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     

Numerical solutions of the coupled Burgers’ equation by the Galerkin quadratic B‐spline finite element method

In this paper, a coupled Burgers’ equation has been numerically solved by a Galerkin quadratic B‐spline FEM. The performance of the method has been examined on three test problems. Results obtained by the method have been compared with known exact solution and other numerical results in the literature. A Fourier stability analysis of the method is also investigated. Copyright © 2013 John Wiley & Sons, Ltd.     

A two‐grid characteristic finite volume element method for semilinear advection‐dominated diffusion equations

A two‐grid finite volume element method, combined with the modified method of characteristics, is presented and analyzed for semilinear time‐dependent advection‐dominated diffusion equations in two space dimensions. The solution of a nonlinear system on the fine‐grid space (with grid size h) is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse‐grid space (with grid size H) and a linear system on the fine‐grid space. An optimal error estimate in H¹ ‐norm is obtained for the two‐grid method. It shows that the two‐grid method achieves asymptotically optimal approximation, as long as the mesh sizes satisfy h = O(H²). Numerical example is presented to validate the usefulness and efficiency of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Numerical investigation of the solution of Fisher's equation via the B‐spline Galerkin method

The Galerkin method is used with quadratic B‐spline base functions to obtain the numerical solutions of Fisher's equation which is a one dimensional reaction‐diffusion equation. To observe the effects of reaction and diffusion, four test problems related to pulse disturbance, step disturbance, super‐speed wave and strong reaction are studied. A comparison is performed between the obtained numerical results and some earlier studies. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

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     

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     

Finite element method by using quartic B‐splines

In this article, we discuss a kind of finite element method by using quartic B‐splines to solve Dirichlet problem for elliptic equations. Bivariate spline proper subspace of S(Δ) satisfying homogeneous boundary conditions on Type‐2 triangulations and quadratic B‐spline interpolating boundary functions are primarily constructed. Linear and nonlinear elliptic equations are solved by Galerkin quartic B‐spline finite element method. Numerical examples are provided to illustrate the proposed method is flexible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 818–828, 2011     

A locally conservative Eulerian‐Lagrangian control‐volume method for transient advection‐diffusion equations

Characteristic methods generally generate accurate numerical solutions and greatly reduce grid orientation effects for transient advection‐diffusion equations. Nevertheless, they raise additional numerical difficulties. For instance, the accuracy of the numerical solutions and the property of local mass balance of these methods depend heavily on the accuracy of characteristics tracking and the evaluation of integrals of piecewise polynomials on some deformed elements generally with curved boundaries, which turns out to be numerically difficult to handle. In this article we adopt an alternative approach to develop an Eulerian‐Lagrangian control‐volume method (ELCVM) for transient advection‐diffusion equations. The ELCVM is locally conservative and maintains the accuracy of characteristic methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general. Numerical experiments show that the ELCVM is favorably comparable with well‐regarded Eulerian‐Lagrangian methods, which were previously shown to be very competitive with many well‐perceived methods. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Minimum time‐step criteria for the Galerkin finite element methods applied to one‐dimensional parabolic partial differential equations

The finite element method has been well established for numerically solving parabolic partial differential equations (PDEs). Also it is well known that a too large time step should not be chosen in order to obtain a stable and accurate numerical solution. In this article, accuracy analysis shows that a too small time step should not be chosen either for some time‐stepping schemes. Otherwise, the accuracy of the numerical solution cannot be improved or can even be worsened in some cases. Furthermore, the so‐called minimum time step criteria are established for the Crank‐Nicolson scheme, the Galerkin‐time scheme, and the backward‐difference scheme used in the temporal discretization. For the forward‐difference scheme, no minimum time step exists as far as the accuracy is concerned. In the accuracy analysis, no specific initial and boundary conditions are invoked so that such established criteria can be applied to the parabolic PDEs subject to any initial and boundary conditions. These minimum time step criteria are verified in a series of numerical experiments for a one‐dimensional transient field problem with a known analytical solution. The minimum time step criteria developed in this study are useful for choosing appropriate time steps in numerical simulations of practical engineering problems. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A new stabilized finite element method for advection‐diffusion‐reaction equations

In this article, a new stabilized finite element method is proposed and analyzed for advection‐diffusion‐reaction equations. The key feature is that both the mesh‐dependent Péclet number and the mesh‐dependent Damköhler number are reasonably incorporated into the newly designed stabilization parameter. The error estimates are established, where, up to the regularity‐norm of the exact solution, the explicit‐dependence of the diffusivity, advection, reaction, and mesh size (or the dependence of the mesh‐dependent Péclet number and the mesh‐dependent Damköhler number) is revealed. Such dependence in the error bounds provides a mathematical justification on the effectiveness of the proposed method for any values of diffusivity, advection, dissipative reaction, and mesh size. Numerical results are presented to illustrate the performance of the method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 616–645, 2016     

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     

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.     

A posteriori error estimates for the generalized Schwarz method of a new class of advection‐diffusion equation with mixed boundary condition

In this paper, a posteriori error estimates for the generalized Schwarz method with mixed boundary condition on the interfaces for advection‐diffusion equation with second‐order boundary value problems are proved using theta time scheme combined with Galerkin spatial method. Furthermore, a asymptotic behavior in Sobolev norm is deduced using Benssoussan‐Lions' algorithm.     

An upwind finite volume scheme and its maximum‐principle‐preserving ADI splitting for unsteady‐state advection‐diffusion equations

We develop an upwind finite volume (UFV) scheme for unsteady‐state advection‐diffusion partial differential equations (PDEs) in multiple space dimensions. We apply an alternating direction implicit (ADI) splitting technique to accelerate the solution process of the numerical scheme. We investigate and analyze the reason why the conventional ADI splitting does not satisfy maximum principle in the context of advection‐diffusion PDEs. Based on the analysis, we propose a new ADI splitting of the upwind finite volume scheme, the alternating‐direction implicit, upwind finite volume (ADFV) scheme. We prove that both UFV and ADFV schemes satisfy maximum principle and are unconditionally stable. We also derive their error estimates. Numerical results are presented to observe the performance of these schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 211–226, 2003     

Superconvergence of the direct discontinuous Galerkin method for convection‐diffusion equations

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one‐dimensional linear convection‐diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2k‐th and ‐th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a ‐th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017     

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion‐acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear GRLW equation by Petrov–Galerkin method in which the element shape functions are cubic and weight functions are quadratic B‐splines. The proposed method is implemented to three reference problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi‐discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of the linearized scheme we show that the scheme is unconditionally stable. To verify practicality and robustness of the new scheme error norms L₂, L_∞ and three invariants I₁, I₂, and I₃ are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective.     

Inf‐sup stability of Petrov‐Galerkin immersed finite element methods for one‐dimensional elliptic interface problems

In this article, we analyze the Petrov‐Galerkin immersed finite element method (PG‐IFEM) when applied to one‐dimensional elliptic interface problems. In the PG‐IFEM (T. Hou, X. Wu and Y. Zhang, Commun. Math. Sci., 2 (2004), 185‐205, and S. Hou and X. Liu, J. Comput. Phys., 202 (2005), 411‐445), the classic immersed finite element (IFE) space was taken as the trial space while the conforming linear finite element space was taken as the test space. We first prove the inf‐sup condition of the PG‐IFEM and then show the optimal error estimate in the energy norm. We also show the optimal estimate of the condition number of the stiffness matrix. The results are extended to two dimensional problems in a special case.     

Continuous Galerkin finite element methods for a forward‐backward heat equation

A space‐time finite element method is introduced to solve a model forward‐backward heat equation. The scheme uses the continuous Galerkin method for the time discretization. An error analysis for the method is presented. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 257–265, 1999     

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.