3D Convection Diffusion Equation Research Articles

The cell-centered positivity-preserving finite volume scheme for 3D convection–diffusion equation on distorted meshes

PurposeThis paper aims to construct positivity-preserving finite volume schemes for the three-dimensional convection–diffusion equation that are applicable to arbitrary polyhedral grids.Design/methodology/approachThe cell vertices are used to define the auxiliary unknowns, and the primary unknowns are defined at cell centers. The diffusion flux is discretized by the classical nonlinear two-point flux approximation. To ensure the fully discrete scheme has positivity-preserving property, an improved discretization method for the convection flux was presented. Besides, a new positivity-preserving vertex interpolation method is derived from the linear reconstruction in the discretization of convection flux. Moreover, the Picard iteration method may have slow convergence in solving the nonlinear system. Thus, the Anderson acceleration of Picard iteration method is used to solve the nonlinear system. A condition number monitor of matrix is employed in the Anderson acceleration method to achieve better robustness.FindingsThe new scheme is applicable to arbitrary polyhedral grids and has a second-order accuracy. The results of numerical experiments also confirm the positivity-preserving of the discretization scheme.Originality/value1. This article presents a new positivity-preserving finite volume scheme for the 3D convection–diffusion equation. 2. The new discretization scheme of convection flux is constructed. 3. A new second-order interpolation algorithm is given to eliminate the auxiliary unknowns in flux expressions. 4. An improved Anderson acceleration method is applied to accelerate the convergence of Picard iterations. 5. This scheme can solve the convection–diffusion equation on the distorted meshes with second-order accuracy.

An efficient spline-based DQ method for 2D/3D Riesz space-fractional convection–diffusion equations

This paper proposes an efficient spline-based DQ method for the 2D and 3D convection–diffusion equations (CDEs) with Riesz fractional derivative in space, which have been widely used to describe the anomalous solute transport in complex media. Firstly, a spline-based differential quadrature (DQ) formula is developed to approximate the Riesz derivative by using cubic B-splines as trial functions, which allows us to approximate the fractional derivatives with high accuracy and small computational cost. We then utilize it to discretize the fractional derivatives in the governing equation and a cubic B-spline DQ scheme is further established by applying the finite difference (FD) scheme to the resulting system of ordinary differential equations. A brief implementation of the proposed DQ method is also presented. To examine the effectiveness of this spline-based DQ method, numerical tests are finally done on some benchmark problems and the simulation of rotating Gaussian hill in convection-dominated flow governed by fractional derivatives. The advantages in computational accuracy and efficiency are illustrated by comparing the results with the other algorithms in open literature.

A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh

This paper is contributed to explore how a Crank-Nicolson weak Galerkin finite element method (WG-FEM) addresses the singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. The problem and some asymptotic behavior results are given for the exact solution and its derivatives with the parameter ε. These results are essential for proving the uniform convergence of the proposed WG-FEM. A tensor product Shishkin mesh featuring piece-wise discontinuous bilinear polynomials is employed to handle the layers for uniform convergence at different parameter values of ε. An error estimate ‖uN−INu‖WG is presented, where INu denotes the vertex-edge-cell interpolation of the solution u, and ‖⋅‖WG denotes the Weak Galerkin norm. The optimal uniform convergence order is demonstrated for semi-discrete and fully discrete schemes. Various numerical experiments are conducted to validate the optimal order of convergence demonstrated by the proposed method.

Fast successive permutation iterative algorithms for solving convection diffusion equation

The solution of convection diffusion problem is a complex task, requiring the solving of linear equations which can take up a significant amount of computational time. This issue can be mitigated through the use of iterative methods. We present a class of fast successive permutation iterative algorithms with relaxation factors (ω-SPI) to solve 1D, 2D and 3D convection diffusion equations. The stability, convergence and optimal value of the relaxation factor are proved. In order to test the efficiency and accuracy of our proposed iterative algorithms, we present several numerical experiments. The results demonstrate that the new algorithms are more accurate and faster than traditional methods such as full-implicit, Gauss-Seidel and SOR algorithms.

Quantifying parameter ranges for high fidelity simulations for prescribed accuracy by Lax–Wendroff method

Explicit second order central difference (CD2) based Lax–Wendroff (LW) method is evaluated for large eddy simulation here, using global spectral analysis (GSA) for its ability to compute fluid flows up to a specified accuracy. While the LW method has been analyzed for numerical stability in the past, rigorous quantification for performing accurate simulations including error dynamics has not been reported before. This is addressed here by determining optimal numerical parameters for prescribed accuracies based on the GSA of 2D convection–diffusion equation (CDE), which is the main focus of the present research. First, two LW methods are reported based on the application of the method for convection and both convection and diffusion operators, respectively, with the latter also having third derivative dispersive and fourth order diffusion terms. Using GSA, the optimal scheme is determined by comparing the two schemes for accuracy. The accuracy of GSA for wave propagation is also demonstrated by the validation of the q-waves determined from the analysis with the numerical simulations of 2D CDE. Finally, motivated by a one-to-one correspondence of the Navier–Stokes equation with the linear CDE established in “Effects of numerical anti-diffusion in closed unsteady flows governed by two-dimensional Navier–Stokes equation- (Suman et al., 2020)”, optimal simulation parameters are determined for the accuracy in representing physical diffusion of the scheme and validated using simulations of the canonical lid-driven cavity problem for a sub- and a super-critical Reynolds number.

Exponential Basis Approximated Fuzzy Components High-Resolution Compact Discretization Technique for 2D Convection–Diffusion Equations

Exponential Basis Approximated Fuzzy Components High-Resolution Compact Discretization Technique for 2D Convection–Diffusion Equations

Analysis of Particle-Resolved CFD Results for Dispersion in Packed Beds

Dispersion is the spreading of a solute while it is moved by a flowing medium. The study of dispersion in catalytic chemical reactors is fundamental to their design, since dispersion influences the reactant and product transport within the bed. In this paper, longitudinal and transverse dispersion of an inert tracer in slender packed beds of spheres and spherocylinders is studied using Computational Fluid Dynamics simulations. The focus is on the analysis of dispersion from full field data. The purpose is to develop a methodology that can later also be used to characterize dispersion from full field experimental data such as MRI measurements. Results obtained by means of particle-resolved CFD simulations are discussed. Spatial distributions and residence times are analyzed and the results are interpreted by comparison to results obtained from 1D and 2D convection-diffusion equations.

Global spectral analysis of the Lax–Wendroff-central difference scheme applied to Convection–Diffusion equation

In this work, the Global Spectral Analysis (GSA) is applied to the convective Lax–Wendroff based discretization of linear convection–diffusion problem in both 1D and 2D. Contrary to standard numerical analysis approaches, two important physical processes (convection and diffusion) are treated together, thus making GSA a function of multiple non-dimensional numerical parameters, namely, the non-dimensional wavenumber (kh), the Courant–Friedrich–Lewy number (Nc) and the Peclet number (Pe). All the three quantities impact the stability of the numerical scheme by affecting both numerical amplification factor as well as numerical diffusion. Likewise, numerical phase speed and numerical group velocity all become expressions of all three non-dimensional parameters. Leveraging the GSA, an accurate map (property charts) of acceptable range of parameter space is evidenced to obtain the spatio-temporal numerical solution that is stable as well as Dispersion Relation Preserving (DRP). The numerical property charts are shown to be useful to calibrate numerical solutions of both 1D and the 2D convection–diffusion equations on uniform meshes. Finally, as demonstrated while solving the Navier–Stokes equation for a Taylor–Green vortex problem, property charts allow to explain numerical behaviors often observed with real complex flow problems.

Artificial neural networks for solving elliptic differential equations with boundary layer

In this paper, we consider the artificial neural networks for solving the elliptic differential equation with boundary layer, in which the gradient of the solution changes sharply near the boundary layer. The solution of the boundary layer problems poses a huge challenge to both traditional numerical methods and artificial neural network methods. By theoretically analyzing the changing rate of the weights of the first hidden layer near the boundary layer, a mapping strategy is added in traditional neural network to improve the convergence of the loss function. Numerical examples are carried out for the 1D and 2D convection–diffusion equation with boundary layer. The results demonstrate that the modified neural networks significantly improve the ability in approximating the solutions with sharp gradient.

Two-Dimensional-One-Dimensional Alternating Direction Schemes for Coastal Systems Convection-Diffusion Problems

The initial boundary value problem for the 3D convection-diffusion equation corresponding to the mathematical model of suspended matter transport in coastal marine systems and extended shallow water bodies is considered. Convective and diffusive transport operators in horizontal and vertical directions for this type of problem have significantly different physical and spectral properties. In connection with the above, a two-dimensional–one-dimensional splitting scheme has been built—a three-dimensional analog of the Peaceman–Rachford alternating direction scheme, which is suitable for the operational suspension spread prediction in coastal systems. The paper has proved the theorem of stability solution with respect to the initial data and functions of the right side, in the case of time-independent operators in special energy norms determined by one of the splitting scheme operators. The accuracy has been investigated, which, as in the case of the Peaceman–Rachford scheme, with the special definition of boundary conditions on a fractional time step, is the value of the second order in dependency of time and spatial steps. The use of this approach makes it possible to obtain parallel algorithms for solving grid convection-diffusion equations which are economical in the sense of total time of problem solution on multiprocessor systems, which includes time for arithmetic operations realization and the one required to carry of information exchange between processors.

Shallow water model for pollutant distribution in the Ypacarai Lake

In this paper, we analyze the distribution of a non-reactive contaminant in Ypacarai Lake. We propose a shallow-water model that considers wind-induced currents, inflow and outflow conditions in the tributaries, and bottom effects due to the lakebed. The hydrodynamic is based on the depth-averaged Navier-Stokes equations considering wind stresses as force terms which are functions of the wind velocity. Bed (bottom) stress is based on Manning's equation, the lakebed characteristics, and wind velocities. The contaminant transportation is modeled by a 2D convection-diffusion equation taking into consideration water level. Comparisons between the simulation of the model, analytical solutions, and laboratory results confirm that the model captures the complex dynamic phenomenology of the lake. In the simulations, one can see the regions with the highest risk of accumulation of contaminants. It is observed the effect of each term and how it can be used them to mitigate the impact of the pollutants.

Compact higher order discretization of 3D generalized convection diffusion equation with variable coefficients in nonuniform grids

A higher-order compact (HOC) discretization of generalized 3D convection-diffusion equation (CDE) in nonuniform grid is presented. Even in the presence of cross-derivative terms, the discretization uses only nineteen point stencil. Extension of this newly proposed discretization to semi-linear and convection-diffusion-reaction problems is seen to be straightforward and this inherent advantage is thoroughly exploited. The scheme being designed on a transformation free coordinate system is found to be efficient in capturing boundary layers and preserve the nonoscillatory property of the solution. The proposed method is tested using several benchmark linear and nonlinear problems from the literature. Additionally, problems with sharp gradients are solved. These diverse numerical examples demonstrate the accuracy and efficiency of the scheme proposed. Further, the numerical rate of convergence is seen to approach four confirming theoretical estimation.

Conservative characteristic finite difference method based on ENO and WENO interpolation for 2D convection–diffusion equations

Conservative characteristic finite difference method based on ENO and WENO interpolation for 2D convection–diffusion equations

Discretization of the 2D Convection–Diffusion Equation Using Discrete Exterior Calculus

While the Discrete Exterior Calculus (DEC) discretization of the diffusive term of the Trans- port Equation is well understood, the DEC discretization of the convective term, as well as its stabilization, is an ongoing area of research. In this paper, we propose a local discretization for this term based on DEC and geometric arguments, assuming the particle velocity field is prescribed at the vertices of the primal mesh. Since this formulation is similar to that of the Finite Element Method with linear interpolation functions (FEML), this numerical scheme can be stabilized us- ing known stabilization techniques, such as Artificial Diffusion. Using this stabilizaton technique, numerical tests are carried out on simple problems with domains discretized with coarse and fine simplicial meshes to show numerical convergence for stationary and transient problems.

Nonlinear Finite Volume Scheme Preserving Positivity for 2D Convection-Diffusion Equations on Polygonal Meshes

In this paper, a nonlinear finite volume scheme preserving positivity for solving 2D steady convection-diffusion equation on arbitrary convex polygonal meshes is proposed. First, the nonlinear positivity-preserving finite volume scheme is developed. Then, in order to avoid the computed solution beyond the upper bound, the cell-centered unknowns and auxiliary unknowns on the cell-edge are corrected. We prove that the present scheme can avoid the numerical solution beyond the upper bound. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results show that our scheme preserves the above conclusion and has second-order accuracy for solution.

Effects of numerical anti-diffusion in closed unsteady flows governed by two-dimensional Navier-Stokes equation

Numerical methods producing acceptable results for a long time abruptly blow up, without providing any indication of localized onset of sudden numerical instability. This has been identified as focusing problem in literature. It is noted that the scale selection of error does not depend on the relevant excited physical space-time scales. While this has been encountered in weather prediction studies, it is not widely reported from the solution of Navier-Stokes equation (NSE). Recently, in “Focusing phenomenon in numerical solution of two-dimensional Navier-Stokes equation, In: Pirozzoli S., Sengupta T. (eds) High-Performance Computing of Big Data for Turbulence and Combustion, CISM International Centre for Mechanical Sciences (Courses and Lectures), vol 592. Springer, Cham (2019)”, focusing was demonstrated for a steady fluid flow and its mechanism was identified from global spectral analysis (GSA) of 2D convection diffusion equation (CDE). Focusing was shown to be due to the anti-diffusion caused by the discretization of diffusion term for the chosen numerical scheme. The present work consolidates the one-to-one correspondence between numerical anti-diffusion of 2D CDE and focusing for unsteady flows by solving flow inside a 2D lid driven cavity (LDC) for the Reynolds number of 10,000. We also present a method to remove numerical anti-diffusion using multi-dimensional filters. Detailed analysis of space-time discretization with filters is also provided to explain the cure of focusing.

Approximation of 3D convection diffusion equation using DQM based on modified cubic trigonometric B-splines

This paper presents an approximate solution of 3D convection diffusion equation (CDE) using DQM based on modified cubic trigonometric B-spline (CTB) basis functions. The DQM based on CTB basis functions is used to integrate the derivatives of space variables which transformed the CDE into the system of first order ODEs. The resultant system of ODEs is solved using SSPRK (5,4) method. The solutions are approximated numerically and also presented graphically. The accuracy and efficiency of the method are validated by comparing the solutions with existing numerical solutions. The stability analysis of the method is also carried out.

Stability and error estimates of a new high-order compact ADI method for the unsteady 3D convection–diffusion equation

In this paper, a new high-order compact ADI method for the unsteady convection–diffusion equation in three dimension(3D) is considered. Collecting the truncation error of the finite difference operator by the recursion method, we derive a new high-order compact finite difference scheme for the unsteady 1D convection–diffusion equation firstly. Then, based on the ADI method, applying a correction technique to reduce the error of the splitting term, a high-order compact ADI method for the unsteady 3D convection–diffusion equation is proposed, in which we solve a series of 1D problems with strictly diagonal dominant tri-diagonal structures instead of the high-dimensional ones. The scheme is proved to be unconditional stable. Moreover, the regularity and error estimate of the numerical solution are derived. Finally, some numerical examples are performed, which confirm the theoretical prediction and show that much better computational accuracy results can be got by applying the new scheme.

Research on Locating Model of Heavy Metal Pollutants Source Based on SFPI Method and 2D Convection-Diffusion Equation

As an important field in environmental science, the research on the locating issue of heavy metals pollutants source has increasingly attracted attention of scholars and society. In order to locate the pollutants source, we should firstly figure out the spatial distribution of heavy metals pollutants and pollution degree of different subareas in a limited area. Based on the datasheets from CUMCM 2011, we obtain the spatial distribution figures of heavy metals on MATLAB platform. Then, we introduce SFPI method to build an evaluation model of pollution degree in different subareas, with analyzing of the result of which, we can approximately conclude that the location of heavy metals pollutants source may be located in Industrial Area and Traffic Area. Further analyzing, to position the pollutants source more precisely, we introduce 2D convection-diffusion equation to describe the pollutant pathway of heavy metals, solve the equation with FDM algorithm on Maple platform, and obtain the propagation function of heavy metals concentration. Finally, we furtherly modify the propagation function and then figure out the precise coordinates of the pollution source and the corresponding subareas.

An exponential cubic B-spline algorithm for multi-dimensional convection-diffusion equations

We present a method viz. “exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM)” to approximate the numerical solution of 2D and 3D convection-diffusion equations (CDEs). This method is used to approximate the special derivatives whereas time derivative is approximated by using “an optimal five stages and fourth order Runge-Kutta (SSP-RK54)” scheme. Four examples are chosen to test out the efficiency and accuracy of the method. It is found that the present method turns out more accurate results compared to the results given by other methods. There is wide scope of the present method to be used for numerical solution of several PDEs governing physical, chemical and biological problems.