Three-dimensional Convection-diffusion Research Articles

Smaller stencil preconditioners for linear systems in RBF-FD discretizations

AbstractRadial basis function finite difference (RBF-FD) discretization has recently emerged as an alternative to classical finite difference or finite element discretization of (systems) of partial differential equations. In this paper, we focus on the construction of preconditioners for the iterative solution of the resulting linear systems of equations. In RBF-FD, a higher discretization accuracy may be obtained by increasing the stencil size. This, however, leads to a less sparse and often also worse conditioned stiffness matrix which are both challenges for subsequent iterative solvers. We propose to construct preconditioners based on stiffness matrices resulting from RBF-FD discretization with smaller stencil sizes compared to the one for the actual system to be solved. In our numerical results, we focus on RBF-FD discretizations based on polyharmonic splines (PHS) with polynomial augmentation. We illustrate the performance of smaller stencil preconditioners in the solution of the three-dimensional convection-diffusion equation.

Adaptive parameter based matrix splitting iteration method for the large and sparse linear systems

For the large and sparse linear systems, we utilize the efficient splittings of the system matrix and introduce an intermediate variable. The main contribution of this paper is that the adaptive parameter based matrix splitting iteration method is constructed from the view of numerical optimization and hybrid procedure to solve the derived equation instead. The novel method adopts the prediction and correction two-step iteration and the global convergence results are established under some suitable conditions. Compared with some well-known methods, numerical experiments show the efficiency and effectiveness of the new proposal with application to the three-dimensional convection-diffusion equation and the image restoration problems.

A General Alternating-Direction Implicit Framework with Gaussian Process Regression Parameter Prediction for Large Sparse Linear Systems

This paper proposes an efficient general alternating-direction implicit (GADI) framework for solving large sparse linear systems. The convergence property of the GADI framework is discussed. Most of the existing ADI methods can be viewed as particular schemes of the developed framework. Meanwhile the GADI framework can derive new ADI methods. Moreover, as the algorithm efficiency is sensitive to the splitting parameters, we offer a data-driven approach, the Gaussian process regression (GPR) method based on the Bayesian inference, to predict the GADI framework's relatively optimal parameters. The GPR method requires a small training data set to learn the regression prediction mapping, which has sufficient accuracy and high generalization capability. It allows us to efficiently solve linear systems with a one-shot computation, and does not require any repeated computations to obtain relatively optimal splitting parameters. Finally, we use the three-dimensional convection-diffusion equation and continuous Sylvester matrix equation to examine the performance of our proposed methods. Numerical results demonstrate that the proposed framework is faster tens to thousands of times than the existing ADI methods, such as (inexact) Hermitian and skew-Hermitian splitting type methods in which the consumption of obtaining relatively optimal splitting parameters is ignored. Due to the efficiency of the developed methods, we can solve much larger linear systems which these existing ADI methods have been not reached.

Prediction-correction matrix splitting iteration algorithm for a class of large and sparse linear systems

For the large and sparse linear systems, we utilize the efficient splittings of the system matrix and introduce an intermediate variable. The main contribution of this paper is that a prediction-correction matrix splitting iteration algorithm is constructed from the view of numerical optimization to solve the derived equation instead, which is inspired by the idea of adaptive parameter update. The novel algorithm adopts the prediction and correction two-step iteration, which uses information with delay to define the iterations. The global convergence results are established and the algorithm enjoys at least a Q-linear convergence rate under some suitable conditions. Further, a preconditioned version is also presented. Compared with some well-known algorithms, numerical experiments show the efficiency and effectiveness of the new proposal with application to the three-dimensional convection-diffusion equation and the image restoration problems.

The formation of single-channel and multiple-channel rivers on large slopes

We investigated the formation of different types of rivers depending on slope, total discharge, and grain size. Calculations were performed using numerical package CCHE2D, developed by the National Center for Computational Hydroscience and Engineering. The model is based on the Navier-Stokes equations for depth-integrated two-dimensional turbulent flow and the three-dimensional convection-diffusion equation of sediment transport. For each model we use the same river geometry, suspended load concentration, and bedload transport rate. We distinguish three types of rivers using two methods for classification (single-channel, multiple-channels and transitional). We found that the trend line for transitional rivers is an increasing function of Q in space (Q, d) and that for large S the number of multichannel rivers decreases.

Numerical modelling of sedimentary structures in rivers on Earth and Titan

The differences in the evolution of rivers on Earth and on Titan are investigated. Dynamical analysis of the rivers was performed using a numerical package CCHE2D developed by the National Center for Computational Hydroscience and Engineering, University of Mississippi. The model is based on the Navier-Stokes equations for depth-integrated two-dimensional turbulent flow and the three-dimensional convection-diffusion equation of sediment transport. The model enables investigation of the evolution of rivers as a function of total discharge and other parameters of the river. Series of short (from one to several hours) and long (up to 67 days) simulations were performed. We have found that three different liquid hydrocarbons considered for Titan’s rivers give similar velocity fields. It was also found that the suspended load is the main means of transport in Titan’s rivers, while in terrestrial ones, for the same discharge, the bedload could be of the same order as suspended load. Moreover, we suppose that for specific boundary conditions, the evolution of rivers on the Earth promotes the development of braided rivers, while for the same conditions evolution on Titan favours regular meandering rivers.

A Novel Method to Deduce a High-Order Compact Difference Scheme for the Three-Dimensional Semilinear Convection-Diffusion Equation with Variable Coefficients

In this article, a new family of fourth-order compact difference schemes for the three-dimensional semilinear convection-diffusion equation with variable coefficients is presented. Like the finite-volume method, a dual partition is introduced. Combining with the Simpson integral formula and parabolic interpolation, fourth-order schemes are derived based on two different types of dual partitions. Moreover, a sixth-order finite-difference discretization strategy is developed, which is based on the fourth-order compact discretization and Richardson extrapolation technique. This extrapolation technique can achieve a sixth-order-accurate solution on fine grids directly, without the need for interpolation. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order schemes and extrapolation formulas.

The use of incomplete LU decomposition in modeling convection-diffusion processes in an anisotropic medium

The three-dimensional stationary convection-diffusion problem with mixed derivatives, describing the convection-diffusion processes in an anisotropic medium, is considered. The solution of this problem in an anisotropic medium is of significant interest for various models. A finite-difference upwind approximation of this problem is performed on a thirteen-point stencil. Sufficient conditions for the matrix of the system of linear algebraic equations (SLAE) obtained after natural ordering of the grid domain to be an M-matrix are proved. The SLAE is numerically solved by the method of incomplete LU decomposition, which converges for M-matrices. Numerical experiments were performed, which showed the efficiency of this method for solving the convection-diffusion problem both in isotropic and anisotropic media. A drawback of the method is the restriction on the coefficients multiplying mixed derivatives, which enters into the set of sufficient conditions for the matrix of the operator to be an M-matrix.

Cross-stream diffusion under pressure-driven flow in microchannels with arbitrary aspect ratios: a phase diagram study using a three-dimensional analytical model

This article presents a three-dimensional analytical model to investigate cross-stream diffusion transport in rectangular microchannels with arbitrary aspect ratios under pressure-driven flow. The Fourier series solution to the three-dimensional convection-diffusion equation is obtained using a double integral transformation method and associated eigensystem calculation. A phase diagram derived from the dimensional analysis is presented to thoroughly interrogate the characteristics in various transport regimes and examine the validity of the model. The analytical model is verified against both experimental and numerical models in terms of the concentration profile, diffusion scaling law, and mixing efficiency with excellent agreement (with <0.5% relative error). Quantitative comparison against other prior analytical models in extensive parameter space is also performed, which demonstrates that the present model accommodates much broader transport regimes with significantly enhanced applicability.

Incremental Identification of Transport Coefficients in Convection-Diffusion Systems

In this paper, an incremental approach for the identification of a model for transport coefficients in convection-diffusion systems on the basis of high-resolution measurement data is presented. The transport is represented by a convection term with known convective velocity and by a diffusion term with an unknown, generally state-dependent transport coefficient. The identification of the transport model for this transport coefficient constitutes an ill-posed nonlinear inverse problem. We present a novel decomposition approach in which this inverse problem is split into a sequence of inverse subproblems. In the first identification step of this incremental approach a source is estimated by solving an affine-linear inverse problem by means of the conjugate gradient method. In the second identification step a nonlinear inverse problem has to be solved to reconstruct a transport coefficient. A Newton-type method using the conjugate gradient method in its inner iteration is used to solve this nonlinear inverse problem of coefficient estimation. Finally, in the third identification step a transport model structure is proposed and identified on the basis of the model-free transport coefficient reconstructed in the two previous steps. The ill-posedness of each inverse problem is examined by using artificially perturbed transient simulation data and appropriate regularization techniques. The identification methodology is illustrated for a three-dimensional convection-diffusion equation which has its origin in the modeling and simulation of energy transport in a laminar wavy film flow.

The Neumann stability of high-order symmetric schemes for convection-diffusion problems

We find a series of sufficient conditions for the Neumann stability of the leapfrog-Euler scheme of arbitrary even order in the spatial coordinates for the three-dimensional convection-diffusion equation.

Convergence of Fourth Order Compact Difference Schemes for Three‐Dimensional Convection‐Diffusion Equations

We consider a Dirichlet boundary-value problem for the three-dimensional convection-diffusion equations with constant coefficients in the unit cube. A high order compact finite difference scheme is constructed on a 19-point stencil using the Steklov averaging operators. We prove that the finite difference scheme converges in discrete $W_2^m(\omega)$-norm with the convergence rate $O(h^{s-m})$, where the real parameter $s$ satisfies the condition $\max (1.5, m)<s \le m+4$, $m=0,1,2$, and the exact solution belongs to the Sobolev space $W_2^s(\Omega)$.

Universal Parallel Solver for Convection-Diffusion Equations

A parallel unconditionally stable solver for three-dimensional convection-diffusion equations is proposed by applying the upwind Crank-Nicolson difference schemes combined with alternating bar parallelization. This solver can be applied numerically to any variation of convection-diffusion equations with Dirichlet boundary conditions. Making use of a fractional step iteration technique for linear systems, this approach yields good runtime performance. To validate the accuracy and efficiency of the method, sample experiments are done on a software tool, Codie4D, which was implemented using the MPICH library.

Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems

We study efficient iterative methods for the large sparse non-Hermitian positive definite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix. These methods include a Hermitian/skew-Hermitian splitting (HSS) iteration and its inexact variant, the inexact Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer HSS iteration. Theoretical analyses show that the HSS method converges unconditionally to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved. Numerical examples are presented to illustrate the effectiveness of both HSS and IHSS iterations. In addition, a model problem of a three-dimensional convection-diffusion equation is used to illustrate the advantages of our methods.

Reduced Systems for Three-Dimensional Elliptic Equations with Variable Coefficients

We consider large sparse nonsymmetric linear systems arising from finite difference discretization of three-dimensional (3D) convection-diffusion equations with variable coefficients. We show that performing one step of cyclic reduction yields a system of equations which is well conditioned and for which fast convergence can be obtained. A certain block ordering strategy is applied, and analytical results concerning symmetrizability conditions and bounds on convergence rates are given. The analysis is accompanied by numerical examples.

Iterative Solution of Cyclically Reduced Systems Arising from Discretization of the Three-Dimensional Convection-Diffusion Equation

We consider the system of equations arising from finite difference discretization of a three-dimensional convection-diffusion model problem. This system is typically nonsymmetric. We show that performing one step of cyclic reduction, followed by reordering of the unknowns, yields a system of equations for which the block Jacobi method generally converges faster than for the original system, using lexicographic ordering. The matrix representing the system of equations can be symmetrized for a large range of the coefficients of the underlying partial differential equation, and the associated iteration matrix has a smaller spectral radius than the one associated with the original system. In this sense, the three-dimensional problem is similar to the one-dimensional and two-dimensional problems, which have been studied by Elman and Golub. The process of reduction, the suggested orderings, and bounds on the spectral radii of the associated iteration matrices are presented, followed by a comparison of the reduced system with the full system and by details of the numerical experiments.

An explicit fourth-order compact finite difference scheme for three-dimensional convection-diffusion equation

We present an explicit fourth-order compact finite difference scheme for approximating the three-dimensional convection–diffusion equation with variable coefficients. This 19-point formula is defined on a uniform cubic grid. We compare the advantages and implementation costs of the new scheme with the standard 7-point scheme in the context of basic iterative methods. Numerical examples are used to verify the fourth-order convergence rate of the scheme and to show that the Gauss–Seidel iterative method converges for large values of the convection coefficients. Some algebraic properties of the coefficient matrices arising from different discretization schemes are compared. We also comment on the potential use of the fourth-order compact scheme with multilevel iterative methods. © 1998 John Wiley & Sons, Ltd.

BLENDING HEIGHTS FOR WINDS OBLIQUE TO CHECKERBOARDS

Analysis of the two-dimensional checkerboardproblem of many alternating surfaces with different properties andvarious scales and aspect ratios is extended to the general case ofwinds oblique to the surface pattern. Relevant periodic solutions ofa three-dimensional atmospheric convection-diffusion equation arefound, and used to estimate blending heights for both concentrationand flux.

Accuracy Evaluation of Deformed Elements in Finite-Element Soluiton of the Three-Dimensional Convection-Diffusion Equation. Approach by the Error Analysis Technique.

The finite-element method is one of the widely used numerical methods for complex geometries. However, no research on the accuracy of various kinds of deformed elements used in the finiteelement analysis has been made thus far. In this study, the error analysis technique initially proposed by one of the authors is applied for three-dimensional convection-diffusion problems. Firstly, the Galerkin formulation and the modified Galerkin formulation are investigated. Then, the numerical accuracy of various kinds of deformed elements is evaluated theoretically. A regular tetrahedral element and nonuniform tetrahedral and hexahedral elements are investigated and we find that a regular tetrahedral element gives the highest accuracy. Finally, we show the general agreement between the error analyses and the numerical simulation results.

ACCURATE DISCRETIZATION OF INCOMPRESSIBLE THREE-DIMENSIONAL NAVIER-STOKES EQUATIONS

In this article an accurate discretization of three-dimensional incompressible Navier-Stokes equations is derived with the finite analytic method (FAM). The finite analytic method incorporates the local analytic solutions to formulate the discrete algebraic representations of partial differential equations. A very accurate 27-point finite analytic discretization scheme can be derived from local analytic solutions of linearized three-dimensional convection-diffusion equations. Also, a computationally efficient 19-point finite analytic discretization scheme is derived utilizing the superposition of the local analytic solutions of linearized two-dimensional convection-diffusion equations. The accuracy of finite analytic discretization is analyzed. It is shown that using the 27-point scheme as an accurate benchmark case, the 19-paint finite analytic numerical solution is computationally efficient and accurately simulates both convection and diffusion. In this study it is also shown that, due to the analytic n...