An efficient explicit/implicit domain decomposition method for convection‐diffusion equations

A nonoverlapping domain decomposition method for some time‐dependent convection‐diffusion equations is presented. It combines predictor‐corrector technique, modified upwind differences with explicit/implicit coupling to provide intrinsic parallelism, and unconditional stability while improving the accuracy. Both rigorous mathematical analysis and numerical experiments are carried out to illustrate the stability, accuracy, and parallelism. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Analytic solution of natural convection flow of a non‐newtonian fluid between two vertical flat plates by using decomposition method

An analysis has been performed to study the natural convection of a non‐Newtonian fluid between two infinite parallel vertical flat plates and the effects of the non‐Newtonian nature of fluid on the heat transfer are studied. The governing boundary layer and temperature equations for this problem are reduced to an ordinary form and are solved by Adomian decomposition method (ADM) and numerical method. Velocity and temperature profiles are shown graphically. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and non‐linear equations and so can be found widely applicable in engineering and science. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1384–1395, 2010     

A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time

In this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter ? leads to a parabolic right boundary layer. A collocation method consisting of cubic B ‐spline basis functions on an appropriate piecewise‐uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter‐uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds.     

Stability of a combined finite element ‐ finite volume discretization of convection‐diffusion equations

We consider a time‐dependent and a stationary convection‐diffusion equation. These equations are approximated by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix‐Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the nonstationary case, we use an implicit Euler approach for time discretization. This scheme is shown to be L²‐stable uniformly with respect to the diffusion coefficient. In addition, it turns out that stability is unconditional in the time‐dependent case. These results hold if the underlying grid satisfies a condition that is fulfilled, for example, by some structured meshes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 402–424, 2012     

An optimization‐based domain decomposition method for the Boussinesq equations

We study an optimization based domain decomposition method for the Boussinesq equations governing natural convection problems. Domain decomposition is cast into a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the interface between solid and fluid subdomains. We showthat solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined. Finite element approximations of the solutions of the optimality system are examined. The domain decomposition method is also reformulated as a nonlinear least‐squares problem and the results of some numerical experiments are given. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 1–25, 2002     

The tanh method: a tool for solving certain classes of non‐linear PDEs

The tanh (or hyperbolic tangent) method is a powerful technique to look for travelling waves when dealing with one‐dimensional non‐linear wave and evolution equations. In particular, this method is well suited for those problems where dispersion, convection and reaction–diffusion play an important role. To show the strength of this method we study a coupled set (the so‐called Boussinesq equations) which arises in the theory of non‐linear dispersive water waves. As a result, a solitary wave profile is found which generalizes an earlier result, the famous Korteweg‐de Vries solitary wave solution. Copyright © 2005 John Wiley & Sons, Ltd.     

An Eulerian‐Lagrangian substructuring domain decomposition method for unsteady‐state advection‐diffusion equations

We develop an Eulerian‐Lagrangian substructuring domain decomposition method for the solution of unsteady‐state advection‐diffusion transport equations. This method reduces to an Eulerian‐Lagrangian scheme within each subdomain and to a type of Dirichlet‐Neumann algorithm at subdomain interfaces. The method generates accurate and stable solutions that are free of artifacts even if large time‐steps are used in the simulation. Numerical experiments are presented to show the strong potential of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:565–583, 2001     

An Explicit-Implicit Predictor-Corrector Domain Decomposition Method for Time Dependent Multi-Dimensional Convection Diffusion Equations

The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms. In this work, we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique, modified upwind differences with explicitimplicit coupling, the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy. Moreover, for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictorcorrector or stabilized schemes. These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments.     

The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Parallel characteristic finite difference method for convection–diffusion equations

Based on the overlapping domain decomposition, an efficient parallel characteristic finite difference scheme is proposed for solving convection‐diffusion equations numerically. We give the optimal convergence order in error estimate analysis, which shows that we just need to iterate once or twice at each time level to reach the optimal convergence order. Numerical experiments also confirm the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 854–866, 2011     

A space‐time spectral method for one‐dimensional time fractional convection diffusion equations

In this paper, we propose a space‐time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space‐time spectral‐Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.     

Implicit–explicit multistep characteristic finite element methods for nonlinear convection–diffusion equations

Implicit–explicit multistep characteristic methods are given for convection‐dominated diffusion equations. Multistep difference along characteristics of the one‐order hyperbolic part of the equation is used for discretization in time, and finite element method is used to discrete the space variables. The resulting schemes are consistent, stable and very efficient. Optimal‐rate of convergence is proved. Also, a note is given for a paper published earlier© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A numerical solution for advection‐diffusion equation using dual reciprocity method

This article describes a numerical method based on the boundary integral equation and dual reciprocity method(DRM) for solving the one‐dimensional advection‐diffusion equations. The concept of DRM is used to convert the domain integral to the boundary that leads to an integration free method. The time derivative is approximated by the time‐stepping method. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of the new approach. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations

In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in \(L^2\)-norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.     

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

We propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Solving 2D time‐fractional diffusion equations by a pseudospectral method and Mittag‐Leffler function evaluation

Two‐dimensional time‐fractional diffusion equations with given initial condition and homogeneous Dirichlet boundary conditions in a bounded domain are considered. A semidiscrete approximation scheme based on the pseudospectral method to the time‐fractional diffusion equation leads to a system of ordinary fractional differential equations. To preserve the high accuracy of the spectral approximation, an approach based on the evaluation of the Mittag‐Leffler function on matrix arguments is used for the integration along the time variable. Some examples along with numerical experiments illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

A uniformly optimal‐order error estimate for a bilinear finite element method for degenerate convection‐diffusion equations

We prove an optimal‐order error estimate in a degenerate‐diffusion weighted energy norm for bilinear Galerkin finite element methods for two‐dimensional time‐dependent convection‐diffusion equations with degenerate diffusion. In the estimate, the generic constants depend only on certain Sobolev norms of the true solution but not the lower bound of the diffusion. This estimate, combined with a known stability estimate of the true solution of the governing partial differential equations, yields an optimal‐order estimate of the Galerkin finite element method, in which the generic constants depend only on the Sobolev norms of the initial and right side data. Preliminary numerical experiments were conducted to verify these estimates numerically. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Block‐centered upwind multistep difference method and convergence analysis for numerical simulation of oil reservoir

The three‐dimensional displacement of two‐phase flow in porous media is a preliminary problem of numerical simulation of energy science and mathematics. The mathematical model is formulated by a nonlinear system of partial differential equations to describe incompressible miscible case. The pressure is defined by an elliptic equation, and the concentration is defined by a convection‐dominated diffusion equation. The pressure generates Darcy velocity and controls the dynamic change of concentration. We adopt a conservative block‐centered scheme to approximate the pressure and Darcy velocity, and the accuracy of Darcy velocity is improved one order. We use a block‐centered upwind multistep method to solve the concentration, where the time derivative is approximated by multistep method, and the diffusion term and convection term are treated by a block‐centered scheme and an upwind scheme, respectively. The composite algorithm is effective to solve such a convection‐dominated problem, since numerical oscillation and dispersion are avoided and computational accuracy is improved. Block‐centered method is conservative, and the concentration and the adjoint function are computed simultaneously. This physical nature is important in numerical simulation of seepage fluid. Using the convergence theory and techniques of priori estimates, we derive optimal estimate error. Numerical experiments and data show the support and consistency of theoretical result. The argument in the present paper shows a powerful tool to solve the well‐known model problem.     

Parallel characteristic finite element method for time‐dependent convection–diffusion problem

Based on the overlapping‐domain decomposition and parallel subspace correction method, a new parallel algorithm is established for solving time‐dependent convection–diffusion problem with characteristic finite element scheme. The algorithm is fully parallel. We analyze the convergence of this algorithm, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration times and sub‐domains overlapping degree. Both theoretical analysis and numerical results suggest that only one or two iterations are needed to reach to optimal accuracy at each time step. Copyright © 2010 John Wiley & Sons, Ltd.     

Modification of upwind finite difference fractional step methods by the transient state of the semiconductor device

The mathematical model of the three‐dimensional semiconductor devices of heat conduction is described by a system of four quasi‐linear partial differential equations for initial boundary value problem. One equation of elliptic form is for the electric potential; two equations of convection‐dominated diffusion type are for the electron and hole concentration; and one heat conduction equation is for temperature. Upwind finite difference fractional step methods are put forward. Some techniques, such as calculus of variations, energy method multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates and techniques are adopted. Optimal order estimates in L² norm are derived to determine the error in the approximate solution.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008