Fast solution algorithms for exponentially tempered fractional diffusion equations

In this article, a fast‐iterative method and a fast‐direct method is proposed for solving one‐dimensional and two‐dimensional tempered fractional diffusion equations with constant coefficients. The proposed iterative method is accelerated by circulant preconditioning which is shown to converge superlinearly while the proposed direct method is based on circulant and skew‐circulant representation for Toeplitz matrix inversion. In one‐dimensional case, the operation cost of the proposed methods are both shown to be with memory requirement in each time step, where is the number of spatial nodes. With the alternating direction implicit method, it is proven that the proposed fast solution algorithms can be extended to handle two‐dimensional tempered fractional diffusion equations with operation cost and memory requirement in each time step, where the number of spatial nodes in ‐direction and ‐direction both equal to . Numerical examples are provided to illustrate the effectiveness and efficiency of the proposed methods.     

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 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.     

An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience

A class of time‐dependent singularly perturbed convection‐diffusion problems with retarded terms arising in computational neuroscience is considered. In particular, a numerical scheme for the parabolic convection‐diffusion problem where the second‐order derivative with respect to the spatial direction is multiplied by a small perturbation parameter and the shifts are of is constructed. The Taylor series expansion is used to tackle the shift terms. The continuous problem is semidiscretized using the Crank‐Nicolson finite difference method in the temporal direction and the resulting set of ordinary differential equations is discretized using a midpoint upwind finite difference scheme on an appropriate piecewise uniform mesh, which is dense in the boundary layer region. It is shown that the proposed numerical scheme is second‐order accurate in time and almost first‐order accurate in space with respect to the perturbation parameter . To validate the computational results and efficiency of the method some numerical examples are encountered and the numerical results are compared with some existing results. It is observed that the numerical approximations are fairly good irrespective of the size of the delay and the advance till they are of . The effect of the shifts on the boundary layer has also been observed.     

Smallest domination number and largest independence number of graphs and forests with given degree sequence

For a sequence d of nonnegative integers, let and be the sets of all graphs and forests with degree sequence d, respectively. Let , , , and where is the domination number and is the independence number of a graph G. Adapting results of Havel and Hakimi, Rao showed in 1979 that can be determined in polynomial time. We establish the existence of realizations with , and with and that have strong structural properties. This leads to an efficient algorithm to determine for every given degree sequence d with bounded entries as well as closed formulas for and .     

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.     

Numerical solutions to a BBM‐Burgers model with a nonlocal viscous term

In this paper, we numerically investigate the BBM‐Burgers equation with a nonlocal viscous term (1) where is the Riemann‐Liouville half derivative. In particular, we implement different numerical schemes to approximate the solution and its asymptotical behavior. Also, we compare our numerical results with those given in 2013, 2014 for similar models.     

Optimal error estimate of a conformal Fourier pseudo‐spectral method for the damped nonlinear Schrödinger equation

In this article, a Fourier pseudospectral method, which preserves the conforal conservation la, is proposed for solving the damped nonlinear Schrödinger equation. Based on the energy method and the semi‐norm equivalence between the Fourier pseudospectral method and the finite difference method, a priori estimate for the new method is established, which shows that the proposed method is unconditionally convergent with order of in the discrete ‐norm, where is the time step and is the number of collocation points used in the spectral method. Some numerical results are addressed to confirm our theoretical analysis.     

Local error estimates of the finite element method for an elliptic problem with a Dirac source term

The solutions of elliptic problems with a Dirac measure right‐hand side are not in dimension and therefore the convergence of the finite element solutions is suboptimal in the ‐norm. In this article, we address the numerical analysis of the finite element method for the Laplace equation with Dirac source term: we consider, in dimension 3, the Dirac measure along a curve and, in dimension 2, the punctual Dirac measure. The study of this problem is motivated by the use of the Dirac measure as a reduced model in physical problems, for which high accuracy of the finite element method at the singularity is not required. We show a quasioptimal convergence in the ‐norm, for on subdomains which exclude the singularity; in the particular case of Lagrange finite elements, an optimal convergence in ‐norm is shown on a family of quasiuniform meshes. Our results are obtained using local Nitsche and Schatz‐type error estimates, a weak version of Aubin‐Nitsche duality lemma and a discrete inf‐sup condition. These theoretical results are confirmed by numerical illustrations.     

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.     

Higher order numerical approximation for time dependent singularly perturbed differential‐difference convection‐diffusion equations

In this article, we develop a higher order numerical approximation for time dependent singularly perturbed differential‐difference convection‐diffusion equations. A priori bounds on the exact solution and its derivatives, which are useful for the error analysis of the numerical method are given. We approximate the retarded terms of the model problem using Taylor's series expansion and the resulting time‐dependent singularly perturbed problem is discretized by the implicit Euler scheme on uniform mesh in time direction and a special hybrid finite difference scheme on piecewise uniform Shishkin mesh in spatial direction. We first prove that the proposed numerical discretization is uniformly convergent of , where and denote the time step and number of mesh‐intervals in space, respectively. After that we design a Richardson extrapolation scheme to increase the order of convergence in time direction and then the new scheme is proved to be uniformly convergent of . Some numerical tests are performed to illustrate the high‐order accuracy and parameter uniform convergence obtained with the proposed numerical methods.     

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.     

Superconvergence analysis of Crank‐Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation

The purpose of this article is to apply nonconforming finite element(FE) to solve a generalized nonlinear Schrödinger equation. First, a new important property of nonconforming FE (see ( 2.3 ) of Lemma 2 below) is proved by use of BHX lemma and the integral identities techniques. Second, a linearized Crank‐Nicolson fully discrete scheme is constructed and the superclose error estimate of order for original variable u in broken H¹‐norm is also derived by using the properties of element and the splitting argument for nonlinear terms, while previous works always only obtain convergent error estimates with this element. Furthermore, the global superconvergence is arrived at by the interpolated postprocessing technique. Finally, two numerical experiments are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step.     

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.     

Two unconditionally stable and convergent difference schemes with the extrapolation method for the one‐dimensional distributed‐order differential equations

The Grünwald formula is used to solve the one‐dimensional distributed‐order differential equations. Two difference schemes are derived. It is proved that the schemes are unconditionally stable and convergent with the convergence orders and in maximum norm, respectively, where and are step sizes in time, space and distributed order. The extrapolation method is applied to improve the approximate accuracy to the orders and respectively. An illustrative numerical example is given to confirm the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 591–615, 2016     

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     

Error estimates to smooth solutions of semi‐discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws

In this article, we focus on error estimates to smooth solutions of semi‐discrete discontinuous Galerkin (DG) methods with quadrature rules for scalar conservation laws. The main techniques we use are energy estimate and Taylor expansion first introduced by Zhang and Shu in (Zhang and Shu, SIAM J Num Anal 42 (2004), 641–666). We show that, with (piecewise polynomials of degree k) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree , error estimates of are obtained for general monotone fluxes, and optimal estimates of are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree , error estimates of are obtained for general monotone fluxes, and are obtained for monotone and sufficiently smooth numerical fluxes. Numerical results validate our analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 467–488, 2017     

Maximum norm stability and error estimates for the evolving surface finite element method

We show convergence in the natural and norm for a semidiscretization with linear finite elements of a linear parabolic partial differential equations on evolving surfaces. To prove this, we show error estimates for a Ritz map, error estimates for the material derivative of a Ritz map and a weak discrete maximum principle.     

A block‐centered finite difference method for fractional Cattaneo equation

In this article, a block‐centered finite difference method for fractional Cattaneo equation is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norm with optimal order of convergence both for pressure and velocity are established on nonuniform rectangular grids. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.     

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

In this study, with the aid of Wolfram Mathematica 11, the modified exp ‐expansion function method is used in constructing some new analytical solutions with novel structure such as the trigonometric and hyperbolic function solutions to the well‐known nonlinear evolutionary equation, namely; the two‐component second order KdV evolutionary system. Second, the finite forward difference method is used in analyzing the numerical behavior of this equation. We consider equation (6.5) and (6.6) for the numerical analysis. We examine the stability of the two‐component second order KdV evolutionary system with the finite forward difference method by using the Fourier‐Von Neumann analysis. We check the accuracy of the finite forward difference method with the help of and norm error. We present the comparison between the exact and numerical solutions of the two‐component second order KdV evolutionary system obtained in this article which and support with graphics plot. We observed that the modified exp ‐expansion function method is a powerful approach for finding abundant solutions to various nonlinear models and also finite forward difference method is efficient for examining numerical behavior of different nonlinear models.