Integrated fast and high‐accuracy computation of convection diffusion equations using multiscale multigrid method

We present an explicit sixth‐order compact finite difference scheme for fast high‐accuracy numerical solutions of the two‐dimensional convection diffusion equation with variable coefficients. The sixth‐order scheme is based on the well‐known fourth‐order compact (FOC) scheme, the Richardson extrapolation technique, and an operator interpolation scheme. For a particular implementation, we use multiscale multigrid method to compute the fourth‐order solutions on both the coarse grid and the fine grid. Then, an operator interpolation scheme combined with the Richardson extrapolation technique is used to compute a sixth‐order accurate fine grid solution. We compare the computed accuracy and the implementation cost of the new scheme with the standard nine‐point FOC scheme and Sun–Zhang's sixth‐order method. Two convection diffusion problems are solved numerically to validate our proposed sixth‐order scheme. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A Fast Augmented Lagrangian Method for Euler's Elastica Models

In this paper, a fast algorithm for Euler's elastica functional is proposed, in which the Euler's elastica functional is reformulated as a constrained minimization problem. Combining the augmented Lagrangian method and operator splitting techniques, the resulting saddle-point problem is solved by a serial of subproblems. To tackle the nonlinear constraints arising in the model, a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution. We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic, real-world and medical images for image denoising, image inpainting and image zooming problems.     

Analysis of a decoupled time‐stepping algorithm for reduced MHD system modeling magneto‐convection

In this article, we analyze the stability and error estimate of a decoupled algorithm for a magneto‐convection problem. Magneto‐convection is assumed to be modeled by a coupled system of reduced magneto‐hydrodynamic (RMHD) equations and convection‐diffusion equation. The proposed algorithm applies the second‐order backward difference formula in time and finite element in space. To obtain a noniterative decouple algorithm from the fully discrete nonlinear system, we use a second‐order extrapolation in time to the nonlinear terms such that their skew symmetry properties are preserved. We prove the stability of the algorithm and derive error estimates without assuming any stability conditions. The algorithm is unconditionally stable and requires the solution of one RMHD problem and one convection‐diffusion equation per time step. Numerical test is presented that illustrates the accuracy and efficiency of the algorithm.     

Convergence of an operator splitting method on a bounded domain for a convection-diffusion-reaction system

We solve a convection-diffusion-sorption (reaction) system on a bounded domain with dominant convection using an operator splitting method. The model arises in contaminant transport in groundwater induced by a dual-well, or in controlled laboratory experiments. The operator splitting transforms the original problem to three subproblems: nonlinear convection, nonlinear diffusion, and a reaction problem, each with its own boundary conditions. The transport equation is solved by a Riemann solver, the diffusion one by a finite volume method, and the reaction equation by an approximation of an integral equation. This approach has proved to be very successful in solving the problem, but the convergence properties where not fully known. We show how the boundary conditions must be taken into account, and prove convergence in L1,loc of the fully discrete splitting procedure to the very weak solution of the original system based on compactness arguments via total variation estimates. Generally, this is the best convergence obtained for this type of approximation. The derivation indicates limitations of the approach, being able to consider only some types of boundary conditions. A sample numerical experiment of a problem with an analytical solution is given, showing the stated efficiency of the method.     

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.     

A combined BDF‐semismooth Newton approach for time‐dependent Bingham flow

This article is devoted to the numerical simulation of time‐dependent convective Bingham flow in cavities. Motivated by a primal‐dual regularization of the stationary model, a family of regularized time‐dependent problems is introduced. Well posedness of the regularized problems is proved, and convergence of the regularized solutions to a solution of the original multiplier system is verified. For the numerical solution of each regularized multiplier system, a fully discrete approach is studied. A stable finite element approximation in space together with a second‐order backward differentiation formula for the time discretization are proposed. The discretization scheme yields a system of Newton differentiable nonlinear equations in each time step, for which a semismooth Newton algorithm is used. We present two numerical experiments to verify the main properties of the proposed approach. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Implicit–explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension

Nonlinear convection–diffusion equations with nonlocal flux and possibly degenerate diffusion arise in various contexts including interacting gases, porous media flows, and collective behavior in biology. Their numerical solution by an explicit finite difference method is costly due to the necessity of discretizing a local spatial convolution for each evaluation of the convective numerical flux, and due to the disadvantageous Courant–Friedrichs–Lewy (CFL) condition incurred by the diffusion term. Based on explicit schemes for such models devised in the study of Carrillo et al. a second‐order implicit–explicit Runge–Kutta (IMEX‐RK) method can be formulated. This method avoids the restrictive time step limitation of explicit schemes since the diffusion term is handled implicitly, but entails the necessity to solve nonlinear algebraic systems in every time step. It is proven that this method is well defined. Numerical experiments illustrate that for fine discretizations it is more efficient in terms of reduction of error versus central processing unit time than the original explicit method. One of the test cases is given by a strongly degenerate parabolic, nonlocal equation modeling aggregation in study of Betancourt et al. This model can be transformed to a local partial differential equation that can be solved numerically easily to generate a reference solution for the IMEX‐RK method, but is limited to one space dimension.     

A fast singly diagonally implicit runge–kutta method for solving 1D unsteady convection‐diffusion equations

In this article, a fast singly diagonally implicit Runge–Kutta method is designed to solve unsteady one‐dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three‐stage singly diagonally implicit Runge–Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth‐order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 788–812, 2014     

Multilevel numerical solutions of convection‐dominated diffusion problems by spline wavelets

In this article, we utilize spline wavelets to establish an adaptive multilevel numerical scheme for time‐dependent convection‐dominated diffusion problems within the frameworks of Galerkin formulation and Eulerian‐Lagrangian localized adjoint methods (ELLAM). In particular, we shall use linear Chui‐Quak semi‐orthogonal wavelets, which have explicit expressions and compact supports. Therefore, both the diffusion term and boundary conditions in the convection‐diffusion problems can be readily handled. Strategies for efficiently implementing the scheme are discussed and numerical results are interpreted from the viewpoint of nonlinear approximation. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Line search SQP method with a flexible step acceptance procedure

This paper describes a new algorithm for nonlinear programming with inequality constraints. The proposed approach solves a sequence of quadratic programming subproblems via the line search technique and uses a new globalization strategy. An increased flexibility in the step acceptance procedure is designed to promote long productive steps for fast convergence. Global convergence is proved under some reasonable assumptions and preliminary numerical results are presented.     

A shift‐adaptive meshfree method for solving a class of initial‐boundary value problems with moving boundaries in one‐dimensional domain

A new shift‐adaptive meshfree method for solving a class of time‐dependent partial differential equations (PDEs) in a bounded domain (one‐dimensional domain) with moving boundaries and nonhomogeneous boundary conditions is introduced. The radial basis function (RBF) collocation method is combined with the finite difference scheme, because, unlike with Kansa's method, nonlinear PDEs can be converted to a system of linear equations. The grid‐free property of the RBF method is exploited, and a new adaptive algorithm is used to choose the location of the collocation points in the first time step only. In fact, instead of applying the adaptive algorithm on the entire domain of the problem (like with other existing adaptive algorithms), the new adaptive algorithm can be applied only on time steps. Furthermore, because of the radial property of the RBFs, the new adaptive strategy is applied only on the first time step; in the other time steps, the adaptive nodes (obtained in the first time step) are shifted. Thus, only one small system of linear equations must be solved (by LU decomposition method) rather than a large linear or nonlinear system of equations as in Kansa's method (adaptive strategy applied to entire domain), or a large number of small linear systems of equations in the adaptive strategy on each time step. This saves a lot in time and memory usage. Also, Stability analysis is obtained for our scheme, using Von Neumann stability analysis method. Results show that the new method is capable of reducing the number of nodes in the grid without compromising the accuracy of the solution, and the adaptive grading scheme is effective in localizing oscillations due to sharp gradients or discontinuities in the solution. The efficiency and effectiveness of the proposed procedure is examined by adaptively solving two difficult benchmark problems, including a regularized long‐wave equation and a Korteweg‐de Vries problem. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1622–1646, 2016     

On entropy stable schemes for degenerate parabolic multispecies kinematic flow models

Entropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.     

Numerical approximation of the conservative Allen–Cahn equation by operator splitting method

In this paper, a second‐order fast explicit operator splitting method is proposed to solve the mass‐conserving Allen–Cahn equation with a space–time‐dependent Lagrange multiplier. The space–time‐dependent Lagrange multiplier can preserve the volume of the system and keep small features. Moreover, we analyze the discrete maximum principle and the convergence rate of the fast explicit operator splitting method. The proposed numerical scheme is of spectral accuracy in space and of second‐order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, efficiency, mass conservation, and stability of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.     

A defect correction method for the time‐dependent Navier‐Stokes equations

A method for solving the time dependent Navier‐Stokes equations, aiming at higher Reynolds' number, is presented. The direct numerical simulation of flows with high Reynolds' number is computationally expensive. The method presented is unconditionally stable, computationally cheap, and gives an accurate approximation to the quantities sought. In the defect step, the artificial viscosity parameter is added to the inverse Reynolds number as a stability factor, and the system is antidiffused in the correction step. Stability of the method is proven, and the error estimations for velocity and pressure are derived for the one‐ and two‐step defect‐correction methods. The spacial error is O(h) for the one‐step defect‐correction method, and O(h²) for the two‐step method, where h is the diameter of the mesh. The method is compared to an alternative approach, and both methods are applied to a singularly perturbed convection–diffusion problem. The numerical results are given, which demonstrate the advantage (stability, no oscillations) of the method presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations

In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two‐dimensional linear and nonlinear time‐fractional modified anomalous subdiffusion equations and time‐fractional convection–diffusion equation. The fractional derivative of problems is described in the Riemann–Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique. Copyright © 2016 John Wiley & Sons, Ltd.     

An alternative framework to Lagrangian relaxation approach for job shop scheduling

A new Lagrangian relaxation (LR) approach is developed for job shop scheduling problems. In the approach, operation precedence constraints rather than machine capacity constraints are relaxed. The relaxed problem is decomposed into single or parallel machine scheduling subproblems. These subproblems, which are NP-complete in general, are approximately solved by using fast heuristic algorithms. The dual problem is solved by using a recently developed “surrogate subgradient method” that allows approximate optimization of the subproblems. Since the algorithms for subproblems do not depend on the time horizon of the scheduling problems and are very fast, our new LR approach is efficient, particularly for large problems with long time horizons. For these problems, the machine decomposition-based LR approach requires much less memory and computation time as compared to a part decomposition-based approach as demonstrated by numerical testing.     

广义非线性最小二乘问题的两个新方法

1．引言讨论如下的广义非线性最小二乘问题其中为常数（i＝1~m），W由于此问题的特殊形式，将此问题转化为如下两个子问题进行求解比较有效[1]子问题1．对每一固定的X，解得子问题2。对子问题1的解，解对两个子问题的求解，[1]中给出了一种有效的方法。然而在两个子问题的已有求解方法中，关于方法收敛速度的讨论非常少见，本文给出了求解这两个子问题的两个算法，并证明了算法的超线性收敛性．为书写简单，以下约定：一个符号在（，L）处的值略去（，L），如V‘F＝＊‘列X，L）等·一个具有上标k和＊的符号分别表示其在（x‘，t‘）和…     

Nonstandard finite difference schemes for a class of generalized convection–diffusion–reaction equations

In this work, a class of nonstandard finite difference (NSFD) schemes are proposed to approximate the solutions of a class of generalized convection–diffusion–reaction equations. First, in the case of no diffusion, two exact finite difference schemes are presented using the method of characteristics. Based on these two exact schemes, a class of exact schemes are presented by introducing a parameter α. Second, since the forms of these exact schemes are so complicated that they are not convenient to use, a class of NSFD schemes are derived from the exact schemes using numerical approximations. It follows that, under certain conditions about denominator function of time‐step sizes, these NSFD schemes are elementary stable and the solutions are positive and bounded. Third, by means of the Mickens' technique of subequations, a new class of implicit NSFD schemes are constructed for the full convection–diffusion–reaction equations. It is shown that, under certain parameters set, these NSFD schemes are capable of preserving the non‐negativity and boundedness of the analytical solutions. Finally, some numerical simulations are provided to verify the validity of our analytical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1288–1309, 2015     

On the convergence of an inexact Gauss–Newton trust-region method for nonlinear least-squares problems with simple bounds

We introduce an inexact Gauss–Newton trust-region method for solving bound-constrained nonlinear least-squares problems where, at each iteration, a trust-region subproblem is approximately solved by the Conjugate Gradient method. Provided a suitable control on the accuracy to which we attempt to solve the subproblems, we prove that the method has global and asymptotic fast convergence properties. Some numerical illustration is also presented.