In this paper for the first time, two kinds of energy-preserving finite element approximation schemes, which are based upon the standard finite element method (FEM) and the mixed FEM, respectively, are developed and analyzed for a class of nonlinear wave equations. The energy conservation and the optimal convergence properties are obtained for both finite element schemes in their respective norms,

This paper deals with the asymptotic boundedness, stability and strong convergence of the split-step theta method for highly nonlinear neutral stochastic delay integro differential equation. Under the generalized coercive condition, we prove that the split-step theta approximation strongly converges to the exact solution of the underlying equation, for θ∈[1/2,1]. Besides, we also prove that when θ∈(1/2

The greedy randomized coordinate descent (GRCD) method is an effective iterative method for solving large linear least-squares problems. In this work, we construct a class of relaxed greedy randomized coordinate descent (RGRCD) methods by introducing a relaxation parameter in the probability criterion. Then, we prove the convergence properties of these methods when the coefficient matrix of the linear

The projection methods with vanilla inertial extrapolation step for variational inequalities have been of interest to many authors recently due to the improved convergence speed contributed by the presence of inertial extrapolation step. However, it is discovered that these projection methods with inertial steps lose the Fejér monotonicity of the iterates with respect to the solution, which is being

In this paper, we develop a novel second order in time, decoupled, energy stable finite element scheme for simulation of Cahn-Hilliard-Hele-Shaw system. The idea of scalar auxiliary variable approach is introduced to handle the nonlinear bulk. An operator-splitting strategy is utilized to fully decouple the coupled Cahn-Hilliard equation and Darcy equation. A full discretization is built in the framework

This work studies a parabolic-parabolic chemotactic PDE's system which describes the evolution of a biological population “U” and a chemical substance “V”, using a Generalized Finite Difference Method, in a two dimensional bounded domain with regular boundary. In a previous paper [12], the authors asserted global classical solvability and periodic asymptotic behavior for the continuous system in 2D

We mainly focus on developing a reduced-order extrapolated finite difference iterative (ROEFDI) method for the Riemann-Liouville tempered fractional derivative (RLTFD) equation. For this reason, we firstly establish a finite difference iterative (FDI) scheme in matrix-form for the RLTFD equation and provide the stability and errors for the FDI solutions. We then develop the ROEFDI method with a few

The performed numerical analysis reveals that Wynn's identity for the compass 1/(N−C)+1/(S−C)=1/(W−C)+1/(E−C)=1/η (here C stands for centre, the other letters correspond to the four directions of the compass) gives the long sought criterion, the minimal |η| or -criterion, for the choice of the optimal Padé approximant. The work of this method is illustrated by calculation of multipoint Padé approximation

Two types of first-order decoupled conservative schemes are firstly proposed for the strongly coupled nonlinear Schrödinger system by using pseudospectral method in space and coordinate increment discrete gradient method in time. And then, in order to improve the solution accuracy in time, the composition methods are employed to construct second- and fourth-order schemes. The proposed schemes are efficient

Optimization problems involving the sum of three convex functions have received much attention in recent years, where one is differentiable with Lipschitz continuous gradient, one is composed of a linear operator and the other is proximity friendly. The primal-dual fixed point algorithm is a simple and effective algorithm for such problems. To exploit the second-order derivatives information of the

The advantage of WENO-JS scheme (1996) [22] over the WENO-LOC scheme (1994) [27] is that the WENO-LOC non-linear weights do not achieve the desired order of convergence in smooth monotone regions and at regions containing critical points. In this article, this drawback is overcome with the WENO-LOC smoothness indicators by constructing ‘WENO-Z type’ non-linear weights with a novel global smoothness

A stable least residue method for solving a nonlinear fractional integro-differential equation with a weakly singular kernel in the reproducing kernel space is proposed. To solve the equation, the multiwavelets bases in the reproducing kernel space are constructed based on the cubic Legendre wavelets in L2[0,1]. The best approximate solution could be obtained by solving the normal equation. Meanwhile

To discretize the distributed-order term of two-dimensional nonlinear Riesz space fractional diffusion equation, we consider the high accuracy Gauss-Legendre quadrature formula. By combining an s-stage implicit Runge-Kutta method in temporal direction with a spectral Galerkin method in spatial direction, we construct a numerical method with high global accuracy. If the nonlinear function satisfies

In this paper, two new efficient energy dissipative difference schemes for the strongly coupled nonlinear damped space fractional wave equations are first set forth and analyzed, which involve a two-level nonlinear difference scheme, and a three-level linear difference scheme based on invariant energy quadratization formulation. Then the discrete energy dissipation properties, solvability, unconditional

Based on the joint bidiagonalization (JBD) process of the matrix pair {A,L}, an iterative regularization algorithm, called JBDQR, is proposed and developed for large scale linear discrete ill-posed problems in general-form Tikhonov regularization. It is proved that the JBDQR iterates take the form of attractive filtered generalized singular value decomposition (GSVD) expansions, where the filters are

We discrete the ergodic semilinear stochastic partial differential equations in space dimension d≤3 with additive noise, spatially by a spectral Galerkin method and temporally by an exponential Euler scheme. It is shown that both the spatial semi-discretization and the spatio-temporal full discretization are ergodic. Further, convergence orders of the numerical invariant measures, depending on the

Modeling an infinite domain arises while simulating physical applications frequently. It is like attempting to model the effects and properties of the ocean in a bucket. In this paper, we develop a spectral Galerkin method based on exponential Jacobi functions (EJF) in unbounded domains. We also establish some basic results on exponential Jacobi orthogonal approximations, which serve as the mathematical

For a class of tempered fractional integro-differential equation of the Caputo type, a comparative study of three numerical schemes is presented in this paper. The schemes discussed are Linear, Quadratic and Quadratic-Linear schemes. Four numerical examples are considered to discuss error estimate and convergence order of the numerical schemes for different values of step-size h and the parameter λ

In the past years several unconditionally positivity preserving numerical methods for production-destruction equations arising in a wide variety of real life applications were developed. The BBKS schemes suggested by Bruggemann et al. (2007) [4] and Broekhuizen et al. (2008) [3] modifications of Heun's method and conserve all linear invariants while still maintaining positivity independent of the time

We consider the strong convergence of the numerical methods for solving stochastic subdiffusion problem driven by an integrated space-time white noise. The time fractional derivative is approximated by using the L1 scheme and the time fractional integral is approximated with the Lubich's first order convolution quadrature formula. We use the Euler method to approximate the noise in time and use the

In this paper, a second order accurate time and space difference method is proposed to solve the nonlinear two-sided space-fractional diffusion equation with variable coefficients. We combine the alternating direction implicit method and Bi-conjugate gradient stabilized iterative solver to develop an efficient numerical method for a two dimensional, nonlinear, two-sided space-fractional diffusion equation

In this paper, we develop a spectral method with essential imposition of nonhomogeneous Neumann boundary condition on quadrilaterals. Results on irrational orthogonal approximation are established, which facilitate the numerical solutions of partial differential equations of the sort. As applications, we provide spectral schemes for two model problems, and prove their spectral accuracy. Efficient numerical

In this paper, two-grid algorithms based on expanded mixed finite element method are presented for solving two-dimensional semilinear time fractional reaction-diffusion equations. To obtain the fully discrete scheme, the classical L1 scheme is considered in the time direction, and the expanded mixed finite element method is used to approximate spatial direction. Then the error estimates and stability

In this paper we develop a family of central schemes for the one and two-dimensional systems of Euler equations with gravitational source term. The proposed schemes are unstaggered, second-order, central finite volume schemes that avoid solving Riemann problems at the cell interfaces and avoid switching between an original and a staggered grid. The main feature of the schemes developed here is that

In this paper, we revisit the multilinear PageRank problem. Under the framework of tensor, we establish several new and tighter uniqueness conditions for the multilinear PageRank vector. Meanwhile, a refined error bound for the inverse iteration as well as the new perturbation bounds under different norms, which improve the existing ones in the current literature, are developed with feasible computations

This paper is concerned with the construction and analysis of a high-order compact finite difference method for a class of time-fractional sub-diffusion equations under the Robin boundary condition. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α∈(0,1). A (3−α)th-order numerical formula (called the L2 formula

In this study, the well-known Cauchy problem of elliptic type equation possibly with variable coefficient is contemplated while a part of right-hand side source is unknown as well whereas overspecified boundary data is imposed on boundary. It is a supposition that the right-hand side source can be observed as a sum of two-parts which are independent of each other and at the same time, each of them

We introduce second derivative two-step collocation methods for the numerical integration of ordinary differential equations. In these methods, the solution of the problem in each step depends on the numerical solution in some points in the two previous steps. These methods are obtained using the collocation approach by relaxing some of the collocation conditions to obtain methods with desirable stability

This paper presents a multi-objective optimization approach for choosing an appropriate regularization parameter in Tikhonov-type regularization of discrete ill-posed problems. Finding the regularization parameter is formulated as a multi-objective optimization problem and some transformations are introduced to construct a new problem equivalent to it. Then, by using the standard multi-objective optimization

In this study, a new multiscale algorithm was proposed to solve the boundary value problems of second order differential equations. A multiscale basis consisting of two sets of multiscale functions was constructed in the reproducing kernel space, and the proposed multiscale basis was proved to be orthonormal. The ε− approximate solution was defined, and then it was proved to be the optimal solution

In this paper, the conservative Fourier spectral scheme is presented for higher order Klein-Gordon-Schrödinger (KGS) system with periodic boundary condition. First, using the Fourier collocation scheme in space, we obtain an conservative Fourier spectral scheme for the higher order KGS system. We prove that the proposed method satisfies the discrete mass and energy conservation laws exactly. The existence

In this paper, by constructing equivalently the absolute value equation (AVE) as the new block equation of two-by-two, we present a new SOR-like (NSOR) method for solving the AVE. The convergence and optimal parameters of NSOR method for solving the AVE are studied in detail. Numerical experiments are presented to show that the NSOR method is implementable and efficient.

We present and analyse a numerical method for understanding the low-inertia dynamics of an open, inextensible viscoelastic rod - a long and thin three dimensional object - representing the body of a long, thin microswimmer. Our model allows for both elastic and viscous, bending and twisting deformations and describes the evolution of the midline curve of the rod as well as an orthonormal frame which

Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the inverse power-law kernels deteriorates the accuracy and efficiency of many existing numerical methods. We develop two efficient first- and second-order implicit-explicit

In this paper, we study and analyse the hybridized discontinuous Galerkin method for the reaction-diffusion problem. We present a mixed formulation based on some operators with discontinuous piecewise polynomials in a shape regular partition. We prove the well posedness of the mixed scheme and establish some error estimates with optimal order of convergence. Finally, we give some numerical simulations

A class of tempered fractional differential equations with terminal value problems are investigated in this paper. Discretized collocation methods on piecewise polynomials spaces are proposed for solving these equations. Regularity results are constructed on weighted spaces and convergence order is studied. Several examples are supported the theoretical parts and compared with other methods.

Dynamical low-rank approximation to the solutions of matrix differential equations leads to differential equations for the factors of a low-rank factorization of the matrices. Error bounds depending on the Lipschitz constant of the problem become not satisfactory in the case of parabolic problems with a linear stiff term and a smooth nonstiff nonlinearity. In this paper, we provide sharper error bounds

In this paper, variable order Burgers' equations with proportional delays in time and space are studied. The variable order derivatives are defined in the sense of Atangana and Baleanu. Two types of variable order Atangana and Baleanu definitions are presented here. The nonstandard weighted average finite difference method is developed to study numerically the proposed models problem in one and two

Eigenvector derivatives of eigenproblems analytically dependent on parameters are required in diverse technical fields such as structural design, system identification. Modal methods, a kind of popular methods in engineering problems, calculate eigenvector derivatives by expressing them as linear combinations of eigenvectors of eigenproblems. In this paper, an incomplete modal method is presented to

In this paper, we propose a fast compact implicit integration factor (FcIIF) method with non-uniform time meshes for solving the two-dimensional nonlinear Riesz space-fractional reaction-diffusion equation. The weighted and shifted Grüwald-Letnikov (WSGD) approximation is employed to the spatial discretization of the equation, and a system of nonlinear ordinary differential equations (ODEs) in matrix

In this paper, we study the immersed finite element (IFE) method for the multi-layer porous wall model for the drug-eluting stents. Firstly, under some realistic assumptions on the initial data and the body force, we deduce the long time priori stability for the variational problem for all t∈R+. Then, based on these results, the uniform optimal error estimates for both the semi-discrete IFE method

The main target of the paper is solving a system of two-dimensional Volterra integral equations (2-DVIE) and receiving convincing solutions. We change the problem to a system of simpler equations by Legendre wavelets. By merging properties of wavelets to system and reducing the number of variables we build more powerful tools than some other methods. 3 examples and proof of convergence verify the applicability

This paper studies fixed-step convergence of implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties. Classical, index-1 differential algebraic equation, and singular perturbation convergence analyses results are given. For all these problems IMEX GLMs from the class of interest converge with

This paper aims to introduce an asymptotically exact a posteriori error estimator for non-selfadjoint Steklov eigenvalue problem arising from inverse scattering by using the complementary technique, which provides an asymptotically exact estimate for eigenpair of non-selfadjoint Steklov eigenvalue problem. Besides, as its applications, we design a novel cascadic adaptive method for non-selfadjoint

Recently there has been a growing interest in designing efficient numerical methods for the solution of fractional differential equations. The solutions of such equations in general exhibit a weak singularity near the initial time. In this paper, we propose finite difference/spectral methods to solve the coupled nonlinear time-space fractional Schrödinger equations with non-smooth solutions in the

In this paper, we consider several splitting schemes for unsteady problems for the most common phase-field models. The fully implicit discretization of such problems would yield at each time step a nonlinear problem that involves second- or higher-order spatial operators. We derive new factorization schemes that linearize the equations and split the higher-order operators as a product of second-order

This report extends the mathematical support of a subgrid artificial viscosity (SAV) method to simulate the incompressible Navier-Stokes equations to better performing a linearly extrapolated BDF2 (BDF2LE) time discretization. The method considers the viscous term as a combination of the vorticity and the grad-div stabilization term. SAV method introduces global stabilization by adding a term, then

A generalized collocation method for solving a weakly singular Fredholm integro-differential equation with Kalman kernel is proposed in reproducing kernel space. To obtain the generalized collocation method, the multiwaves basis in reproducing kernel space Wn+1[0,b] is constructed based on Legendre multiwaves in L2[0,1]. Using the multiwaves basis, we propose ε-approximate solutions and use the method

In this paper we introduce the semiregularity property for a family of decompositions of a polyhedron into a natural class of prisms. In such a family, prismatic elements are allowed to be very flat or very long compared to their triangular bases, and the edges of quadrilateral faces can be nonparallel. Moreover, the triangular faces of each element are either parallel or skew to each other. To estimate

In this paper, we present a continuous and discrete time weak Galerkin finite element method (WG-FEM) for solving two dimensional time fractional coupled Burgers' equations. The stability is proved for discrete time WG-FEM and the optimal order error in L2-norm is obtained based on fractional derivative definition, fractional integral definition and dual argument technique for continues and discrete

In the construction of cell-centered finite volume schemes for general diffusion equations with discontinuous coefficients on distorted meshes, auxiliary unknowns are usually introduced to resolve discontinuities across cell-edges. In general, it is difficult to express unconditionally auxiliary unknowns as a convex combination of primary unknowns around. As we know, all existing methods have to impose

We introduce a new generalized Caputo-type fractional derivative which generalizes Caputo fractional derivative. Some characteristics were derived to display the new generalized derivative features. Then, we present an adaptive predictor corrector method for the numerical solution of generalized Caputo-type initial value problems. The proposed algorithm can be considered as a fractional extension of

Comparison principles are developed for piecewise linear finite element approximations of quasilinear elliptic partial differential equations. We consider the analysis of a class of nonmonotone Leray-Lions problems featuring both nonlinear solution and gradient dependence in the principal coefficient, and a solution dependent lower-order term. Sufficient local and global conditions on the discretization

We herein propose a new method that combines the inexact Newton method with a procedure to obtain a feasible inexact projection for solving constrained smooth and nonsmooth equations. Local convergence theorems are established under the assumption of smoothness or semi-smoothness of a function that defines the equations and the regularity of the solution. In particular, we demonstrate that a sequence

In this paper, we develop an effective numerical method for solving the fractional sub-diffusion equation with Neumann boundary conditions. The time fractional derivative is approximated by the L1 scheme on graded meshes, the spatial discretization is done by using the compact finite difference methods. By adding some corrected terms, the fully discrete alternating direction implicit (ADI) method is

This paper aims to construct structure-preserving numerical schemes for the two-dimensional space fractional Klein-Gordon-Schrödinger equation, which are based on the newly developed partitioned averaged vector field methods. First, we derive an equivalent equation, and reformulate the equation as an infinite dimensional canonical Hamiltonian system by virtue of the variational derivative of the functional

This paper is concerned with the numerical solutions of nonlinear Schrödinger equation (NLSE) in one-dimensional unbounded domain. The unbounded problem is truncated by the third-order absorbing boundary conditions (ABCs), and the corresponding initial-boundary value problem is solved by a three-level linearized difference scheme. By introducing auxiliary functions to simplify the difference scheme

In this paper, we discuss the construction of a class of implicit-explicit (IMEX) methods for systems of ordinary differential equations which their right hand side can be split into two parts; nonstiff or mildly stiff part and stiff part. The proposed methods treat the non-stiff part by an explicit second derivative diagonally implicit multistage integration method (SDIMSIM) and the stiff part by

In this work we show that numerical integration during sliding motion, for piecewise-smooth systems, can be performed without the need to project the approximate solutions on the constraints' surface. We give a general algorithm, an order of approximation result, and show the effectiveness of the technique on several examples. Comparison between projected and non-projected methods and the standard

A continuation of previous authors' work on adaptive parameter estimation for linear dynamical systems having irrational transfer function is presented in this work. An original modification of the gradient algorithm, inspired by the variable structure control techniques and additionally featuring a time-varying adaptation gain, is presented and analyzed using Lyapunov techniques. The exposition is