Solving Convection-diffusion Equations Research Articles

The second-order modified upwind PPM characteristic difference method and analysis for solving convection-diffusion equations

The second-order modified upwind PPM characteristic difference method and analysis for solving convection-diffusion equations

Combining wall interactions, fluid momentum balances and the Maxwell-Stefan equations for gas transport in porous media: An alternative approach

This paper presents a modelling framework for incorporating wall interactions into the Maxwell-Stefan equations for ideal gases in porous media. In contrast to already existing models in literature, such as the Dusty Gas, Binary Friction and Modified Binary Friction models, the framework makes use of N−1 (with N being the number of gas species) independent mass balances for the gas species in combination with a total fluid momentum balance. This approach has the advantage that already existing numerical schemes for fluid momentum balances, such as Darcy’s law or the Brinkman equations, can be used with only minor modifications. It also allows for separate initialisation of the velocity and pressure fields, something that is often beneficial when solving convection–diffusion equations numerically. The derived model is validated against both stationary and time-dependent experimental data from literature for binary Ar-He systems in porous media, using Knudsen diffusivities to define the wall friction terms.

High-Effectiveness and -Accuracy Difference Scheme Based on Nonuniform Grids for Solving Convection–Diffusion Equations with Boundary Layers

In this paper, some rational high-accuracy compact finite difference schemes on nonuniform grids (NRHOC) are introduced for solving convection–diffusion equations. The derived NRHOC schemes not only can suppress the oscillatory property of numerical solutions but can also obtain a high-accuracy approximate solution, and they can effectively solve the convection–diffusion problem with boundary layers by flexibly adjusting the discrete grid, which can be obtained with the singularity in the computational region. Three numerical experiments with boundary layers are conducted to verify the accuracy of the proposed NRHOC schemes. We compare the computed results with the analytical solutions, the results of the rational high-accuracy compact finite difference schemes on uniform grids (RHOC), and the other schemes in the literature. For all test problems, good computed results are obtained with the presented NRHOC schemes. It is shown that the presented NRHOC schemes have a better resolution for the solution of convection-dominated problems.

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.

A stable radial basis function partition of unity method for solving convection–diffusion equations on surfaces

The radial basis function partition of unity (RBF-PU) method is used to solve various types of partial differential equations, since it is based on scattered node sets that can be easily implemented. In this paper, a framework for the RBF-PU method is proposed to solve the convection–diffusion equations on closed surfaces. The surface gradient and Laplace–Beltrami operators are discretized based on the RBF-PU method using the polyharmonic spline radial basis functions augmented with polynomials. Since the RBF-PU method is a local meshless method, an advantage is that the differential matrix obtained by discretizing the surface differential operator is sparse. Additionally, solving convection-dominated problems or transport equations directly using RBF-PU method can produce numerical oscillations. For this reason, we add hyperviscosity term to improve the stability of the RBF-PU method for solving the convection–diffusion equations on surfaces, and numerical experiments show that such a combination has higher-order convergence rates. Finally, we solve the discontinuous source problem on the torus and the Turing system with convection term on different surfaces to demonstrate the effectiveness of the proposed method.

Comparative Study of Uniform and Graded Meshes for Solving Convection-Diffusion Equation with Quadratic Source

Due to its fundamental nature, the problems of convection-diffusion are discussed in various aviation, science and engineering applications. Among major applications are in the study of the dynamics of aircraft wake vortex and its interaction with turbulent jet which is a very serious hazard in aviation. Other applications include those in the investigation of intrusive sampling of jet engine exhaust gases, and the effectiveness of hot fluid injection in the removal of ice on aircraft wings. The numerical solutions of convection-diffusion require propermeshing schemes. Among major meshes in computational fluid dynamics are those of uniform, piecewise-uniform, graded, and hybrid over which the solutions of discretized governing equations are found. Bad solutions as spurious fluctuations, over-or under-predictions, and excessive computation time might be the results of unwitting application of the meshes. Accentuating comparative effectiveness of two meshes, namely uniform mesh and graded mesh with mesh expansion factor, this paper takes the solution of aconvection-diffusion equation with quadratic source term into account. The problem is solved by assigning several values of mesh expansion factor to graded mesh, while mesh number is kept constant. The factors are calculated based on the generalization oftheir logarithmically linear relationship with low Peclet numbers derived in previous work. Based on the values of Peclet number, five test cases are considered. Graded mesh is proven relatively more robust, particularly due the solution on the mesh beingfree from spurious fluctuation. Furthermore, the accuracy level of the solution of up to two order of magnitude higher is obtained. The mesh expansion factor therefore contributes to a stable and highly accurate solution corresponding to all interested Peclet numbers.

The mass‐conserving domain decomposition method for convection diffusion equations with variable coefficients

In this paper, a conserved domain decomposition method for solving convection–diffusion equations with variable coefficients is analyzed. The interface fluxes over the subdomains are firstly obtained by the explicit fluxes scheme. Secondly, the interior solutions and fluxes over each subdomains are computed by the modified upwind implicit scheme. Then, the interface fluxes are corrected by the obtained solutions. We prove rigorously that our scheme is mass conservative, unconditionally stable, and of second‐order convergence in spatial step. Numerical examples test the theoretical analysis and efficiencies. Lastly, we extend our scheme to the nonlinear convection–diffusion equations and give the error estimate.

Mass conservative characteristic finite difference method for convection–diffusion equations

In the paper, the space second-order conservative characteristic finite difference method for solving convection–diffusion equations is developed and analysed. Applying the directional derivative and mass correction technique, the convection–diffusion equations are changed into the parabolic equations, where the convection term and unsteady term are considered as one term. The splitting implicit solution-flux coupled scheme computes the solutions and fluxes on the staggered meshes. The mass conservation, stability and convergence of our proposed characteristic difference method are analysed theoretically. Numerical experiments are presented to illustrate mass conservation and convergence.

CONVERGENCE ANALYSIS OF NEW ADDITIVE SCHWARZ METHOD FOR SOLVING NONSELFADJOINT ELLIPTIC PROBLEMS

In this paper, we present a two-level additive Schwarz method for solving a system arising from the discretization of the nonselfadjoint elliptic equation. By employing the Cauchy-Schwarz-type inequality and stable decomposition under the energy norm, we obtain the optimal convergence theory for the proposed method. It shows that the convergence rate is bounded and independent of the fine mesh size and the number of subdomains. Some numerical results are reported to verify our theoretical result. Moreover, we demonstrate the benefit compared to the classical two-level additive Schwarz algorithm for solving convection-diffusion equations.

Wavelet-optimized compact finite difference method for convection–diffusion equations

Abstract In this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case.

Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations

This article proposes a space–time meshless approach based on the transient radial polynomial series function (TRPSF) for solving convection–diffusion equations. We adopted the TRPSF as the basis function for the spatial and temporal discretization of the convection–diffusion equation. The TRPSF is constructed in the space–time domain, which is a combination of n–dimensional Euclidean space and time into an n + 1–dimensional manifold. Because the initial and boundary conditions were applied on the space–time domain boundaries, we converted the transient problem into an inverse boundary value problem. Additionally, all partial derivatives of the proposed TRPSF are a series of continuous functions, which are nonsingular and smooth. Solutions were approximated by solving the system of simultaneous equations formulated from the boundary, source, and internal collocation points. Numerical examples including stationary and nonstationary convection–diffusion problems were employed. The numerical solutions revealed that the proposed space–time meshless approach may achieve more accurate numerical solutions than those obtained by using the conventional radial basis function (RBF) with the time-marching scheme. Furthermore, the numerical examples indicated that the TRPSF is more stable and accurate than other RBFs for solving the convection–diffusion equation.

A semi-Lagrangian discontinuous Galerkin (DG) – local DG method for solving convection-diffusion equations

In this paper, we propose an efficient high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for solving linear convection-diffusion equations. The method generalizes our previous work on developing the SLDG method for transport equations [5], making it capable of handling additional diffusion and source terms. Within the DG framework, the solution is evolved along the characteristics; while the diffusion term is discretized by the local DG (LDG) method and integrated along characteristics by implicit Runge-Kutta methods together with source terms. The proposed method is named the ‘SLDG-LDG’ method and enjoys many attractive features of the DG and SL methods. These include the uniformly high order accuracy (e.g. third order) in space and in time, compact, mass conservative, and stability under large time stepping size. An L2 stability analysis is provided when the method is coupled with the first order backward Euler discretization. Effectiveness of the method are demonstrated by a group of numerical tests in one and two dimensions.

The theory of steady state current for chronoamperometric and cyclic voltammetry on rotating disk electrodes for EC’ and ECE reactions

A theory is presented for the shape of current curves measured at a rotating-disc electrode for the cases where the substrate reacts via for the E, EC′ and ECE reaction mechanisms respectively. An analytical expression of concentration and current is presented by solving convection-diffusion equations for the E and EC’ reaction mechanism. The closed-form of an analytical expression of concentration and current for ECE reaction mechanisms are also reported by solving the system of nonlinear reaction-convection-diffusion equations using homotopy-perturbation method. The solutions enable us to obtain the current response in chronoamperometry and steady-state voltammetry. Analytical results are also compared with the simulation result. An excellent agreement is observed between the series of solutions obtained by simulation solution. The approximate analytical expression derived here is useful to predict the dynamical behaviour of the system. The effect of the parameters on concentration and current are discussed and presented graphically.

A HBM approach for temperature and heat flux convection–diffusion equations and nonlinear problems

Solving convection diffusion equation is widely required in many fields of science, technology and engineering. The calculation is usually difficult and time-consuming. In this paper, a highly efficient method, the half boundary method (HBM), is proposed to solve the convection diffusion equation. The main idea of HBM is to reduce the order of the convection–diffusion equations by introducing a new variable and constructing the relations of the variables between the nodes inside the area and the nodes on half of the boundaries. Using the relations, the temperature and heat flux at any point can be calculated simultaneously, after obtaining the variables on the half of the boundaries. Because the unknown variables exist on only half of the boundaries, the computing matrix is reduced to only second order regardless of the number of nodes, the internal storage in the HBM is even less than that required in the finite volume method, making the HBM extremely fast and efficient. The validity and accuracy of the proposed method are investigated. Numerical studies for steady and unsteady, as well as nonlinear convection–diffusion equations, were carried out. The results show that HBM is more accurate than the finite volume method under identical grids conditions.

Cubic quasi-interpolation spline collocation method for solving convection–diffusion equations

In this paper, we use a cubic spline collocation method to solve a two dimensional convection–diffusion equation. More precisely, we approximate first and second order partial derivatives by those of cubic spline quasi-interpolants to produce a system of first order ordinary differential equations. The resulting system can be solved using MATLAB’s ode solver. Error estimates of quasi-interpolants which are used are given with full discussion. Furthermore, numerical examples are presented to show the validity of our methods.

对流扩散方程的非振荡守恒特征差分法

对流扩散方程的非振荡守恒特征差分法

A type of high order schemes for steady convection-diffusion problems

Based on the viewpoint that the flow direction can be accounted for by adding more weigh of the contribution of the upstream grid point in evaluating the transport fluxes at the cell faces for the convection-dominated flows, a type of high order schemes are proposed for solving convection-diffusion equations. These schemes share the same main algorithm which is developed by integrating the convection-diffusion equation expressed as the first order partial differential equations over selected regions and applying the numerical quadrature with a weighting parameter for approximating the resulting integrals. Interestingly, the error analysis allows us to determine the expressions of the weighting parameter for a series of schemes with the third order scheme as the lowest and the infinite order as the highest order scheme for the source free convection-diffusion problems. Numerical results show that the highest order scheme achieves almost the same accuracy as the exact solution, and does not induce any unphysical oscillation for the convection-dominated flows. Even the third order scheme shows obvious advantages over the traditional finite volume method (FVM) and QUICK scheme in dealing with the multi-dimensional convection-diffusion problems.

Inverse Lax–Wendroff procedure for numerical boundary conditions of convection–diffusion equations

We consider numerical boundary conditions for high order finite difference schemes for solving convection–diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax–Wendroff procedure [28] , in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, has been quite successful. In this paper, we extend this methodology to convection–diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection–diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier–Stokes equations, to demonstrate the good performance of our numerical boundary conditions.

Alternating evolution discontinuous Galerkin methods for convection–diffusion equations

In this work, we propose a high order alternating evolution discontinuous Galerkin (AEDG) method to solve convection-diffusion equations. The difficulties related to numerical fluxes of DG methods for diffusion problems have been a major issue of investigation in the literature. The AEDG scheme presented here is based on an alternating evolution system for convection-diffusion equations, and therefore no numerical fluxes are needed in the scheme formulation. Moreover, the method is shown to be consistent, conservative and stable. Numerical experiments are provided to show the goodness of the proposed method.

MLPG method based on moving kriging interpolation for solving convection–diffusion equations with integral condition

A formulation of the meshless local Petrov–Galerkin (MLPG) method based on the moving kriging interpolation (MK) is presented in this paper. The method is used for solving time-dependent convection–diffusion equations in two-dimensional spaces with the Dirichlet, Neumann, and non-local boundary conditions on a square domain. The method is developed based on the moving kriging interpolation method for constructing shape functions which have the Kronecker delta property. In the method, the test function in each sub-domain is chosen as the indicator function. The Crank–Nicolson method is chosen for temporal discretization. Two test problems are presented which demonstrate the easiness and accuracy of this method as shown by the relative error.