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The present comment concerns some doubtful results included in the above paper. [ABSTRACT FROM AUTHOR]

The multilevel fast multipole method (MLFMM) is widely used to accelerate the solutions of acoustic and electromagnetic scattering problems. In the expansions and translation operators of the MLFMM for 3‐D scattering problems, some special functions are used, including spherical Bessel functions, spherical harmonics and Wigner 3j$$ 3j $$ symbol. This makes it difficult to analyze the truncation errors. In this paper, we first give sharp bounds for the truncation errors of the expansions used in the MLFMM, then derive the overall error formula of the MLFMM and estimate its upper bound, the result is finally applied to the cube octree structure. Some numerical examples are performed to validate the proposed results. The method in this paper can also be used to the MLFMM for other 3‐D problems, such as potential problems, elastostatic problems, Stokes flow problems and so on. [ABSTRACT FROM AUTHOR]

A homogenization approach is one of effective strategies to solve multiscale elliptic problems approximately. The finite element heterogeneous multiscale method (FEHMM) which is based on the finite element makes possible to simulate such process numerically. In this paper we introduce a FEHMM scheme for multiscale elliptic problems based on nonconforming elements. In particular we use the noconforming element with the periodic boundary condition introduced in the companion paper. Theoretical analysis derives a priori error estimates in the standard Sobolev norms. Several numerical results which confirm our analysis are provided. [ABSTRACT FROM AUTHOR]

This paper investigates the strong convergence of a fully discrete numerical method for the stochastic partial differential equations driven by multiplicative noise. The fully discrete space–time approximation consists of the symmetric interior penalty discontinuous Galerkin method for the spatial discretization and the implicit Euler method for the temporal discretization. Rather than the usual semi group analysis techniques, in this paper, we present an analysis framework in the variational formulation by introducing new weak variational approximation techniques. Some error estimates in a strong sense are established for the proposed fully discrete scheme. The optimal convergence rates are then obtained in both space and time. Numerical results for the nonlinear stochastic partial differential equations are finally presented to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Studying ordinary or partial differential equations or integrals using traditional asymptotic analysis, unfortunately, fails to extract the exponentially small terms and fails to derive some of their asymptotic features. In this paper, we discuss how to characterize an asymptotic behavior of a singular linear differential equation by the methods in exponential asymptotics. This paper is particularly concerned with the formulation of the series representation of a general second‐order differential equation. It provides a detailed explanation of the asymptotic behavior of the differential equation and its relation between the prefactor functions and the singulant of the expansion of the equation. Through having this relationship, one can directly uncover and investigate invisible exponentially small terms and Stokes phenomenon without doing more work for the particular type of equations. Here, we demonstrate how these terms and form of the expansion can be computed straight‐away, and, in a manner, this can be extended to the derivation of the potential Stokes and anti‐Stokes lines. [ABSTRACT FROM AUTHOR]

In this paper, we consider a numerical method to solve scattering problems with multi‐periodic layers with different periodicities. The main tool applied in this paper is the Bloch transform. With this method, the problem is written into an equivalent coupled family of quasi‐periodic problems. As the Bloch transform is only defined for one fixed period, the inhomogeneous layer with another period is simply treated as a non‐periodic one. First, we approximate the refractive index by a periodic one where its period is an integer multiple of the fixed period, and it is decomposed by finite number of quasi‐periodic functions. Then the coupled system is reduced into a simplified formulation. A convergent finite element method is proposed for the numerical solution, and the numerical method has been applied to several numerical experiments. At the end of this paper, relative errors of the numerical solutions will be shown to illustrate the convergence of the numerical algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi‐implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H(curl) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider an explicit fully‐discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity‐tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform moment bounds for the numerical approximations. To the best of our knowledge, the main contribution of this work is the first result in the literature which establishes strong convergence for an explicit fully‐discrete finite element approximation of the CHC equation. Finally, numerical results are finally reported to confirm the previous theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider the development and analysis of a new explicit compact high‐order finite difference scheme for acoustic wave equation formulated in divergence form, which is widely used to describe seismic wave propagation through a heterogeneous media with variable media density and acoustic velocity. The new scheme is compact and of fourth‐order accuracy in space and second‐order accuracy in time. The compactness of the scheme is obtained by the so‐called combined finite difference method, which utilizes the boundary values of the spatial derivatives and those boundary values are obtained by one‐sided finite difference approximation. An empirical stability analysis has been conducted to obtain the Courant‐Friedrichs‐Levy (CFL) condition, which confirmed the conditional stability of the new scheme. Four numerical examples have been conducted to validate the convergence and effectiveness of the new scheme. The application of the new scheme to a realistic wave propagation problem with a Perfect Matched Layer is validated in this paper as well. [ABSTRACT FROM AUTHOR]

This article deals with the error analysis of a Galerkin‐mixed finite element methods for the advection–reaction–diffusion Brinkman flow in porous media. Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest‐order Raviart–Thomas element, the lowest‐order Nédélec edge element and piece‐wise constant discontinuous Galerkin element are used for the velocity, vorticity and pressure, respectively. The existing error estimate of this lowest‐order finite element method is only O(h)$$ O(h) $$ for all variables in spatial direction, which is not optimal for the concentration variable. This paper focuses on a new and optimal error estimate of a linearized backward Euler Galerkin‐mixed FEMs, where the second‐order accuracy for the concentration in spatial directions is established unconditionally. The key to our optimal error analysis is a new negative norm estimate for Nédélec edge element. Moreover, based on the computed numerical concentration, we propose a simple one‐step recovery technique to obtain a new numerical velocity, vorticity and pressure with second‐order accuracy. Numerical experiments are provided to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper focuses on numerical algorithms to value American options under regime switching. The prices of such options satisfy a set of complementary parabolic problems on an unbounded domain. Based on our previous experience, the pricing model could be truncated into a linear complementarity problem (LCP) over a bounded domain. In addition, we transform the resulting LCP into an equivalent variational problem (VP), and discretize the VP by an Euler‐finite element method. Since the variational matrix in the discretized system is P‐matrix, a primal‐dual active set (PDAS) algorithm is proposed to evaluate the option prices efficiently. As a specialty of PDAS, the optimal exercise boundaries in all regimes are obtained without further computation cost. Finally, numerical simulations are carried out to test the performance of our proposed algorithm and compare it to existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose an a posteriori error estimate of the weak Galerkin finite element method (WG‐FEM) solving the Stokes problems with variable coefficients. Its error estimator, based on the property of Stokes' law conservation, Helmholtz decomposition and bubble functions, yields global upper bound and local lower bound for the approximation error of the WG‐FEM. Error analysis is proved to be valid under the mesh assumptions of the WG‐FEM and the way can be extended to other FEMs with the property of Stokes' law conservation, for example, discontinuous Galerkin (DG) FEMs. Finally, we verify the performance of error estimator by performing a few numerical examples. [ABSTRACT FROM AUTHOR]

This paper deals with the a posteriori error analysis for an augmented mixed discontinuous formulation for the stationary Stokes problem. By considering an appropriate auxiliary problem, we derive an a posteriori error estimator. We prove that this estimator is reliable and locally efficient, and consists of just five residual terms. Numerical experiments confirm the theoretical properties of the augmented discontinuous scheme as well as of the estimator. They also show the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution. [ABSTRACT FROM AUTHOR]

In this paper, the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments are investigated. Firstly, the variation formulation is derived by applying Green's formula and Galerkin finite element method to spatial direction of the original equation. Next, semidiscrete and fully discrete schemes are obtained and the convergence is analyzed in L2$$ {L}^2 $$‐norm rigorously. Moreover, the stability analysis shows that the semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are also obtained under which the analytic solution is asymptotically stable. Finally, some numerical experiments are provided to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

The nonlocal Cahn‐Hilliard equation has attracted much attention these years. Despite the advantage of describing more practical phenomena for modeling phase transitions of microstructures in materials, the nonlocal operator in the equation brings a lot of extra computational costs compared with the local Cahn‐Hilliard equation. Thus high order time integration schemes are needed in numerical simulations. In this paper, we propose two classes of exponential time differencing (ETD) schemes for solving the nonlocal Cahn‐Hilliard equation. We first use the Fourier collocation method to discretize the spatial domain, and then the ETD‐based multistep and Runge‐Kutta schemes are adopted for the time integration. In particular, some specific multistep and Runge‐Kutta schemes up to fourth order are constructed. We rigorously establish the energy stabilities of the multistep schemes up to fourth order and the second order Runge‐Kutta scheme, which show that the first order ETD and the second order Runge‐Kutta schemes unconditionally decrease the original energy. We also theoretically prove the mass conservations of the proposed schemes. Several numerical experiments in two and three dimensions are carried out to test the temporal convergence rates of the schemes and to verify their mass conservations and energy stabilities. The long time simulations of coarsening dynamics are also performed to verify the power law for the energy decay. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid numerical scheme based on combining exponential time differencing (ETD) and local radial basis function collocation method was constructed. Model problems with different boundary conditions were considered, and the resulting linear system was carefully analyzed. The relation between the number of points employed in the local radial basis function collocation method and the condition number of the coefficient matrix was given. For application, three typical differential equations were investigated, that is, the Allen–Cahn equation for checking the maximum‐preserving property, the combustion equation for checking the monotonicity‐preserving property, and the Gray–Scott system for checking the robustness of the proposed scheme. Numerical examples show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Recently developed concept of dissipative measure‐valued solution for compressible flows is a suitable tool to describe oscillations and singularities possibly developed in solutions of multidimensional Euler equations. In this paper we study the convergence of the first‐order finite volume method based on the exact Riemann solver for the complete compressible Euler equations. Specifically, we derive entropy inequality and prove the consistency of numerical method. Passing to the limit, we show the weak and strong convergence of numerical solutions and identify their limit. The numerical results presented for the spiral and the Richtmyer‐Meshkov problem are consistent with our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, the Biot's consolidation model is considered, and Darcy‐Forchheimer equation is used to describe the relationship between the fluid velocity and pressure. The model is nonlinear and the unknown variables are the solid displacement, the fluid velocity and pressure. The Bernardi‐Raugel element is used for displacement and the RT mixed element is used for the fluid velocity and pressure. The parameter‐robust BR‐RT0‐P0 method presented here is uniformly stable not only with respect to the Lamé constant λ$$ \lambda $$, but also with respect to the constrained specific storage coefficient c0$$ {c}_0 $$. We demonstrate the well‐posedness of the BR‐RT0‐P0 finite element solutions, and give the error estimates of the finite element approximations using the monotonicity of the nonlinear Forchheimer term. The error estimates are divided into two cases, that is, c0$$ {c}_0 $$ is positive and c0$$ {c}_0 $$ is nonnegative. Numerical experiments are presented to verify theoretical analysis. Moreover we focus on the variation of pressure by several poroelasticity problems. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the numerical approximation of a three‐parameter family of Boussinesq systems. The systems have been proposed as models of the propagation of long internal waves along the interface of a two‐layer system of fluids with rigid‐lid condition for the upper layer and under a Boussinesq regime for the flow in both layers. We first present some theoretical properties of the systems on well‐posedness, conservation laws, Hamiltonian structure, and solitary‐wave solutions, using the results for analogous models for surface wave propagation. Then the corresponding periodic initial‐value problem is discretized in space by the spectral Fourier Galerkin method and for each system, error estimates for the semidiscrete approximation are proved. The spectral semidiscretizations are numerically integrated in time by a fourth‐order Runge–Kutta‐composition method based on the implicit midpoint rule. Numerical experiments illustrate the accuracy of the fully discrete scheme, in particular its ability to simulate accurately solitary‐wave solutions of the systems. [ABSTRACT FROM AUTHOR]

The time‐fractional nonlinear stochastic fourth‐order reaction diffusion equation perturbed by the noise is paid close attention by the conforming finite element method in this paper. The semi‐ and fully discrete schemes are obtained. Further, the convergence orders of the semi‐ and fully discrete schemes in L2$$ {L}^2 $$ norm are given detailed proof. The numerical tests are gotten to verify the theoretical result. [ABSTRACT FROM AUTHOR]

This paper develops a high‐order discontinuous Galerkin (DG) method for the Camassa‐Holm‐Kadomtsev‐Petviashvili (CH‐KP) type equations on Cartesian meshes. The significant part of the simulation for the CH‐KP type equations lies in the treatment for the integration operator ∂−1$$ {\partial}^{-1} $$. Our proposed DG method deals with it element by element, which is efficient and applicable to most solutions. Using the instinctive energy of the original PDE as a guiding principle, the DG scheme can be proved as an energy stable numerical scheme. In addition, the semi‐discrete error estimates results for the nonlinear case are derived without any priori assumption. Several numerical experiments demonstrate the capability of our schemes for various types of solutions. [ABSTRACT FROM AUTHOR]

Let ℒ be a bounded lattice and α : ℒ → ℒ be its automorphism. In this paper, we study zero‐divisor graph of ℒ with respect to an automorphism α. It is a simple undirected graph and denoted by Γα(ℒ). Some combinatorial structures such as coloring, diameter and girth were given for Γα(ℒ). [ABSTRACT FROM AUTHOR]

The main objective of this paper is to numerically investigate the results of a mathematical model for unsteady magnetohydrodynamics (MHD) boundary layer flow over a porous stretching surface. The analysis of non‐Newtonian Maxwell nanofluid is presented involving the influence of porous media, thermal radiation, viscous dissipation, and joule heating through the Keller box method. Partial slip and convective conditions are also enacted near the boundary. After using the similarity technique on the governing system of nonlinear partial differential equations, the Keller box method is then implemented to find out the numerical solution for copper–water Cu–H2O and molybdenum disulfide MoS2–H2O nanofluids. The impact of various governing flow parameters is interpreted numerically and illustrated graphically on the interaction of particles. Additionally, numerical results are further utilized to calculate the skin friction coefficient and heat transfer rate at the boundary. Finally, in a limiting case, the acquired numerical results are compared with existing results. The remarkable finding of the present study is that Cu–H2O based nanofluid is detected as a superior thermal conductor instead of MoS2–H2O based nanofluid. [ABSTRACT FROM AUTHOR]

In this paper, we investigate novel solutions of fractional‐order option pricing models and their fundamental mathematical analyses. The main novelties of the paper are the analysis of the existence and uniqueness of European‐type option pricing models providing to give fundamental solutions to them and a discussion of the related analyses by considering both the classical and generalized Mittag‐Leffler kernels. In recent years, the generalizations of classical fractional operators have been attracting researchers' interest globally and they also have been needed to describe the dynamics of complex phenomena. In order to carry out the mentioned analyses, we take the Laplace transforms of either classical or generalized fractional operators into account. Moreover, we evaluate the option prices by giving the models' fractional versions and presenting their series solutions. Additionally, we make the error analysis to determine the efficiency and accuracy of the suggested method. As per the results obtained in the paper, it can be seen that the suggested generalized operators and the method constructed with these operators have a high impact on obtaining the numerical solutions to the option pricing problems of fractional order. This paper also points out a good initiative and tool for those who want to take these types of options into account either individually or institutionally. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonlocal‐in‐time Allen‐Cahn equation (NiTACE), which incorporates a nonlocal operator in time with a finite nonlocal memory. Our objective is to examine the well‐posedness of the NiTACE by establishing the maximal Lp$$ {L}^p $$ regularity for the nonlocal‐in‐time parabolic equations with fractional power kernels. Furthermore, we derive a uniform energy bound by leveraging the positive definite property of kernel functions. We also develop an energy‐stable time stepping scheme specifically designed for the NiTACE. Additionally, we analyze the discrete maximum principle and energy dissipation law, which hold significant importance for phase field models. To ensure convergence, we verify the asymptotic compatibility of the proposed stable scheme. Lastly, we provide several numerical examples to illustrate the accuracy and effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical approximation for a two‐dimensional multiterm reaction‐subdiffusion equation, where we adopt an alternating direction implicit (ADI) method combined with the L1 approximation for the multiterm time Caputo fractional derivatives of orders between 0 and 1. Stability and convergence of the full‐discrete L1‐ADI scheme are established. The final convergence in time direction is point‐wise, that is, O(τtkα1−1+τ2α1)$$ O\left(\tau {t}_k^{\alpha_1-1}+{\tau}^{2{\alpha}_1}\right) $$ at t=tk$$ t={t}_k $$. Numerical results are given to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms. [ABSTRACT FROM AUTHOR]

By postprocessing quadratic and eight‐node serendipity finite element solutions on arbitrary triangular and quadrilateral meshes, we obtain new quadratic/serendipity finite volume element solutions for solving anisotropic diffusion equations. The postprocessing procedure is implemented in each element independently, and we only need to solve a 6‐by‐6 (resp. 8‐by‐8) local linear algebraic system for triangular (resp. quadrilateral) element. The novelty of this paper is that, by designing some new quadratic dual meshes, and adding six/eight special constructed element‐wise bubble functions to quadratic/serendipity finite element solutions, we prove that the postprocessed solutions satisfy local conservation property on the new dual meshes. In particular, for any full anisotropic diffusion tensor, arbitrary triangular and quadrilateral meshes, we present a general framework to prove the existence and uniqueness of new quadratic/serendipity finite volume element solutions, which is better than some existing ones. That is, the existing theoretical results are improved, especially we extend the traditional rectangular assumption to arbitrary convex quadrilateral mesh. As a byproduct, we also prove that the new solutions converge to exact solution with optimal convergence rates under H1$$ {H}^1 $$ and L2$$ {L}^2 $$ norms on primal arbitrary triangular/quasi–parallelogram meshes. Finally, several numerical examples are carried out to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance. [ABSTRACT FROM AUTHOR]

In this paper, a second‐order time discretizing block‐centered finite difference method is introduced to solve the compressible wormhole propagation. The optimal second‐order error estimates for the porosity, pressure, velocity, concentration and its flux are established carefully in different discrete norms on non‐uniform grids. Then by introducing Lagrange multiplier, a novel bound‐preserving scheme for concentration is constructed. Finally, numerical experiments are carried out to demonstrate the correctness of theoretical analysis and capability for simulations of compressible wormhole propagation. [ABSTRACT FROM AUTHOR]

The Boussinesq equation has some application in fluid dynamics, water sciences and so forth. In the current paper, we study an improved Boussinesq model. First, a finite difference approximation is employed to discrete the derivative of the temporal variable. Then, we study the existence and uniqueness of solution of the semi‐discrete scheme according to the fixed point theorem. In addition, the unconditional stability and convergence of the semi‐discrete scheme are presented. Then, we construct the fully discrete formulation based upon the radial basis function‐finite difference method. The convergence rate and stability of the fully‐discrete scheme are analyzed. In the end, some examples in 1D and 2D cases are studied to corroborate the capability of the proposed scheme. [ABSTRACT FROM AUTHOR]

An efficient numerical method with high accuracy both in time and in space is proposed for solving d$$ d $$‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an s$$ s $$‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order s+1$$ s+1 $$ in time, when the implicit Runge‐Kutta method with classical order p$$ p $$ (p≥s+1$$ p\ge s+1 $$) is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter α$$ \alpha $$. This improves the previous result which depends on the fractional parameter α$$ \alpha $$. Numerical experiments verify and complement our theoretical results. [ABSTRACT FROM AUTHOR]

Retraction: Gnitchogna R, Atangana A. New two step Laplace Adam‐Bashforth method for integer a noninteger order partial differential equations. Numer. Methods Partial Differ. Eq. 2017; 34:1739–1758. https://doi.org/10.1002/num.22216. The above article, published online on 13 October 2017 in Wiley Online Library (wileyonlinelibrary.com), has been retracted by agreement between the journal Editor in Chief, Clayton Webster, and Wiley Periodicals, LLC. The article was re‐evaluated by the journal's editorial team and additional independent reviewers and a retraction has been agreed due to fundamental flaws in the paper which make the findings unreliable. The authors disagreed with the decision to retract the paper. [ABSTRACT FROM AUTHOR]

The aim of the paper is to develop and analyze a spectrally accurate pseudospectral method in time and space to find the approximate solution of the viscous Burgers' equation. The method is employed in time and space both at Chebyshev‐ Gauss‐ Lobbato (CGL) points. The approximate solution is represented in terms of basis functions. The spectral coefficients are found in such a way that the residual becomes minimum. The given problem is reduced to a system of nonlinear algebraic equations, which is solved by Newton‐Raphson's method. Error estimates for interpolating polynomials are derived. The computational experiments are carried out to corroborate the theoretical results and to compare the present method with existing methods in the literature. [ABSTRACT FROM AUTHOR]

This paper mainly analyzes conservation laws and convergence of splitting conformal multisymplectic scheme for solving the two‐dimensional damped nonlinear fractional Schrödinger equation. In order to obtain conformal multisymplectic scheme, first using Strang splitting method, the original problem is split into conservative multisymplectic and dissipative multisymplectic systems. The conservative multisymplectic system is numerically solved and the dissipative part is solved exactly. And then there shows the corresponding conservation laws and numerical scheme, in which the implicit midpoint method is used in time and the Fourier pseudospectral method is used in space, it also preserves conformal multisymplectic, the global conformal symplectic and global conformal mass conservation laws. Most important of all, we discuss convergence of the proposed scheme which is second‐order accuracy in time and spectral accuracy in space. Finally, the validity and accuracy of the theoretical results are verified by several numerical examples. [ABSTRACT FROM AUTHOR]

The main target of this paper is to present an efficient method to solve a nonlinear free boundary mathematical model of prostate tumor. This model consists of two parabolics, one elliptic and one ordinary differential equations that are coupled together and describe the growth of a prostate tumor. We start our discussion by using the front fixing method to fix the free domain. Then, after employing a nonclassical finite difference and the collocation methods on this model, their stability and convergence are proved analytically. Finally, some numerical results are considered to show the efficiency of the mentioned methods. [ABSTRACT FROM AUTHOR]

A novel Hermite radial basis function‐based differential quadrature (H‐RBF‐DQ) method is presented in this paper based on 2D variable order time fractional advection–diffusion equations with Neumann boundary conditions. The proposed method is designed to treat accurately for derivative boundary conditions, which considerably improve the approximation results and extend the range of applicability for the method of RBF‐DQ. The advantage of the present method is that the Hermite interpolation coefficients are only dependent of the point distribution yielding a substantially better imposition of boundary conditions, even for time evolution. The proposed algorithm is thoroughly validated and is demonstrated to handle the fractional calculus problems with both Dirichlet and Neumann boundaries very well. [ABSTRACT FROM AUTHOR]

Physical laws governing population dynamics are generally expressed as differential equations. Research in recent decades has incorporated fractional‐order (non‐integer) derivatives into differential models of natural phenomena, such as reaction–diffusion systems. In this paper, we develop a method to numerically solve a multi‐component and multi‐dimensional space‐fractional system. For space discretization, we apply a Fourier spectral method that is suited for multidimensional partial differential equation systems. Efficient approximation of time‐stepping is accomplished with a locally one dimensional exponential time differencing approach. We show the effect of different fractional parameters on growth models and consider the convergence, stability, and uniqueness of solutions, as well as the biological interpretation of parameters and boundary conditions. [ABSTRACT FROM AUTHOR]

The LL*‐method is a least‐squares finite element approach producing an approximation by solving dual problem corresponding to the given partial differential equations. Due to the unique structure of LL* approximation, it has advantages if the problem has low regularities and when L2‐approximation needs to be established. As a drawback, piecewise polynomial type approximation often generates artifacts such as spurious oscillations near where shocks or discontinuities occur in solution. Allowing discontinuous piecewise polynomial approximation in LL* seems to exacerbate this trouble. This paper presents a stabilized LL*‐method that is designed to effectively reduce these oscillatory behavior. The consistency and error convergence of proposed method are analyzed and numerical examinations are performed. [ABSTRACT FROM AUTHOR]

In this paper, a linearized mass and energy conservative mixed finite element method (MFEM) is proposed for solving the nonlinear Klein–Gordon–Schrödinger equations. Optimal error estimates without grid‐ratio condition are derived by some rigorous analysis and an error splitting technique, that is, one is the temporal error which is only τ‐dependent and the other is the spatial error which is only h‐dependent. Furthermore, the superclose results are obtained by using the idea of combination of interpolation and projection. Besides, a so‐called "lifting" approach also play an important role to obtain the superclose results. With the above achievements, the global superconvergent properties are deduced through the interpolated post processing operators. Finally, three numerical examples are given to validate the convergence order, unconditional stability, mass, and energy conservation. [ABSTRACT FROM AUTHOR]

In this paper, we derive a residual‐based a posteriori error estimates for nonconforming finite element approximation to parabolic interface problems. The present approach does not involve the Helmholtz decomposition while analyzing the reliability of the estimator. The constants involved in the estimators are independent of the jump of the diffusion coefficient across the interface, and the quasi‐monotonocity assumption on the diffusion coefficient is relaxed. The reliability bound of the estimator consists of the element residual, the edge flux jump and the edge solution jump. The efficiency of the estimator is analyzed by employing a coarsening strategy introduced by Chen and Feng's study. We derive both global upper bound and a local lower bound for the error and an adaptive space–time algorithm is prescribed using the derived estimators. Numerical results illustrating the behavior of the estimators are provided. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast Alikhanov algorithm with nonuniform time steps for a two dimensional distributed‐order time–space fractional advection–dispersion equation. First, an efficient fast Alikhanov algorithm on the general nonuniform time steps for the evaluation of Caputo fractional derivative is presented to sharply reduce the computational work and storage, and are applied to the distributed‐order time fractional derivative or multi‐term time fractional derivative under the nonsmooth regularity assumptions. And a generalized discrete fractional Grönwall inequality is extended to multi‐term fractional derivative or distributed‐order fractional derivative for analyzing theoretically our algorithm. Then the stability and convergence of time semi‐discrete scheme are investigated. Furthermore, we derive the corresponding fully discrete scheme by finite element method and discuss its convergence. At last, the given numerical examples adequately confirm the correctness of theoretical analysis and compare the computing effectiveness between the fast algorithm and the direct method. [ABSTRACT FROM AUTHOR]

A meshless finite point method (FPM) is developed in this paper for the numerical solution of the nonlinear improved Boussinesq equation. A time discrete technique is used to approximate time derivatives, and then a linearized procedure is presented to deal with the nonlinearity. To achieve stable convergence numerical results in space, the stabilized moving least squares approximation is used to obtain the shape function, and then the FPM is adopted to establish the linear system of discrete algebraic equations. To enhance the accuracy and convergence order in time, the Richardson extrapolation is finally incorporated into the FPM. Numerical results show that the FPM is fourth‐order accuracy in both space and time and can obtain highly accurate results in simulating the propagation of a single solitary wave, the interaction of two solitary waves, the solitary wave break‐up and the solution blow‐up phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the development and study of a numerical method based on the idea of kernel‐based approximation and finite difference discretization to obtain the solution for the time‐fractional diffusion equation. Using the theory of reproducing kernel, reproducing kernel functions with a polynomial form will be established in polynomial reproducing kernel spaces spanned by the Chebychev basis polynomials. In the numerical method, first the time‐fractional derivative term in the aforementioned equation is approximated by using the finite difference scheme. Then, by the help of collocation method based on reproducing kernel approximation, we will illustrate how to derive the numerical solution in polynomial reproducing kernel space. Finally, to support the accuracy and efficiency of the numerical method, we provide several numerical examples. In numerical experiments, the quality of approximation is calculated by absolute error and discrete error norms. [ABSTRACT FROM AUTHOR]

The determination of an unknown time‐dependent zero‐order coefficient in a fourth‐order parabolic problem is investigated form the integral‐type observation in this paper. Under certain assumptions, the well‐posedness of the weak solution to the inverse problem is obtained. Furthermore, its global solvability can be proved by applying some transformations. For the numerical reconstruction of the unknown coefficient, three methods are established, including the transformations used above, the time‐discrete method with the cubic spline function method, and the optimization method. The convergence and error estimates for the time‐discrete method are derived rigorously. Furthermore, the corresponding predictor–corrector scheme is introduced to determine the unknown quantity numerically. For the optimization method, the minimizer of the objective functional is applied to approximate the unknown coefficient. The convergence rates of the optimization problem is proved under some suitable source condition. In addition, the Fréchet derivative of the objective functional is obtained and utilized to establish the conjugate gradient algorithm. The numerical example shows that all the three methods can be applied to numerically recover the unknown quantity efficiently. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the study of a kernel‐based method in pseudo‐spectral (PS) mode for multi‐term and distributed order time‐fractional diffusion equations. Using the theory of reproducing kernel, reproducing kernel functions will be established in reproducing kernel Hilbert space. In the proposed method, a finite difference scheme is used in temporal space to achieve a semi‐discrete configuration. Then, with the help of the kernel‐based PS method, we will illustrate how to derive the numerical solution. Finally, to support the accuracy and efficiency of the proposed method, we provide several numerical examples. In numerical experiments, the quality of the approximation is calculated by absolute error and discrete error norms. [ABSTRACT FROM AUTHOR]

We present a C0 interior penalty finite element method for the sixth‐order phase field crystal equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. We remark that the novelty of this paper lies in the fact that this is the first C0 interior penalty finite element method developed for the phase field crystal equation. Additionally, the error analysis presented develops a detailed methodology for analyzing time dependent problems utilizing the C0 interior penalty method. We furthermore benchmark our method against numerical experiments previously established in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we develop a gradient recovery based linear (GRBL) finite element method (FEM) and a Hessian recovery based linear FEM for second‐order elliptic equations in non‐divergence form. The elliptic equation is casted into a symmetric non‐divergence weak formulation, in which second‐order derivatives of the unknown function are involved. We use gradient and Hessian recovery operators to calculate the second‐order derivatives of linear finite element approximations. Although, thanks to low degrees of freedom of linear elements, the implementation of the proposed schemes is easy and straightforward, the performances of the methods are competitive. The unique solvability and the H2$$ {H}^2 $$ seminorm error estimate of the GRBL scheme are rigorously proved. Optimal error estimates in both the L2$$ {L}^2 $$ norm and the H1$$ {H}^1 $$ seminorm have been proved when the coefficient is diagonal, which have been confirmed by numerical experiments. Superconvergence in errors has also been observed. Moreover, our methods can handle computational domains with curved boundaries without loss of accuracy from approximation of boundaries. Finally, the proposed numerical methods have been successfully applied to solve fully nonlinear Monge–Ampère equations. [ABSTRACT FROM AUTHOR]

In this paper, a fully coupled system of transient Navier–Stokes fluid flow model and unsteady variable coefficient advection–diffusion–reaction transport model has been studied through subgrid multiscale stabilized finite element method. In particular algebraic approach of approximating the subscales has been considered to arrive at the stabilized variational formulation of the coupled system and standard expressions for the stabilization parameters have been proposed. The unknown subgrid scales are considered to be time dependent. The consideration of the fluid viscosity coefficient depending upon the concentration of the solute mass makes this coupling strong. Fully implicit backward Euler scheme has been employed for time discretization. Stability analysis of the stabilized formulation has been conducted. Furthermore detailed derivations of both apriori$$ apriori $$ and aposteriori$$ aposteriori $$ error estimates for the stabilized finite element scheme have been carried out. The performance of the proposed scheme is validated for benchmark problems as well as the credibility of the stabilized method is also established well through various numerical experiments. [ABSTRACT FROM AUTHOR]