-dimensional discretizations is straightforward. By increasing the accuracy of one-dimensional schemes or by increasing the number of grid points in the grid, the isotropic characteristics of the waves in the multi-dimensional case can be improved. These two practices, however, are not always effective since an increase in accuracy may require larger stencils which may introduce spurious waves at the boundaries of the domain, while by increasing of the resolution of the grid one may increase the computational time. It is necessary then to analyze the schemes in the multi-dimensional case and design specific optimizations with the specific objective of reducing the numerical anisotropy, and at the same time of conserving the dispersion characteristics of the corresponding one-dimensional schemes. Various attempts to reduce the numerical anisotropy in finite differencing applied to various model equations were presented and discussed.
Future directions should focus on optimizations of existing compact finite difference schemes in terms of reducing the numerical anisotropy, or derivations of novel compact schemes with low numerical anisotropy. Optimizations and derivations of finite volume schemes (in terms of reducing the numerical anisotropy) applied to either structured or unstructured grids should also be taken into account, especially in the framework of wave propagation problems. Filtering schemes, as applied, for example, in large eddy simulations to separate the small scales from the large scales, may experience numerical anisotropy since they are effective at high wave number ranges. Optimizations of such filters in terms of reducing the numerical anisotropy is also another future area of research.
Declarations
Acknowledgements
The author would like to thank Ray Hixon, Abdollah Afjeh, Vasanth Allampalli, Shivaji Medida, Daniel Ingraham, and Carmen Sescu for constructive support and encouragement.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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