Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L2-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.

New complex variable meshless method for advection-diffusion problems

In this paper,an improved complex variable meshless method（ICVMM） for two-dimensional advection-diffusion problems is developed based on improved complex variable moving least-square（ICVMLS） approximation.The equivalent functional of two-dimensional advection-diffusion problems is formed,the variation method is used to obtain the equation system,and the penalty method is employed to impose the essential boundary conditions.The difference method for twopoint boundary value problems is used to obtain the discrete equations.Then the corresponding formulas of the ICVMM for advection-diffusion problems are presented.Two numerical examples with different node distributions are used to validate and investigate the accuracy and efficiency of the new method in this paper.It is shown that ICVMM is very effective for advection-diffusion problems,and has a good convergent character,accuracy,and computational efficiency.

Air Pollution Steady-State Advection-Diffusion Equation: The General Three-Dimensional Solution

Atmospheric air pollution turbulent fluxes can be assumed to be proportional to the mean concentration gradient. This assumption, along with the equation of continuity, leads to the advection-diffusion equation. Many models simulating air pollution dispersion are based upon the solution (numerical or analytical) of the advection-diffusion equation as- suming turbulence parameterization for realistic physical scenarios. We present the general steady three-dimensional solution of the advection-diffusion equation considering a vertically inhomogeneous atmospheric boundary layer for arbitrary vertical profiles of wind and eddy-diffusion coefficients. Numerical results and comparison with experimental data are shown.

Comparison between Non-Gaussian Puff Model and a Model Based on a Time-Dependent Solution of Advection-Diffusion Equation

A comparison between a non-Gaussian puff model and an advanced time-dependent model to simulate the pollutant dispersion in the Planetary Boundary Layer is presented. The puff model is based on a general technique for solving the K-equation, using the truncated Gram-Charlier expansion (type A) of the concentration field and finite set equations for the corresponding moments. The other model (named ADMM: Analytical Dispersion Multilayers Model) is an semi- analytical solution to the time-dependent two-dimensional advection-diffusion equation based on a discretization of the PBL in N sub-layers;in each sub-layers the advection-diffusion equation is solved by the Laplace transform technique, considering an average value for eddy diffusivity and the wind speed. A preliminary performance evaluation is shown in the case of continuous emission from an elevated source in a variable boundary layer. Both models were able to correctly reproduce the concentration field measured and so to be used as operative air pollution models.

Solution of Nonlinear Advection-Diffusion Equations via Linear Fractional Map Type Nonlinear QCA

Linear fractional map type (LFMT) nonlinear QCA (NLQCA), one of the simplest reversible NLQCA is studied analytically as well as numerically. Linear advection equation or Time Dependent Schrödinger Equation (TDSE) is obtained from the continuum limit of linear QCA. Similarly it is found that some nonlinear advection-diffusion equations including inviscid Burgers equation and porous-medium equation are obtained from LFMT NLQCA.

Improvement of the One-dimensional Vertical Advection-diffusion Model in Seawater

The classical 1-D vertical advection-diffusion model was improved in this work. The main advantages of the improved model over the previous one are: 1) The applicable condition of the 1-D model is made clear in the improved model, in that it is substantively applicable only to a vertical domain on which two end-member water masses are mixing. 2) The substitution of parameter f(z) in the equation of the classical 1-D model with end-member fraction f 1z makes the model more precisely and easily solved. 3) All the terms in the improved model equation have specific physical meanings, which makes the model easily understood. Practical application of the improved model to predict the vertical profiles of dissolved oxygen and micronutrients in abyssal ocean water of the North Pacific proved that the improvement of the 1-D advection-diffusion model is successful and practicable.

PESEDOSPECTRAL-MULTIWAVELET-GALERKIN METHOD FOR ADVECTION-DIFFUSION PROBLEM WITH COMPLEX BOUNDARY

The element of pesedospectral-multiwavelet-Galerkin method, and how tocombine it with penalty method for treating boundary conditions are given. Multiwavelet bases don'toverlap on the given scale, and possess the same compact set in a group of several functions, sothey can be directly used to the numerical discretion on the finite interval. Numerical tests showthat general boundary conditions can be enforced with the penalty method, and thatpesedospectral-multiwavelet-Galerkin method can well track the solutions' development. This alsoproves that pesedospectral-multiwavelet-Galerkin method is effective.

AN ASYMPTOTIC BEHAVIOR AND A POSTERIORI ERROR ESTIMATES FOR THE GENERALIZED SCHWARTZ METHOD OF ADVECTION-DIFFUSION EQUATION

In this paper, a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved by using the Euler time scheme combined with Galerkin spatial method. Furthermore, an asymptotic behavior in Sobolev norm is de- duced using Benssoussau-Lions＇ algorithm. Finally, the results of some numerical experiments are presented to support the theory.

Analysis of variable coefficient advection-diffusion problems via complex variable reproducing kernel particle method

The complex variable reproducing kernel particle method （CVRKPM） of solving two-dimensional variable coefficient advection-diffusion problems is presented in this paper. The advantage of the CVRKPM is that the shape function of a two-dimensional problem is formed with a one-dimensional basis function. The Galerkin weak form is employed to obtain the discretized system equation, and the penalty method is used to apply the essential boundary conditions. Then the corresponding formulae of the CVRKPM for two-dimensional variable coefficient advection-diffusion problems are obtained. Two numerical examples are given to show that the method in this paper has greater accuracy and computational efficiency than the conventional meshless method such as reproducing the kernel particle method （RKPM） and the element- free Galerkin （EFG） method.

Two Modified QUICK Schemes for Advection-Diffusion Equation of Pollutants on Unstructured Grids

In this paper, two modified QUICK schemes, namely Q-QUICK and UQ-QUICK, for improving the preci-sion of convective flux approximation are verified in advection-diffusion equation of pollutants on unstruc-tured grids. The constructed auxiliary nodes for Q-QUICK/UQ-QUICK are composed of two neighboring nodes plus the next upwind node, the later node is generated from intersection of the line of current neighboring nodes and their corresponding interfaces. 2D unsteady advection-diffusion equation of pollut-ants is conducted for their verifications on unstructured grids. The numerical results show that Q-QUICK and UQ-QUICK have similar computational accuracy to the central difference scheme and similar numerical stability to upwind difference scheme after applying the deferred correction method. In addition, their corre-sponding CPU times are approximately equivalent to those of traditional difference schemes and their abili-ties for adapting high grid deformation are robust.

Finite Element Analysis for Singularly Perturbed Advection-Diffusion Robin Boundary Values Problem

A singularly perturbed advection-diffusion two-point Robin boundary value problem whose solution has a single boundary layer is considered. Based on the piecewise linear polynomial approximation, the finite element method is applied to the problem. Estimation of the error between solution and the finite element approximation is given in energy norm on shishkin-type mesh.

Discontinuous Legendre Wavelet Galerkin Method for One-Dimensional Advection-Diffusion Equation

This paper presents discontinuous Legendre wavelet Galerkin (DLWG) approach for solving one-dimensional advection-diffusion equation (ADE). Variational formulation of this type equation and corresponding numerical fluxes are devised by utilizing the advantages of both the Legendre wavelet bases and discontinuous Galerkin (DG) method. The distinctive features of the proposed method are its simple applicability for a variety of boundary conditions and able to effectively approximate the solution of PDEs with less storage space and execution. The results of a numerical experiment are provided to verify the efficiency of the designed new technique.

An Approximate Riemann Solver for Advection-Diffusion Based on the Generalized Riemann Problem

We construct an approximate Riemann solver for scalar advection-diffusion equations with piecewise polynomial initial data.The objective is to handle advection and diffusion simultaneously to reduce the inherent numerical diffusion produced by the usual advection flux calculations.The approximate solution is based on the weak formulation of the Riemann problem and is solved within a space-time discontinuous Galerkin approach with two subregions.The novel generalized Riemann solver produces piecewise polynomial solutions of the Riemann problem.In conjunction with a recovery polynomial,the Riemann solver is then applied to define the numerical flux within a finite volume method.Numerical results for a piecewise linear and a piecewise parabolic approximation are shown.These results indicate a reduction in numerical dissipation compared with the conventional separated flux calculation of advection and diffusion.Also,it is shown that using the proposed solver only in the vicinity of discontinuities gives way to an accurate and efficient finite volume scheme.

A Numerical Algorithm for the Caputo Tempered Fractional Advection-Diffusion Equation

By transforming the Caputo tempered fractional advection-diffusion equation into the Riemann–Liouville tempered fractional advection-diffusion equation,and then using the fractional-compact Grünwald–Letnikov tempered difference operator to approximate the Riemann–Liouville tempered fractional partial derivative,the fractional central difference operator to discritize the space Riesz fractional partial derivative,and the classical central difference formula to discretize the advection term,a numerical algorithm is constructed for solving the Caputo tempered fractional advection-diffusion equation.The stability and the convergence analysis of the numerical method are given.Numerical experiments show that the numerical method is effective.

Alternating Group Explicit Iterative Methods for One-Dimensional Advection-Diffusion Equation

The finite difference method such as alternating group iterative methods is useful in numerical method for evolutionary equations and this is the standard approach taken in this paper. Alternating group explicit (AGE) iterative methods for one-dimensional convection diffusion equations problems are given. The stability and convergence are analyzed by the linear method. Numerical results of the model problem are taken. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show that the behavior of the method with emphasis on treatment of boundary conditions is valuable.

A Third-Order Scheme for Numerical Fluxes to Guarantee Non-Negative Coefficients for Advection-Diffusion Equations

According to Godunov theorem for numerical calculations of advection equations, there exist no high-er-order schemes with constant positive difference coefficients in a family of polynomial schemes with an accuracy exceeding the first-order. In case of advection-diffusion equations, so far there have been not found stable schemes with positive difference coefficients in a family of numerical schemes exceeding the second-order accuracy. We propose a third-order computational scheme for numerical fluxes to guarantee the non-negative difference coefficients of resulting finite difference equations for advection-diffusion equations. The present scheme is optimized so as to minimize truncation errors for the numerical fluxes while fulfilling the positivity condition of the difference coefficients which are variable depending on the local Courant number and diffusion number. The feature of the present optimized scheme consists in keeping the third-order accuracy anywhere without any numerical flux limiter by using the same stencil number as convemtional third-order shemes such as KAWAMURA and UTOPIA schemes. We extend the present method into multi-dimensional equations. Numerical experiments for linear and nonlinear advection-diffusion equations were performed and the present scheme’s applicability to nonlinear Burger’s equation was confirmed.

High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation

We present a high-order Galerkin method in both space and time for the 1D unsteady linear advection-diffusion equation. Three Interior Penalty Discontinuous Galerkin (IPDG) schemes are detailed for the space discretization, while the time integration is performed at the same order of accuracy thanks to an Arbitrary high order DERivatives (ADER) method. The orders of convergence of the three ADER-IPDG methods are carefully examined through numerical illustrations, showing that the approach is consistent, accurate, and efficient. The numerical results indicate that the symmetric version of IPDG is typically more accurate and more efficient compared to the other approaches.

UNIFORM OPTIMAL-ORDER ESTIMATES FOR FINITE ELEMENT METHODS FOR ADVECTION-DIFFUSION EQUATIONS

This article summarizes our recent work on uniform error estimates for various finite elementmethods for time-dependent advection-diffusion equations.

Application of Advection-Diffusion Routing Model to Flood Wave Propagation： A Case Study on Big Piney River, Missouri USA

Flood wave propagation modeling is of critical importance to advancing water resources management and protecting human life and property. In this study, we investigated how the advection-diffusion routing model performed in flood wave propagation on a 16 km long downstream section of the Big Piney River, MO. Model performance was based on gaging station data at the upstream and downstream cross sections. We demonstrated with advection-diffusion theory that for small differences in watershed drainage area between the two river cross sections, inflow along the reach mainly contributes to the downstream hydrograph＇s rising limb and not to the falling limb. The downstream hydrograph＇s falling limb is primarily determined by the propagated flood wave originating at the upstream cross section. This research suggests the parameter for the advectiondiffusion routing model can be calibrated by fitting the hydrograph falling limb. Application of the advection diffusion model to the flood wave of January 29, 2013 supports our theoretical finding that the propagated flood wave determines the downstream cross section falling limb, and the model has good performance in our test examples.

A POSTERIORI ENERGY-NORM ERROR ESTIMATES FOR ADVECTION-DIFFUSION EQUATIONS APPROXIMATED BY WEIGHTED INTERIOR PENALTY METHODS

We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms： a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfiirth on continuous finite elements, that the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor.