Global spectral analysis for convection-diffusion-reaction equation in one and two-dimensions: Effects of numerical anti-diffusion and dispersion
Introduction
The subject of the present investigation relates to the physical processes of a canonical problem involving a single variable, which is governed by simultaneous action of convection, diffusion and reaction. This variable can be viewed as a passive scalar [1]. Importance of this problem is related to the fact that the results can be of relevance to acoustics [2], astrophysics [3], various branches of mathematical physics related to pattern formation and nonlinear dynamics as given in [4], [5], [6]. From a fluid dynamics perspective, this equation has been used for the study of onset of convection with differentially heated fluid layer [7]; it has been used to develop constitutive equations for ()- turbulence models [8]. Parametric variation of coefficient of viscosity with temperature can be the cause of thermal convection for which this equation can have applications towards heat transfer effects, as described in [9], with similar effects in geothermal studies described in [10], formation of Rayleigh-Benard roll pattern in [11]. The major impetus in studying spatio-temporal patterns in many dynamical systems, including in mathematical biology [12], [13], [14], started with the pioneering work of Turing on the chemical basis of morphogenesis, with spatial pattern formations [15]. This equation has been used in many ecological studies of predator-prey dynamics in [16], [17], [18].
The use of convection-diffusion-reaction (CDR) equation in diverse fields of science and engineering, including in combustion and reactive models [19], [20], [21], [22] makes it important. Specifically in [21], the detailed chemical processes are represented by an equivalent reduced system to track scalars. Two such scalars have been considered; (i) A mixture fraction variable, that tracks the mixing of fuel and oxidizer and (ii) a progress variable, which tracks the global extent of reaction for local mixture of fuel and oxidizer.
For example, the progress variable (C) is governed by a scalar transport equation governed by, where is the variable density, represents the convective velocity field; α is a coefficient of diffusion (may include turbulent diffusion component) and f is the forcing due to chemical reaction.
To characterize the dynamics of CDR equation, some past efforts include those which have used finite element solutions [1], [23], [24], [25]. Of specific interest, is the Fourier analysis reported in [25], where the authors demonstrate that their positivity preserving variational method is superior to Galerkin/Least Squares (GLS) and sub-grid scale (SGS) methods. According to these authors [25], the formulation proposed by them shows superiority for both reaction-dominated and convection-dominated flow regimes, due to minimization of spurious oscillations. The CDR equation continues to be studied in diverse disciplines, as in study of pollution related to hydro-geological dynamics [26], simulating transport of in reservoir rock [27]. However, such complex geometry applications are mostly using finite element method [28] using mixed, first-order, formulation. Recently in [29], von Neumann stability analysis is applied for discontinuous Galerkin methods for generalized nonlinear convection-diffusion-reaction systems. However, to the authors knowledge a full domain spectral analysis considering both space and time discretizations have not been conducted. In the present work, we present analysis of a standard second order scheme, along with analysis for a very high order combined compact difference scheme.
To the knowledge of the present authors, detailed analysis of numerical methods are absent, as in finite difference framework reported here for CDR equation. Such analysis are available for the convection equation [30]; diffusion equation [31] and convection-diffusion equation [32]. In these latter references, the authors have shown for the linear equations that the convection speed, coefficient of diffusion become function of length scales, even though the same are constants in the governing equations. These are shown in the respective non-dimensional parameter spaces given by the wave number, CFL number, Peclet number and critical conditions identified by the global spectral analysis (GSA). Using GSA complete analysis of the adopted numerical methods is performed, which involves discretization in space and time simultaneously, including the treatment of boundary and initial conditions numerically.
The presented analysis is relevant to direct numerical simulation and large eddy simulations of flows which requires carefully chosen numerical parameters for accuracy and robustness. This aspect has been discussed in [33], [34] for large eddy simulation of turbulent compressible flows. It is important to note that in a recent study on focusing [35], one to one correspondence between analysis of convection-diffusion equation and the Navier-Stokes equation has been shown with respect to choice of numerical parameters (CFL and Peclet numbers). This is despite the fact that the analysis is for a linear model equation while the simulations are based on nonlinear equations. Therefore, we expect the present analysis to be relevant to LES/DNS of turbulent flows as it quantifies the accuracy and effectiveness of the numerical scheme for dissipation, signal propagation (group velocity) and reaction rate. These parameters are of prime importance in turbulent flows. Hence, the present analysis will be of use for numerical solution of problems involving convection, diffusion and reaction processes.
The present interest is to extend the GSA in [30], [31], [32] for the CDR equation. Thus, the number of non-dimensional parameters would increase by the introduction of the chemical reaction through the Damkohler number, as defined in the following section. The one-dimensional (1D) CDR equation is analyzed theoretically in the next section. In section 3, spectral analysis of numerical method is presented for the 1D CDR equation. In the following, section 4, we provide the property charts for some of the numerical methods of present interest. This is followed by section 5, where numerical solutions of 1D CDR equation are presented and compared with exact solution and explained with the help of property charts given in section 4. Two-dimensional (2D) CDR equation and its theoretical properties are described in section 6. The property charts for 2D CDR equation are presented in section 7, which are used to explain the numerical solution of 2D CDR equation in section 8. The paper closes with a summary and some conclusions.
Section snippets
Theoretical analysis of linear 1-D convection-diffusion-reaction equation
We consider the linear convection-diffusion-reaction equation Here c, α and s are constants specifying the convection speed, coefficient of diffusion and coefficient of reaction respectively. The first step in GSA is to represent the unknown , in the hybrid spectral plane [36], [37], given by Here is the Fourier amplitude and k represents the wavenumber. Substituting this expression in Eq. (1), the following transformed expression is obtained
Spectral analysis of numerical schemes for 1D CDR equation
Like convection equation [30], diffusion equation [31] and convection-diffusion equation [32], here also the correctness and accuracy of numerical solution will depend upon the choice of space and time discretization methods considered simultaneously. In the following, we discuss only two such combinations, as representatives of low order and high accuracy methods.
Numerical properties by spectral analysis of 1D CDR equation
It is now well understood that numerical solution of any governing equation requires methods which have suitable properties to capture the physical processes, with high fidelity. Apart from scale resolution, one must also ensure that the adopted methods are numerically stable and follow the physical properties faithfully [37], [43]. For problems of fluid flows and wave phenomena, one must adopt methods which follow physical dispersion relation in the numerical sense also. For the 1D CDR
Numerical simulation of 1D unsteady inhomogeneous CDR equation
In this section, the inhomogeneous 1D CDR equation is solved numerically using the above mentioned -NCCD and -- schemes. The equation is solved in a domain: , and is rewritten for the addition of a forcing term given as, For a specific case of and the initial condition of , one can look for the exact solution available for , which takes the form as given in [1] We introduce the non-dimensional parameters
Spectral analysis of linear 2-D convection-diffusion-reaction equation
The same analysis performed for 1D CDR equation is now applied for 2D CDR in this section. We consider the 2D linear CDR equation as follows Here , , α and s are constants specifying the convection speed in x- and y-directions, coefficient of diffusion and coefficient of reaction, respectively. Using the GSA explained for 1D CDR, here the important expressions for the corresponding 2D analysis are mentioned. The physical dispersion relation is obtained
Properties of 2D CDR equation obtained by GSA
GSA is performed for the same two schemes (-NCCD and --) used for the 1D CDR analysis and the property charts are presented in this section. For all the analysis shown, the cell aspect ratio, and the wave propagation angle, have been prescribed. In the following, the property charts are prepared with a view to calibrate these results with a case reported in [23] with very low convection speed () and the reaction coefficient of order one (). To calibrate the two
Numerical simulation of 2D unsteady inhomogeneous CDR equation
The following equation is solved numerically by the two schemes, whose numerical properties have been presented in the previous section (-NCCD and -- schemes).
For the test case solved, the parameters are as follows, which were considered in [23] with the source term, , the diffusion coefficient, , the convection velocity in x-direction as, , the convection velocity in y-direction as,
Summary and conclusion
Convection-diffusion-reaction (CDR) equation plays a central role in problems of many disciplines of engineering, science and finances. As a consequence, importance of analysis of CDR equation and its numerical solution cannot be overemphasized, which has motivated the present research. We have used the global spectral analysis (GSA) of numerical methods to characterize all the three important physical processes in terms of the non-dimensional numerical parameters, such as kh, , Pe and Da. To
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (43)
- et al.
An implicit scheme for solving the convection-diffusion-reaction equation in two dimensions
J. Comput. Phys.
(2000) - et al.
Finite element methods for the Helmholtz equation in an exterior domain: model problems
Comput. Methods Appl. Mech. Eng.
(1991) - et al.
Theory of pattern forming systems under traveling-wave forcing
Phys. Rep.
(2007) - et al.
Travelling-stripe forcing of Turing patterns
Physica D
(2004) - et al.
Thermal convection in rotating spherical shells
Phys. Earth Planet. Inter.
(1997) - et al.
Visualization of roll patterns in Rayleigh-Bénard convection of air in rectangular shallow cavity
Int. J. Heat Mass Transf.
(2001) - et al.
Dynamics of pattern formation in biomimetic systems
J. Theor. Biol.
(2008) - et al.
A moving grid finite element method applied to a model biological pattern generator
J. Comput. Phys.
(2003) - et al.
Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system
J. Differ. Equ.
(2009) - et al.
Computational examples of reaction-convection-diffusion equations solution under the influence of fluid flow: first example
Appl. Math. Model.
(2012)
Monotone scheme and boundary conditions for finite volume simulation of magnetohydrodynamic internal flows at high Hartmann number
J. Comput. Phys.
A positivity preserving variational method for multi-dimensional convection-diffusion-reaction equation
J. Comput. Phys.
Convection-diffusion-reaction of -enriched brine in porous media: a pore-scale study
Comput. Geosci.
A stabilised finite element method for the convection-diffusion-reaction equation in mixed form
Comput. Methods Appl. Mech. Eng.
Error dynamics: beyond von Neumann analysis
J. Comput. Phys.
Error dynamics of diffusion equation: effects of numerical diffusion and dispersive diffusion
J. Comput. Phys.
Spectral analysis of finite difference schemes for convection diffusion equation
Comput. Fluids
Analysis of central and upwind compact schemes
J. Comput. Phys.
A three-point combined compact difference scheme
J. Comput. Phys.
A new combined stable and dispersion relation preserving compact scheme for non-periodic problems
J. Comput. Phys.
Further improvement and analysis of CCD scheme: dissipation discretization and de-aliasing properties
J. Comput. Phys.
Cited by (19)
A new compact scheme-based Lax–Wendroff method for high fidelity simulations
2024, Computers and FluidsGlobal spectral analysis: Review of numerical methods
2023, Computers and FluidsAnalysis and design of a local time stepping scheme for LES acceleration in reactive and non-reactive flow simulations
2022, Journal of Computational PhysicsCitation Excerpt :The application of GSA to the study of numerical schemes used for non-uniform grids [35,36] and for domain-decomposition based methods [37,38] have been demonstrated in the past. While GSA of the linear convection-diffusion [39] and convection-diffusion-reaction [40] equations are already available in the literature, for the sake of brevity, we restrict our analysis of the proposed method to the linear convection equation only. For the extension of the analysis to linear convection-diffusion equation, readers may refer to [41].
Global spectral analysis of the Lax–Wendroff-central difference scheme applied to Convection–Diffusion equation
2022, Computers and FluidsCitation Excerpt :The results from this analysis are extremely useful to design Dispersion Relation Preserving (DRP) schemes. GSA has been successfully used in the past to study the linear convection equation [8], diffusion equation [19] and convection–diffusion-reaction equation [20] for various numerical schemes. Convection–diffusion equation plays an important role in many disciplines of engineering and science, the accurate numerical solution of which cannot be understated and has motivated the present study.