One-dimensional Convection-diffusion Equation Research Articles

Error Estimates for First- and Second-Order Lagrange–Galerkin Moving Mesh Schemes for the One-Dimensional Convection–Diffusion Equation

A new moving mesh scheme based on the Lagrange–Galerkin method for the approximation of the one-dimensional convection–diffusion equation is studied. The mesh movement is prescribed by a discretized dynamical system for the nodal points. This system is related to the velocity and diffusion coefficient in the convection–diffusion equation such that the nodal points follow the convective flow of the model. It is shown that under a restriction of the time step size the mesh movement cannot lead to an overlap of the elements and therefore an invalid mesh. Using a piecewise linear approximation, optimal error estimates in the ℓ∞(L2)∩ℓ2(H01)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^\\infty (L^2) \\cap \\ell ^2(H_0^1)$$\\end{document} norm are proved in case of both, a first-order backward Euler method and a second-order two-step method in time. These results are based on new estimates of the time dependent interpolation operator derived in this work. Preservation of the total mass is verified for both choices of the time discretization. Numerical experiments are presented that confirm the error estimates and demonstrate that the proposed moving mesh scheme can circumvent limitations that the Lagrange–Galerkin method on a fixed mesh exhibits.

Macroscopic finite-difference scheme and modified equationsof the general propagation multiple-relaxation-time lattice Boltzmann model.

In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations(MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation(NACDE) and Navier-Stokes equations(NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, three benchmark problems, the Gauss hill problem, the CDE with nonlinear convection and diffusion terms, and the Taylor-Green vortex flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results not only are in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis.

Learning the flux and diffusion function for degenerate convection-diffusion equations using different types of observations

In recent years, there has been an increasing interest in utilizing deep learning-based techniques to predict solutions to various partial differential equations. In this study, we investigate the identification of an unknown flux function and diffusion coefficient in a one-dimensional convection-diffusion equation. The diffusion function is allowed to vanish on intervals implying that solutions generally possess low regularity, i.e., are discontinuous. Therefore, solutions must be interpreted in the sense of entropy solutions which combine a weak formulation with an additional constraint (entropy condition). We explore a methodology that utilizes symbolic neural networks (S-Nets) in combination with an entropy-consistent discrete numerical scheme (ECDNS). Different types of observation data are explored. Extensive experiments in this paper demonstrate that the proposed method is a robust tool to identify the unknown flux and diffusion function. The flux and diffusion functions are restricted to be rational functions.

Fourth-order multiple-relaxation-time lattice Boltzmann model and equivalent finite-difference scheme for one-dimensional convection-diffusion equations.

In this paper, we first develop a fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is used. We also perform the Chapman-Enskog analysis to recover the CDE from the MRT-LB model. Then an explicit four-level finite-difference (FLFD) scheme is derived from the developed MRT-LB model for the CDE. Through the Taylor expansion, the truncation error of the FLFD scheme is obtained, and at the diffusive scaling, the FLFD scheme can achieve the fourth-order accuracy in space. After that, we present a stability analysis and derive the same stability condition for the MRT-LB model and FLFD scheme. Finally, we perform some numerical experiments to test the MRT-LB model and FLFD scheme, and the numerical results show that they have a fourth-order convergence rate in space, which is consistent with our theoretical analysis.

Numerical Methods for Unsteady Convection-diffusion Problems Based on Combining Compact Difference Schemes with Runge-Kutta Methods

The convection-diffusion equation is of primary importance in understanding transport phenomena within a physical system. However, the currently available methods for solving unsteady convection-diffusion problems are generally not able to offer excellent accuracy in space and time. The one-dimensional unsteady convection-diffusion equation was solved by combing a compact difference scheme with the Runge-Kutta method. The combined method has fourth-order accuracy in space and time. To check the accuracy of the combined method, numerical experiments were carried out and comparisons were performed with the Crank-Nicolson method. The analysis results indicated that the combined method is numerically stable at low wave numbers and small Courant-Friedrichs-Lewy numbers. The combined method has higher accuracy than the Crank-Nicolson method.

Numerical Solution to Unsteady One-Dimensional Convection-Diffusion Problems Using Compact Difference Schemes Combined with Runge-Kutta Methods

The convection-diffusion equation is of primary importance in understanding transport phenomena within a physical system. However, the currently available methods for solving unsteady convection-diffusion problems are generally not able to offer excellent accuracy in both time and space variables. A procedure was given in detail to solve the unsteady one-dimensional convection-diffusion equation through a combination of Runge-Kutta methods and compact difference schemes. The combination method has fourth-order accuracy in both time and space variables. Numerical experiments were conducted and the results were compared with those obtained by the Crank-Nicolson method in order to check the accuracy of the combination method. The analysis results indicated that the combination method is numerically stable at low wave numbers and small CFL numbers. The combination method has higher accuracy than the Crank-Nicolson method.

Least-squares solutions of boundary-value problems in hybrid systems

This paper applies the mathematical framework of the Theory of Functional Connections to solve the boundary-value problems arising from hybrid systems. A hybrid system is simply a sequence of different differential equations. The Theory of Functional Connections is a technique to derive constrained expressions which are functionals with embedded constraints. These functionals allow to transform a large class of constrained optimization problems into unconstrained problems. The initial and most useful application of this technique is in the solution of differential equations where the problem can be posed as an unconstrained optimization problem and solved with simple numerical techniques, i.e., least-squares.The approach developed in this work derives an analytical constrained expression for the entire range of a hybrid system, enforcing both the boundary conditions as well as the continuity conditions across the sequence of differential equations. This reduces the searched solution space of the hybrid system to only admissible solutions. The transformation allows for a least-squares solution of the sequence for linear differential equations and an iterative least-squares solution for nonlinear differential equations.This technique is validated by numerical tests for three differential equation sequences: linear–linear, linear–nonlinear, and nonlinear–nonlinear. The accuracy levels obtained are all at machine-error, which is consistent with the accuracy experienced in past studies on the application of the Theory of Functional Connections to solve single ordinary differential equations. In these tests, TFC produces solutions up to 3 orders of magnitude more accurate than the spectral method and up to 4 orders of magnitude more accurate than the shooting method. Additionally, in all cases except the linear–linear sequence, the TFC method is faster. Finally, the proposed framework is used to solve the one-dimensional convection–diffusion equation to further highlight its utility outside of hybrid systems.

Parameter Estimation in Convection-Diffusion Model

This paper describes a special one-dimensional convection-diffusion equation and analyzes two types of difference schemes. Numerical solutions of the two difference methods for this equation are implemented to estimate the parameters of the velocity component of the fluid and the diffusion coefficient. Good results of parameters estimated are not achieved because of the larger approximation errors by the difference schemes. Then multiple linear regression is applied to estimating the corresponding parameters by using the analytical solution of this special equation. By this means, the better estimated values of the parameters are obtained.

Entropy Schemes for One-Dimensional Convection-Diffusion Equations

In this paper, we extend the entropy scheme for hyperbolic conservation laws to one-dimensional convection-diffusion equation. The operator splitting method is used to solve the convection-diffusion equation that is divided into conservation and diffusion parts, in which the first-order accurate entropy scheme is applied to solve the conservation part and the second accurate central difference scheme is applied to solve the diffusion part. Numerical tests show that the L∞ error achieves about second-order accuracy, but the L1 error reaches about forth-order accuracy.

Rebuttal on “A mathematical model on depth-averaged β-factor in open-channel turbulent flow” Environmental Earth Sciences 77:253 (2018)

In a recent paper published in Environmental Earth Sciences, Jain et al. (Environ Earth Sci 77:253, 2018) proposed an interesting study about the $$\beta$$-factor (the inverse of the turbulent Schmidt number, i.e. the ratio of momentum diffusivity to mass diffusivity in a turbulent flow). They proposed an equation for the depth-averaged $$\beta$$-factor which was used in the Rouse equation to calculate concentration profiles of suspended sediments in open-channel turbulent flows. Despite the interest, the study shows a weakness related to some inconsistencies and contradictions in the method. The main result should be improved. In this note, the weakness in this study will be pointed out. Some used equations are in contradiction with the initial assumptions. The Rouse equation is a solution of the steady-state one-dimensional convection–diffusion equation with a parabolic eddy viscosity which is based on a logarithmic velocity profile. In the same study, Jain et al. (2018) used two different mixing length equations, parabolic and linear relations. The parabolic mixing length and a constant mixing velocity are in contradiction with the used exponential-type eddy viscosity profile. The used equation for coefficient $$\gamma$$ is based on the finite mixing length model and the related assumptions of a linear eddy viscosity and a logarithmic velocity distribution (Nielsen and Teakle in Phys Fluids 16(7):2342–2348, 2004) which are in contradiction with the used exponential-type eddy viscosity profile and velocity distribution. An improved approach and new equations for the depth-averaged $$\beta$$-factor are proposed. The first is based on the equation for the coefficient $$A$$ proposed by Jain et al. (2018) and a second based on a linear function for $$A$$ which gives an equation for $$\beta$$ similar to that of van Rijn (J Hydraul Eng ASCE 1104(11):1613–1641, 1984).

A theoretical model of thermo-dynamic interpretation for the Pettit effect and associated heat transport processes

Based on several fundamental preassumptions, a one-dimensional convection-diffusion equation for heat transport inside liquid wedges causing the Pettit effect is proposed. With a hot or cold thermode placed at the middle of the liquid wedge, the average temperature of the liquid wedge determined from the convection-diffusion equation proposed shows a maximum, which corresponds to a particular liquid flow rate. The state achieved at this maximum temperature is believed to be the most stable for its minimum interfacial energy. The theory suggests a thermodynamic mechanism, which drives the liquid to flow in directions corresponding to those observed in experiments. It is believed that this work improves the thermodynamic interpretation proposed previously since the new-form convection-diffusion equation is more rigorously deduced and is thus more accurate. In addition, the work also presents a detailed theoretical analysis for heat transport. The results show that, in practical situations, the manifested heat-transport behaviors of a liquid wedge are governed by conductive heat transfer because convective heat flow is self-balancing due to the restriction by the law of mass conservation. Meanwhile, based on the asymmetric features of the conductive heat flows transiting within two different halves of the liquid wedge, a closed-loop formed by connecting a hot-thermode-driven liquid wedge and a cold-thermode-driven liquid wedge is proposed such that a hot thermode-cold thermode loop can lead to controllable heat transfer with which targeted heating or cooling may be realized. The effect may reveal the technical principles upon which novel small-size thermal engines, pumps, heaters, and coolers can be built.

A simple approach to one-dimensional convection-diffusion equation and its fractional modification for E reaction arising in rotating disk electrodes

A convection-diffusion equation for E reaction arising in rotating disk electrodes is discussed and solved by Taylor series method and Pade approximation. A fractional modification is suggested to deal with the porous property of the electrodes, the effect of the porous structure on the convection-diffusion process is elucidated.

Analysis of Quantization Error in Financial Pricing via Finite Difference Methods

In this paper, we study the error of a second order finite difference scheme for the one-dimensional convection-diffusion equation. We consider non-smooth initial conditions commonly encountered in financial pricing applications. For these initial conditions, we establish the explicit expression of the quantization error, which is loosely defined as the error of the numerical solution due to the placement of the point of non-smoothness on the numerical grid. Based on our analysis, we study the issue of optimal placement of such non-smoothness points on the grid, and the effect of smoothing operators on quantization errors.

Analysis of Quantization Error in Financial Pricing via Finite Difference Methods

In this paper, we study the error of a second order finite difference scheme for the one-dimensional convection-diffusion equation. We consider nonsmooth initial conditions commonly encountered in financial pricing applications. For these initial conditions, we establish the explicit expression of the quantization error, which is loosely defined as the error of the numerical solution due to the placement of the point of nonsmoothness on the numerical grid. Based on our analysis, we study the issue of optimal placement of such nonsmoothness points on the grid, and the effect of smoothing operators on quantization errors.

Delay-dependent stability analysis of convection-diffusion equations with delay

This paper is concerned with the delay-dependent stability analysis of the one-dimensional and two-dimensional convection-diffusion equations with time delay. Some sufficient and necessary conditions for the asymptotical stability of the equations are proposed. These results lay the foundation for the numerical stability analysis of such kinds of equations.

Macroscopic model and truncation error of discrete Boltzmann method

A derivation procedure to secure the macroscopically equivalent equation and its truncation error for discrete Boltzmann method is proffered in this paper. Essential presumptions of two time scales and a small parameter in the Chapman–Enskog expansion are disposed of in the present formulation. Equilibrium particle distribution function instead of its original non-equilibrium form is chosen as key variable in the derivation route. Taylor series expansion encompassing fundamental algebraic manipulations is adequate to realize the macroscopically differential counterpart. A self-contained and comprehensive practice for the linear one-dimensional convection–diffusion equation is illustrated in details. Numerical validations on the incurred truncation error in one- and two-dimensional cases with various distribution functions are conducted to verify present formulation. As shown in the computational results, excellent agreement between numerical result and theoretical prediction are found in the test problems. Straightforward extensions to more complicated systems including convection–diffusion–reaction, multi-relaxation times in collision operator as well as multi-dimensional Navier–Stokes equations are also exposed in the Appendix to point out its expediency in solving complicated flow problems.

Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport–diffusion equation

In this paper, we prove explicit lower bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type involving the spectral fractional Laplace operator. We notably deduce the following striking result: in the case of the heat equation controlled on the boundary, Miller's conjecture formulated in Miller (2004) [16] is not verified. Moreover, we also give a new lower bound for the minimal time needed to ensure the uniform controllability of the one-dimensional convection–diffusion equation with negative speed controlled on the left boundary, proving that the conjecture formulated in Coron and Guerrero (2005) [2] concerning this problem is also not verified at least for negative speeds.The proof is based on complex analysis, and more precisely on a representation formula for entire functions of exponential type, and is quite related to the moment method.

Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems

The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit–explicit (IMEX) time discretization schemes, for solving one-dimensional convection–diffusion equations with a nonlinear convection. Both Runge–Kutta and multi-step IMEX methods are considered. By the aid of the energy method, we show that the IMEX LDG schemes are unconditionally stable for the nonlinear problems, in the sense that the time-step τ is only required to be upper-bounded by a positive constant which depends on the flow velocity and the diffusion coefficient, but is independent of the mesh size h. We also give optimal error estimates for the IMEX LDG schemes, under the same temporal condition, if a monotone numerical flux is adopted for the convection. Numerical experiments are given to verify our main results.

An r-Adaptive Technique for Unstructured Grids Based on the Segment Spring Analogy Method

A simple and effective r-adaptive technique for unstructured grids based on the segment spring analogy method is proposed. The finite element method and a corresponding error estimate method using second derivatives are used for computation. The traditional segment spring analogy method is modified, based on an idea of controlling the equilibrium length of the fictitious springs, and used for mesh adjustment. The principle of making numerical errors distributed uniformly over all elements is applied. Three numerical examples involving the one-dimensional (1D) convection-diffusion equation, the two-dimensional (2D) linear parabolic equation and the 2D Euler equations are presented and the effectiveness of the proposed r-type grid adaptive technique is examined.

Numerical Method for One-Dimensional Convection-diffusion EquationUsing Radical Basis Functions

In this paper, the meshless method is employed for the numerical solution of the one-dimensional (1D) convection-diffusion equation based on radical basis functions (RBFs). Coupled with the time discretization and the collocation method, the proposed method is a truly meshless method which requires neither domain nor boundary discretization. The algorithm is very simple so it is very easy to implement. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme