High‐order implicit staggered‐grid finite differences methods for the acoustic wave equation

Motivated by the idea that staggered‐grid methods give a greater stability and give energy conservation, this article presents a new family of high‐order implicit staggered‐grid finite difference methods with any order of accuracy to approximate partial differential equations involving second‐order derivatives. In particular, we numerically analyze our new methods for the solution of the one‐dimensional acoustic wave equation. The implicit formulation is based on the plane wave theory and the Taylor series expansion and only involves the solution of tridiagonal matrix equations resulting in an attractive method with higher order of accuracy but nearly the same computation cost as those of explicit formulation. The order of accuracy of the proposal staggered formulas are similar to the methods with conventional grids for a ‐point operator: the explicit formula is th‐order and the implicit formula is th‐order; however, the results demonstrate that new staggered methods are superior in terms of stability properties to the classical methods in the context of solving wave equations.     

A fourth‐order Hermitian box‐scheme with fast solver for the Poisson problem in a cube

We present a fourth‐order Hermitian box‐scheme (HB‐scheme) for the Poisson problem in a cube. A single‐nonstaggered regular grid is used supporting the discrete unknowns u and . The scheme is fourth‐order accurate for u and in norm. The fast numerical resolution uses a matrix capacitance method, resulting in a computational complexity of . Numerical results are reported on several examples including nonseparable problems. The present scheme is the extension to the three‐dimensional case of the HB‐scheme presented in Abbas and Croisille [J Sci Comp 49 (2011), 239–267]. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 609–629, 2015     

Fourth‐order compact and energy conservative scheme for solving nonlinear Klein‐Gordon equation

In this article, a fourth‐order compact and conservative scheme is proposed for solving the nonlinear Klein‐Gordon equation. The equation is discretized using the integral method with variational limit in space and the multidimensional extended Runge‐Kutta‐Nyström (ERKN) method in time. The conservation law of the space semidiscrete energy is proved. The proposed scheme is stable in the discrete maximum norm with respect to the initial value. The optimal convergent rate is obtained at the order of in the discrete ‐norm. Numerical results show that the integral method with variational limit gives an efficient fourth‐order compact scheme and has smaller error, higher convergence order and better energy conservation for solving the nonlinear Klein‐Gordon equation compared with other methods under the same condition. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1283–1304, 2017     

Stability analysis of several first order schemes for the Oldroyd model with smooth and nonsmooth initial data

In this article, we develop several first order fully discrete Galerkin finite element schemes for the Oldroyd model and establish the corresponding stability results for these numerical schemes with smooth and nonsmooth initial data. The stable mixed finite element method is used to the spatial discretization, and the temporal treatments of the spatial discrete Oldroyd model include the first order implicit, semi‐implicit, implicit/explicit, and explicit schemes. The ‐stability results of the different numerical schemes are provided, where the first‐order implicit and semi‐implicit schemes are the ‐unconditional stable, the implicit/explicit scheme is the ‐almost unconditional stable, and the first order explicit scheme is the ‐conditional stable. Finally, some numerical investigations of the ‐stability results of the considered numerical schemes for the Oldroyd model are provided to verify the established theoretical findings.     

The fictitious domain method with L2‐penalty for the Stokes problem with the Dirichlet boundary condition

We consider the fictitious domain method with L²‐penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order in H¹‐norm for the velocity and in L²‐norm for the pressure, where is the penalty parameter. The L²‐norm error estimate for the velocity is upgraded to . Moreover, we derive the a priori estimates depending on for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate in H¹‐norm for the velocity and in L²‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L²‐norm, the convergence rate is improved to . The theoretical results are verified by the numerical experiments.     

Error analysis of mixed finite element method for Poisson‐Nernst‐Planck system

To improve the convergence rate in L² norm from suboptimal to optimal for both electrostatic potential and ionic concentrations in Poisson‐Nernst‐Planck (PNP) system, we propose the mixed finite element method in this article to discretize the electrostatic potential equation, and still use the standard finite element method to discretize the time‐dependent ionic concentrations equations. Optimal error estimates in norm for the electrostatic potential, and in and norms for the ionic concentrations are attained. As a by‐product, the electric field can also achieve a higher approximation order in contrast with the standard finite element method for PNP system. Numerical experiments are performed to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1924–1948, 2017     

A fast‐high order compact difference method for the fractional cable equation

The Cable equation is one of the most fundamental equations for modeling neuronal dynamics. In this article, we consider a high order compact finite difference numerical solution for the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. The resulting finite difference scheme is unconditionally stable and converges with the convergence order of in maximum norm, 1‐norm and 2‐norm. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix‐vector multiplications arising from finite difference discretization. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from required by traditional methods to without using any lossy compression, where and τ is the size of time step, and h is the size of space step. Moreover, we give a compact finite difference scheme and consider its stability analysis for two‐dimensional fractional Cable equation. The applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.     

Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one‐dimensional second‐order wave equation

In this article, we analyze a residual‐based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one‐dimensional second‐order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L² error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862–901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, at a fixed time, converge to the true spatial errors in the L²‐norm under mesh refinement. The order of convergence is proved to be , when p‐degree piecewise polynomials with are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and superconvergent solutions. Our computational results show higher convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L²‐norm converge to unity at rate while numerically they exhibit and rates, respectively. Numerical experiments are shown to validate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1461–1491, 2015     

B‐spline solution of fractional integro partial differential equation with a weakly singular kernel

The main objective of the paper is to find the approximate solution of fractional integro partial differential equation with a weakly singular kernel. Integro partial differential equation (IPDE) appears in the study of viscoelastic phenomena. Cubic B‐spline collocation method is employed for fractional IPDE. The developed scheme for finding the solution of the considered problem is based on finite difference method and collocation method. Caputo fractional derivative is used for time fractional derivative of order α, . The given problem is discretized in both time and space directions. Backward Euler formula is used for temporal discretization. Collocation method is used for spatial discretization. The developed scheme is proved to be stable and convergent with respect to time. Approximate solutions are examined to check the precision and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1565–1581, 2017     

Superconvergence of Ritz‐Galerkin finite element approximations for second order elliptic problems

In this paper, the author derives an ‐superconvergence for the piecewise linear Ritz‐Galerkin finite element approximations for the second‐order elliptic equation equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor and the usual shape functions on each element, called ‐equilateral assumption in this paper. Several examples are presented for the coefficient tensor and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.     

High‐order algorithms for Riesz derivative and their applications (V)

In this article, based on the idea of combing symmetrical fractional centred difference operator with compact technique, a series of even‐order numerical differential formulas (named the fractional‐compact formulas) are established for the Riesz derivatives with order . Properties of coefficients in the derived formulas are studied in details. Then applying the constructed fourth‐order formula, a difference scheme is proposed to solve the Riesz spatial telegraph equation. By the energy method, the constructed numerical algorithm is proved to be stable and convergent with order , where τ and h are the temporal and spatial stepsizes, respectively. Finally, several numerical examples are presented to verify the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1754–1794, 2017     

On the Euler implicit/explicit iterative scheme for the stationary Oldroyd fluid

In this article, we consider the stationary Oldroyd fluid equations from the large time behavior research of the nonstationary equations. Thus, to obtain its numerical solution, we first solve the nonstationary Oldroyd fluid equations via the Euler implicit/explicit finite element method with the integral term discretized by the right‐hand rectangle rule, then increase the total time (i.e., number of time steps) to approximate the solution of the original stationary equations. Under a new uniqueness condition (A2), we prove the exponential stability of the solution pair for the stationary equations and the almost unconditional stability of the numerical method. Furthermore, we also obtain the uniform optimal and error estimates in time integral . Finally, several numerical experiments are provided to verify our theoretical results.     

Least‐squares spectral method for velocity‐vorticity‐pressure form of the Stokes equations

The aim of this article is to present and analyze first‐order system least‐squares spectral method for the Stokes equations in two‐dimensional spaces. The Stokes equations are transformed into a first‐order system of equations by introducing vorticity as a new variable. The least‐squares functional is then defined by summing up the ‐ and ‐norms of the residual equations. The ‐norm in the least‐squares functional is replaced by suitable operator. Continuous and discrete homogeneous least‐squares functionals are shown to be equivalent to ‐norm of velocity and ‐norm of vorticity and pressure for spectral Galerkin and pseudospectral method. The spectral convergence of the proposed methods are given and the theory is validated by numerical experiment. Mass conservation is also briefly investigated. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 661–680, 2016     

Long‐time stability and asymptotic analysis of the IFE method for the multilayer porous wall model

In this article, we study the long‐time stability and asymptotic behavior of the immersed finite element (IFE) method for the multilayer porous wall model for the drug‐eluting stents. First, with the IFE method for the spatial descretization, and the implicit Euler scheme for the temporal discretization, respectively, we deduce the global stability of fully discrete solution. Then, we investigate the asymptotic behavior of the discrete scheme which reveals that the multilayer porous wall model converges to the corresponding elliptic equation if approaches to a steady‐state in both and norms as . Finally, some numerical experiments are given to verify the theoretical predictions.     

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     

Unconditional superconvergence analysis of an H1‐galerkin mixed finite element method for nonlinear Sobolev equations

An efficient H¹‐Galerkin mixed finite element method (MFEM) is presented with and zero order Raviart‐Thomas elements for the nonlinear Sobolev equations. On one hand, the existence and uniqueness of the solutions of the semidiscrete approximation scheme are proved and the super close results of order for the original variable u in a broken H¹ norm and the auxiliary variable in norm are deduced without the boundedness of the numerical solution in ‐norm. Conversely, a linearized Crank‐Nicolson fully discrete scheme with the unconditional super close property is also developed through a new approach, while previous literature always require certain time step conditions (see the references below). Finally, a numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.     

High‐order difference scheme for the solution of linear time fractional klein–gordon equations

In this article, we apply a high‐order difference scheme for the solution of some time fractional partial differential equations (PDEs). The time fractional Cattaneo equation and the linear time fractional Klein–Gordon and dissipative Klein–Gordon equations will be investigated. The time fractional derivative which has been described in the Caputo's sense is approximated by a scheme of order , and the space derivative is discretized with a fourth‐order compact procedure. We will prove the solvability of the proposed method by coefficient matrix property and the unconditional stability and ‐convergence with the energy method. Numerical examples demonstrate the theoretical results and the high accuracy of the proposed scheme. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1234–1253, 2014     

A compact ADI scheme for the two dimensional time fractional diffusion‐wave equation in polar coordinates

In this article, we consider finite difference schemes for two dimensional time fractional diffusion‐wave equations on an annular domain. The problem is formulated in polar coordinates and, therefore, has variable coefficients. A compact alternating direction implicit scheme with accuracy order is derived, where τ, h₁, h₂ are the temporal and spatial step sizes, respectively. The stability and convergence of the proposed scheme are studied using its matrix form by the energy method. Numerical experiments are presented to support the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1692–1712, 2015     

The local superconvergence of the linear finite element method for the poisson problem

Assume that . In this study, the Richardson extrapolation for the tensor‐product block element and the linear finite element theory of the Green's function will be combined to study the local superconvergence of finite element methods for the Poisson equation in a bounded polytopic domain (polygonal or polyhedral domain for ), where a family of tensor‐product block partitions is not required or the solution need not have high global smoothness. We present a special family of partitions satisfying, for any , e is a tensor‐product block whenever where denotes the distance between e and . By the linear finite element theory of the Green's function and the Richardson extrapolation for the tensor‐product block element, we obtain the local superconvergence of the displacement for the linear finite element method over the special family of partitions . © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 930–946, 2014     

Error and stability analysis of numerical solution for the time fractional nonlinear Schrödinger equation on scattered data of general‐shaped domains

In present work, a kind of spectral meshless radial point interpolation (SMRPI) technique is applied to the time fractional nonlinear Schrödinger equation in regular and irregular domains. The applied approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. It is proved the scheme is unconditionally stable with respect to the time variable in and also convergent by the order of convergence , . In the current work, the thin plate spline are used as the basis functions and to eliminate the nonlinearity, a simple predictor‐corrector (P‐C) scheme is performed. It is shown that the SMRPI solution, as a complex function, is suitable one for the time fractional nonlinear Schrödinger equation. The results of numerical experiments are compared to analytical solutions to confirm the reliable treatment of these stable solutions. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1043–1069, 2017