Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains

Abstract

SUMMARYThis paper presents a detailed multi-methods comparison of the spatial errors associated with ÿnite difference, ÿnite element and ÿnite volume semi-discretizations of the scalar advection-di usion equation. The errors are reported in terms of non-dimensional phase and group speed, discrete di usivity, artiÿcial di usivity, and grid-induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew-symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure di usion. It is demonstrated that streamline upwind Petrov-Galerkin and its control-volume ÿnite element analogue, the streamline upwind control-volume method, produce both an artiÿcial di usivity and a concomitant phase speed adjustment in addition to the usual semi-discrete artifacts observed in the phase speed, group speed and di usivity. The Galerkin ÿnite element method and its streamline upwind derivatives are shown to exhibit super-convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second-order behaviour. In Part II of this paper, we consider two-dimensional semi-discretizations of the advection-di usion equation and also assess the a ects of grid-induced anisotropy observed in the non-dimensional phase speed, and the discrete and artiÿcial di usivities. Although this work can only be considered a ÿrst step in a comprehensive multimethods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common analysis framework. Published in 2004 by John Wiley & Sons, Ltd.

Comparison of Galerkin and control volume finite element for advection-diffusion problems

“…The value ÿ = 2= √ 15 is used to 'optimize' the SUPG for dispersion error; for SUCV ÿ = 1 is used. ¶ See Reference [20], for a discussion of how these values are optimal for reducing dispersion errors.…”

“…This aspect is also true of the SUPG and related stabilized FEM formulations, and in all methods with so-called artiÿcial di usion. In the Fourier analysis of Christon et al [20] the spectral nature of artiÿcial di usion is illustrated for many popular numerical schemes, including those discussed here. The better numerical schemes inject appreciable artiÿcial di usion only for short wavelength modes, in the same spectral range where dispersion errors also become signiÿcant.…”

“…For large Peclet number, = O(h= u ), while for vanishing Peclet number, = O(h 2 =Ä), where h denotes a mesh size measure and Ä = k= C. For steady problems the term involving the time step t is omitted. The only modiÿcation to Shakib's formula is the factor ÿ, which can be used to optimize the dispersive properties of the SUCV scheme under pure advection conditions on Cartesian grids [20]. In general, the time term will dominate as t → 0, which has the e ect of annihilating the stabilization.…”

“…The scheme was formulated in an analogous way to SUPG, in the context of a Petrov-Galerkin method. These methods have recently been analysed and compared via Fourier analysis by Christon et al [20] in 1D and Voth et al [21] in 2D, as part of a larger set of methods for advection-di usion problems.…”

Numerical approximation of viscous terms in finite volume models for shallow waters

“…This approach is widely used for the study of discrete models corresponding to various equations (e.g. [9]). In particular, in the case of shallow water equations, Fourier analysis has been extensively employed for various schemes such as finite difference (e.g.…”