A CFL‐free explicit characteristic interior penalty scheme for linear advection‐reaction equations

We develop a CFL‐free, explicit characteristic interior penalty scheme (CHIPS) for one‐dimensional first‐order advection‐reaction equations by combining a Eulerian‐Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal‐order error estimate in the L² norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

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     

A two‐grid method for mixed finite‐element solution of reaction‐diffusion equations

We present a scheme for solving two‐dimensional, nonlinear reaction‐diffusion equations, using a mixed finite‐element method. To linearize the mixed‐method equations, we use a two grid scheme that relegates all the Newton‐like iterations to a grid Δ_H much coarser than the original one Δ_h, with no loss in order of accuracy so long as the mesh sizes obey . The use of a multigrid‐based solver for the indefinite linear systems that arise at each coarse‐grid iteration, as well as for the similar system that arises on the fine grid, allows for even greater efficiency. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 317–332, 1999     

An Eulerian‐Lagrangian discontinuous Galerkin method for transient advection‐diffusion equations

We develop an Eulerian‐Lagrangian discontinuous Galerkin method for time‐dependent advection‐diffusion equations. The derived scheme has combined advantages of Eulerian‐Lagrangian methods and discontinuous Galerkin methods. The scheme does not contain any undetermined problem‐dependent parameter. An optimal‐order error estimate and superconvergence estimate is derived. Numerical experiments are presented, which verify the theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A preliminary investigation on an ELLAM scheme for linear transport equations

We present an Eulerian‐Lagrangian localized adjoint method (ELLAM) for linear advection‐reaction partial differential equations in multiple space dimensions. We carry out numerical experiments to investigate the performance of the ELLAM scheme with a range of well‐perceived and widely used methods in fluid dynamics including the monotonic upstream‐centered scheme for conservation laws (MUSCL), the minmod method, the flux‐corrected transport method (FCT), and the essentially non‐oscillatory (ENO) schemes and weighted essentially non‐oscillatory (WENO) schemes. These experiments show that the ELLAM scheme is very competitive with these methods in the context of linear transport PDEs, and suggest/justify the development of ELLAM‐based simulators for subsurface porous medium flows and other applications. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 22–43, 2003     

Numerical resolution of linear evolution multidimensional problems of second order in time

We present a new class of efficient time integrators for solving linear evolution multidimensional problems of second‐order in time named Fractional Step Runge‐Kutta‐Nyström methods (FSRKN). We show that these methods, combined with suitable spliting of the space differential operator and adequate space discretizations provide important advantages from the computational point of view, mainly parallelization facilities and reduction of computational complexity. In this article, we study in detail the consistency of such methods and we introduce an extension of the concept of R‐stability for Runge‐Kutta‐Nyström methods. We also present some numerical experiments showing the unconditional convergence of a third order method of this class applied to resolve one Initial Boundary Value Problem of second order in time. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 597–620, 2012     

Two‐grid methods for mixed finite‐element solution of coupled reaction‐diffusion systems

We develop 2‐grid schemes for solving nonlinear reaction‐diffusion systems: where p = (p, q) is an unknown vector‐valued function. The schemes use discretizations based on a mixed finite‐element method. The 2‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. The iterative algorithms examined here extend a method developed earlier for single reaction‐diffusion equations. An application to prepattern formation in mathematical biology illustrates the method's effectiveness. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 589–604, 1999     

An Eulerian‐Lagrangian single‐node collocation method for transient advection‐diffusion equations in multiple space dimensions

We developed a nonconventional Eulerian‐Lagrangian single‐node collocation method for transient advection‐diffusion transport partial differential equations in multiple space dimensions. This method greatly reduces the number of unknowns in conventional collocation method, generates accurate numerical solutions, and allows large time steps to be used in numerical simulations. We perform numerical experiments to show the strong potential of the method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 284–301, 2004     

A uniform optimal‐order estimate for an Eulerian‐Lagrangian discontinuous Galerkin method for transient advection–diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for the Eulerian‐Lagrangian discontinuous Galerkin method for unsteady‐state advection–diffusion equations with general inflow and outflow boundary conditions. It is well‐known that these problems admit dynamic fronts with interior and boundary layers. The estimate holds uniformly with respect to the vanishing diffusion coefficient. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Block s‐step Krylov iterative methods

Block (including s‐step) iterative methods for (non)symmetric linear systems have been studied and implemented in the past. In this article we present a (combined) block s‐step Krylov iterative method for nonsymmetric linear systems. We then consider the problem of applying any block iterative method to solve a linear system with one right‐hand side using many linearly independent initial residual vectors. We present a new algorithm which combines the many solutions obtained (by any block iterative method) into a single solution to the linear system. This approach of using block methods in order to increase the parallelism of Krylov methods is very useful in parallel systems. We implemented the new method on a parallel computer and we ran tests to validate the accuracy and the performance of the proposed methods. It is expected that the block s‐step methods performance will scale well on other parallel systems because of their efficient use of memory hierarchies and their reduction of the number of global communication operations over the standard methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Ill‐conditioning in the virtual element method: Stabilizations and bases

In this article, we investigate the behavior of the condition number of the stiffness matrix resulting from the approximation of a 2D Poisson problem by means of the virtual element method. It turns out that ill‐conditioning appears when considering high‐order methods or in presence of “bad‐shaped” (for instance nonuniformly star‐shaped, with small edges…) sequences of polygons. We show that in order to improve such condition number one can modify the definition of the internal moments by choosing proper polynomial functions that are not the standard monomials. We also give numerical evidence that at least for a 2D problem, standard choices for the stabilization give similar results in terms of condition number.     

A nonconventional Eulerian‐Lagrangian single‐node collocation method with Hermite polynomials for unsteady‐state advection‐diffusion equations

We developed a nonconventional Eulerian‐Lagrangian single‐node collocation method (ELSCM) with piecewise‐cubic Hermite polynomials as basis functions for the numerical simulation to unsteady‐state advection‐diffusion transport partial differential equations. This method greatly reduces the number of unknowns in the conventional collocation method, and generates accurate numerical solutions even if very large time steps are taken. The method is relatively easy to formulate. Numerical experiments are presented to show the strong potential of this method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 271–283, 2003.     

Two‐grid methods for time‐harmonic Maxwell equations

In this paper, we develop several two‐grid methods for the Nédélec edge finite element approximation of the time‐harmonic Maxwell equations. We first present a two‐grid method that uses a coarse space to solve the original problem and then use a fine space to solve a corresponding symmetric positive definite problem. Then, we present two types of iterative two‐grid methods, one is to add the kernel of the curl ‐operator in the fine space to a coarse mesh space to solve the original problem and the other is to use an inner iterative method for dealing with the kernel of the curl ‐operator in the fine space and the coarse space, separately. We provide the error estimates for the first two methods and present numerical experiments to show the efficiency of our methods.Copyright © 2012 John Wiley & Sons, Ltd.     

New fifth‐order two‐derivative Runge‐Kutta methods with constant and frequency‐dependent coefficients

Two‐derivative Runge‐Kutta methods are Runge‐Kutta methods for problems of the form y′ = f(y) that include the second derivative y′′ = g(y) = f ′(y)f(y) and were developed in the work of Chan and Tsai. In this work, we consider explicit methods and construct a family of fifth‐order methods with three stages of the general case that use several evaluations of f and g per step. For problems with oscillatory solution and in the case that a good estimate of the dominant frequency is known, methods with frequency‐dependent coefficients are used; there are several procedures for constructing such methods. We give the general framework for the construction of methods with variable coefficients following the approach of Simos. We modify the above family to derive methods with frequency‐dependent coefficients following this approach as well as the approach given by Vanden Berghe. We provide numerical results to demonstrate the efficiency of the new methods using three test problems.     

A single‐node characteristic collocation method for unsteady‐state convection‐diffusion equations in three‐dimensional spaces

We develop a nonconventional single‐node characteristic collocation method with piecewise‐cubic Hermite polynomials for the numerical simulation to unsteady‐state advection‐diffusion transport partial differential equations. This method greatly reduces the number of unknowns in the conventional collocation method, and generates accurate numerical solutions even if very large time steps are taken. The reduction of number of nodes has great potential for problems defined on high space dimensions, which appears in such problems as quantification of uncertainties in subsurface porous media. The method developed here is easy to formulate. Numerical experiments are presented to show the strong potential of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 786–802, 2011     

A uniform estimate for the ELLAM scheme for transport equations

We prove a priori error estimate in a weighted energy norm for the Eulerian‐Lagrangian localized adjoint method (ELLAM) for the transport equations, without any special refinement or numerical stabilization introduced. The estimate holds uniformly with respect to ?. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Analysis and computation of least‐squares methods for a compressible Stokes problem

We develop and analyze a negative norm least‐squares method for the compressible Stokes equations with an inflow boundary condition. Least‐squares principles are derived for a first‐order form of the equations obtained by using ω = ?×u and φ = ? · u as new dependent variables. The resulting problem is incompletely elliptic, i.e., it combines features of elliptic and hyperbolic equations. As a result, well‐posedness of least‐squares functionals cannot be established using the ADN elliptic theory and so we use direct approaches to prove their norm‐equivalence. The article concludes with numerical examples that illustrate the theoretical convergence rates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

An Eulerian‐Lagrangian control volume scheme for two‐dimensional unsteady advection‐diffusion problems

We develop a mass conservative Eulerian‐Lagrangian control volume scheme (ELCVS) for the solution of the transient advection‐diffusion equations in two space dimensions. This method uses finite volume test functions over the space‐time domain defined by the characteristics within the framework of the class of Eulerian‐Lagrangian localized adjoint characteristic methods (ELLAM). It, therefore, maintains the advantages of characteristic methods in general, and of this class in particular, which include global mass conservation as well as a natural treatment of all types of boundary conditions. However, it differs from other methods in that class in the treatment of the mass storage integrals at the previous time step defined on deformed Lagrangian regions. This treatment is especially attractive for orthogonal rectangular Eulerian grids composed of block elements. In the algorithm, each deformed region is approximated by an eight‐node region with sides drawn parallel to the Eulerian grid, which significantly simplifies the integration compared with the approach used in finite volume ELLAM methods, based on backward tracking, while retaining local mass conservation at no additional expenses in terms of accuracy or CPU consumption. This is verified by numerical tests which show that ELCVS performs as well as standard finite volume ELLAM methods, which have previously been shown to outperform many other well‐received classes of numerical methods for the equations considered. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Subspace‐by‐subspace preconditioners for structured linear systems

We consider the iterative solution of symmetric positive‐definite linear systems whose coefficient matrix may be expressed as the outer product of low‐rank terms. We derive suitable preconditioners for such systems, and demonstrate their effectiveness on a number of test examples. We also consider combining these methods with existing techniques to cope with the commonly‐occuring case where the coefficient matrix is the linear sum of elements, some of which are of very low rank. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of algebraic systems arising from fourth‐order compact discretizations of convection‐diffusion equations

We study the properties of coefficient matrices arising from high‐order compact discretizations of convection‐diffusion problems. Asymptotic convergence factors of the convex hull of the spectrum and the field of values of the coefficient matrix for a one‐dimensional problem are derived, and the convergence factor of the convex hull of the spectrum is shown to be inadequate for predicting the convergence rate of GMRES. For a two‐dimensional constant‐coefficient problem, we derive the eigenvalues of the nine‐point matrix, and we show that the matrix is positive definite for all values of the cell‐Reynolds number. Using a recent technique for deriving analytic expressions for discrete solutions produced by the fourth‐order scheme, we show by analyzing the terms in the discrete solutions that they are oscillation‐free for all values of the cell Reynolds number. Our theoretical results support observations made through numerical experiments by other researchers on the non‐oscillatory nature of the discrete solution produced by fourth‐order compact approximations to the convection‐diffusion equation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 155–178, 2002; DOI 10.1002/num.1041