The a posteriori error estimates for finite element approximations to the governing equations of heat and mass transfer in fluidized beds are derived in this work. These are a system of five time dependent coupled nonlinear convection-diffusion-reaction equations. Based on the variational formulation, computable residual based a posteriori error estimates are obtained. The time discretization has been

Efficient and fast explicit methods are proposed to solve nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed methods produce the second-order for linear interpolation and the third-order accuracy for quadratic interpolation, respectively. The convergence analysis is proved by using discrete

This paper proposes a method based on two-dimensional Euler polynomials combined with Gauss-Jacobi quadrature formula. The method is used to solve two-dimensional Volterra integral equations with fractional order weakly singular kernels. Firstly, we prove the existence and uniqueness of the original equation by Gronwall inequality and mathematical induction method. Secondly, we use two-dimensional

For a class of optimal control problems constrained with certain time- and space-fractional diffusive equations, by making use of mixed discretizations of temporal finite-difference and spatial finite-element schemes along with Lagrange multiplier approach, we obtain specially structured block two-by-two linear systems. We demonstrate positive definiteness of the coefficient matrices of these discrete

By combining the defect-correction method with the two-level discretization strategy and the local pressure projection stabilized method, this paper presents and studies two kinds of two-level defect-correction stabilized algorithms for the simulation of 2D/3D steady Navier-Stokes equations with damping, where the lowest equal-order P1−P1 finite elements are used for the velocity and pressure approximations

The epitaxial thin film growth model without slope selection is the L2-gradient flow of energy with a logarithmic potential in terms of the gradient of a height function. A challenge to numerically solving the model is how to treat the nonlinear term to preserve energy stability without compromising accuracy and efficiency. To resolve this problem, we present a high-order energy stable scheme by placing

A new procedure is presented for computing the matrix cosine and sine simultaneously by means of Taylor polynomial approximations. These are factorized so as to reduce the number of matrix products involved. Two versions are developed to be used in single and double precision arithmetic. The resulting algorithms are more efficient than schemes based on Padé approximations for a wide range of norm matrices

In this paper, we deal with the existence and stability of an equilibrium age distribution of the linearly implicit Euler-Riemann method for nonlinear age-structured population model with density dependence, i.e., the Gurtin-MacCamy models. It is shown that a dynamical invariance is replicated by numerical solutions for a long time. With the help of infinite-dimensional Leslie operators, the numerical

In this paper we consider a fractional spectral collocation method for solving linear multi-term fractional differential equations involving Caputo-type fractional derivative. By using integral equation method, the boundary value problem of the fractional differential equation is reformulated as an equivalent Volterra-Fredholm integral equation with weakly singular kernels. Then we apply the fractional

We develop a preconditioned fast divided-and-conquer finite element approximation for the initial-boundary value problem of variable-order time-fractional diffusion equations. Due to the impact of the time-dependent variable order, the coefficient matrix of the resulting all-at-once system does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix

In this paper, we propose a novel numerical approach to construct unconditionally energy stable schemes for the modified phase field crystal (MPFC) model. The new technique is based on the invariant energy quadratization (IEQ) method. The numerical schemes based on IEQ approach lead to time-dependent dense matrices, thus the fast Fourier transform (FFT) is difficult to be applied to solve the systems

In this paper, we construct a circulant-matrix-based new accelerated GSOR (CNAGSOR) iteration method for a class of large and sparse block two-by-two linear systems of generalized saddle-point structure. Theoretical results about the convergence properties and eigenvalues distribution of the preconditioning matrix are studied in detail. Implementations in the image restoration problem and in the PDE-constraint

In this paper, we propose an optimally convergent discontinuous Galerkin (DG) method for nonlinear third-order ordinary differential equations. Convergence properties for the solution and for the two auxiliary variables that approximate the first and second derivatives of the solution are established. More specifically, we prove that the method is L2-stable and provides the optimal (p+1)-th order of

In this article, we implement and analyze a locally modified parametric finite element method for fluid structure interaction problems using LBB-stable finite elements. The variational formulation of the monolithically coupled fluid structure interaction problems is solved using a fully Eulerian framework. A combination of Q2-Q2-(Q1+Q0dc) and P2-P2-(P1+P0dc⁎) finite elements are used to approximate

In this work, we revisit the spectral Petrov-Galerkin method for fractional elliptic equations with the general fractional operators. To prove the optimal convergence of the method, we first present the ultra-weak formulation and establish its well-posedness. Then, based on such a novel formulation, we are able to prove the discrete counterpart and obtain the optimal convergence of the spectral method

We apply the spectral-Lagrangian method of Gamba and Tharkabhushanam for solving the homogeneous Boltzmann equation to compute the low probability tails of the velocity distribution function, f, of a particle species. This method is based on a truncation, Qtr(f,f), of the Boltzmann collision operator, Q(f,f), whose Fourier transform is given by a weighted convolution. The truncated collision operator

Many application problems that lead to solving linear systems make use of preconditioned Krylov subspace solvers to compute their solution. Among the most popular preconditioning approaches are incomplete factorization methods either as single-level approaches or within a multilevel framework. We will present a block incomplete factorization that is based on skillfully blocking the system initially

In this article, a mathematical model representing solution of the nonlinear generalized equal width (GEW) equation has been considered. Here we aim to investigate solutions of GEW equation using a numerical scheme by using sextic B-spline Subdomain finite element method. At first Galerkin finite element method is proposed and a priori bound has been established. Then a semi-discrete and a Crank-Nicolson

In this paper, a novel numerical scheme is proposed to numerically solve the fractional activator-inhibitor system, which is a coupled nonlinear model. In the temporal direction, we employ two families of novel fractional θ-methods, the FBT-θ and FBN-θ methods, in spatial direction, the Legendre spectral method is used. Based on some positivity properties of the coefficients of both methods, the stability

Finding analytical and semi-analytical solutions of two-dimensional nonlinear fractional-order problems arising in mathematical physics is a challenging task for research community. In this work, an innovative scheme is proposed based on the shifted Gegenbauer wavelets. For this purpose first, we introduce shifted Gegenbauer polynomials via suitable transformation. Then, we present three-dimensional

The aim of this paper is to tackle the numerical calculation of the so-called finite-part integrals, including Cauchy principal value integrals and any supersingular integral. The numerical procedure consists of replacing the density function f(t) by the polynomial that is the best fit in a least-squares sense for the values f(zk), where {zk} is a set of distinct points located on an ellipse that surrounds

The purpose of this paper is twofold: to study the superconvergence properties and to present an efficient and reliable a posteriori error estimator for the local discontinuous Galerkin (LDG) method for linear second-order elliptic problems on Cartesian grids. We prove that the LDG solution is superconvergent towards a particular projection of the exact solution. The order of convergence is proved

This article devises a new primal-dual weak Galerkin finite element method for the convection-diffusion equation. Optimal order error estimates are established for the primal-dual weak Galerkin approximations in various discrete norms and the standard L2 norms. A series of numerical experiments are conducted and reported to verify the theoretical findings.

Increased attention has been paid on numerical modeling of interface problems as its wide applications in various aspects of science. Motivated by enhancing the application of the reproducing kernel, we focus on establishing a broken cubic spline spaces and developing a fourth-order numerical scheme for one-dimensional elliptic interface problems in this paper. The method is based on the least-squares

The paper considers the problem of computing the values of the Atangana-Baleanu derivative in Caputo sense which arises while solving fractional partial differential equations. In such case the values of the derivative are needed to be calculated in recurrent manner with increasing value of time or space variable. We consider series expansion and integral representations of the Mittag-Leffler function

In this paper, we propose and analyze two conservative fourth-order compact finite difference schemes for the (1+1) dimensional nonlinear Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise forms of the proposed compact schemes into equivalent vector forms and analyze their conservative and convergence properties. We prove that the proposed schemes

This paper focuses on developing the reduced-order extrapolated model based on the splitting implicit finite difference (SIFD) scheme and the proper orthogonal decomposition (POD) for the two-dimensional (2D) fourth-order nonlinear Rosenau equation. For this purpose, we first construct the SIFD scheme and analyze the stability and convergence. And then, we develop a reduced-order extrapolated SIFD

In this paper, we consider the weak Galerkin finite element approximations of second order linear parabolic problems in two dimensional convex polygonal domains under the low regularities of the solutions. Optimal order error estimates in L2(L2) and L2(H1) norms are shown to hold for both the spatially discrete continuous time and the discrete time weak Galerkin finite element schemes, which allow

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. A particular complimentary error function is identified which matches the discontinuity in the initial condition. The difference between this analytical function and the solution of the parabolic problem is approximated numerically. A coordinate transformation is used so that a

In this work, the study of suspended sediment transport under unsteady, uniform and non equilibrium condition is extended using space fractional diffusion equation (fADE) with parameter α. Semi analytical solutions of this space fADE with realistic boundary conditions are obtained using spectral method developed on Chebyshev orthogonal polynomials. Solutions obtained from this method are compared with

This article aims to investigate the pathwise convergence of the higher order scheme, introduced by Jentzen (2011) [9], for the stochastic Burgers' equation (SBE) driven by space-time white noise. In particular, first and second order derivatives of the non-linear drift term of the SBE are assumed to be defined and bounded in Sobolev spaces using the definition of distribution derivative i.e. Lemma

Interpolation methods for nonlinear finite element discretizations are commonly used to eliminate the computational costs associated with the repeated assembly of the nonlinear systems. While the group finite element formulation interpolates nonlinear terms onto the finite element approximation space, we propose the use of a separate approximation space that is tailored to the nonlinearity. In many

This paper presents a new type of local and parallel multigrid method to solve semilinear elliptic equations. The proposed method does not directly solve the semilinear elliptic equations on each layer of the multigrid mesh sequence, but transforms the semilinear elliptic equations into several linear elliptic equations on the multigrid mesh sequence and some low-dimensional semilinear elliptic equations

The scaling of the exact solution of a hyperbolic balance law generates a family of scaled problems in which the source term does not depend on the current solution. These problems are used to construct a sequence of solutions whose limiting function solves the original hyperbolic problem. Thus this gives rise to an iterative procedure. Its convergence is demonstrated both theoretically and analytically

We propose to solve elliptic interface problems by a meshless finite difference method, where the second order elliptic operator and jump conditions are discretized with the help of the QR decomposition of an appropriately rescaled multivariate Vandermonde matrix with partial pivoting. A prescribed consistency order is achieved on irregular nodes with small influence sets, which allows to place the

In this paper, a posteriori error estimates for the Weak Galerkin finite element methods (WG-FEMs) for second order elliptic problems are derived in terms of an H1−equivalent energy norm. Corresponding estimators based on the helmholtz decomposition yield globally upper and locally lower bounds for the approximation errors of the WG-FEMs. Especially, the error analysis of our methods is proved to be

This study aims to efficiently solve the space-time fractional partial integro-differential equations with spatial-time delays, employing a fast numerical methodology dependent upon the matching polynomial of complete graph and matrix-collocation procedure. This methodology provides a sustainable approach for each computation limit since it arises from the durable graph structure of complete graph

In this paper, we study preconditioners for multilevel Toeplitz linear systems arising from discretization of steady-state and evolutionary advection-diffusion equations, in which upwind scheme and central difference scheme are employed to discretize first-order and second-order terms, respectively. For the steady-state case, the preconditioner is constructed by replacing each of the discrete advection

In this paper, we generalize a coupled system of nonlinear reaction-advection-diffusion equations to a variable-order fractional one by using the Caputo-Fabrizio fractional derivative, which is a non-singular fractional derivative operator. In order to establish an appropriate method for this system, we introduce a new formulation of the discrete Legendre polynomials namely the orthonormal shifted

In this paper, the notion of generalized split feasibility problem (GSFP) is studied in p-uniformly convex Banach spaces. Some special cases of the GSFP are highlighted. A self-adaptive step-size iterative algorithm which converges strongly to solution of the GSFP is proved. The implementation of the method is demonstrated with two numerical examples. Our method does not require prior information of

In this paper, we study high-order compact schemes for semilinear parabolic moving boundary problems. We first convert the original problem into an equivalent one defined on a rectangular region by introducing a linear transformation and the well-known exponential transformation. Next, we derive a compact scheme with fourth-order accuracy in the spatial dimension and second-order accuracy in the temporal

We consider a new second-order hydrostatic reconstruction (HR) based on the bottom topography, the depth-averaged velocity, and the water surface level for the shallow water flows with discontinuous bottom topography. The Audusse's second-order hydrostatic reconstruction (HR) (Audusse et al. (2004) [1]) fails to correctly reflect the wave pattern for the large discontinuous bottom topography. We propose

In this paper, we propose a new finite difference scheme for the 3D Helmholtz problem, which is compact and fourth-order in accuracy. Different from a standard compact fourth-order one, the new scheme is specially established based on minimizing the numerical dispersion, by approximating the zeroth-order term of the equation with a weighted-average for the values at 27 points. To determine optimal

Whether high order temporal integrators can preserve the maximum principle of Allen-Cahn equation has been an open problem in recent years. This work provides a positive answer by designing and analyzing a class of up to fourth order maximum principle preserving integrators for the Allen-Cahn equation. First, the second order finite difference discretization is applied to the Allen-Cahn equation in

In this paper, we construct a new efficient numerical scheme for variable fractional order differential equation of fourth order with delay. Here, we propose to use parametric quintic spline in the spatial dimension and L2−1σ formula for time dimension. The stability, convergence, and solvability are rigorously proved using discrete energy method. Our proposed scheme improves convergence in both aspects

We study two mathematical models consisting of nonlinear systems of partial differential equations, a predator prey and a competitive two-species chemotaxis systems with two chemicals satisfying their corresponding elliptic equations in a smooth bounded domain. By introducing global factors, for different ranges of parameters and by deriving a discretization of the system by means of the Generalized

The main focus of this paper is on studying an order optimal regularization scheme based on the Meyer wavelets method to solve the analytic continuation problem in the high-dimensional complex domain Ω:={x+iy∈CN:x∈RN,‖y‖≤‖y0‖,y,y0∈R+N}. This problem is exponentially ill-posed and suffers from the Hadamard's instability. Theoretically, we first provide an optimal conditional stability estimate for the

Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic time scales. The multimethod approach discretizes each physical process with an appropriate numerical method; the methods are coupled appropriately such that the

In this article, an idea based on moving least squares (MLS) and spectral collocation method is used to estimate the solution of nonlinear stochastic Volterra–Fredholm integral equations (NSVFIEs). The main advantage of the suggested approach is that in some parts where interpolation and integration are necessary, this approach does not require any meshes. Therefore, it is independent of the geometry

In this paper, we construct two efficient numerical schemes by combining the finite difference method and operational matrix method (OMM) to solve Riesz-space fractional diffusion equation (RFDE) and Riesz-space fractional advection-dispersion equation (RFADE) with initial and Dirichlet boundary conditions. We applied matrix transform method (MTM) for discretization of Riesz-space fractional derivative

Gauss quadrature rules associated with a nonnegative measure with support on (part of) the real axis find many applications in Scientific Computing. It is important to be able to estimate the quadrature error when replacing an integral by an ℓ-node Gauss quadrature rule in order to choose a suitable number of nodes. A classical approach to estimate this error is to evaluate the associated (2ℓ+1)-node

Retinex theory shows that the color of a scene perceived by human visual system is only dependent on the intrinsic reflectance and unrelated to the illumination. For images recorded by digital cameras, the intensities are the product of the scene reflectance and illumination. Image intensities are different when taken under different illuminations. The aim of image retinex problem is to decompose the

Various applications in science and engineering depend upon computing real solutions to systems of analytic equations which depend upon real parameters. Locally in the parameter space, the qualitative behavior of the solutions remains the same except at critical parameter values. This article develops a singular value homotopy that aims to compute critical parameter values. Several examples are presented

In this paper, we develop high order numerical schemes for the solution of the initial-boundary value problem of one-dimensional and two-dimensional space fractional diffusion equations of orders belonging to the interval (1,2). Firstly, certain weighted and shifted Grünwald difference (WSGD) operator is used to approximate space Riemann-Liouville fractional derivatives, resulting in a linear system

Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible semi-discretization in space, the class of problems under consideration is modeled by a system of ordinary differential equations where the right-hand side is a

Based on the requirement of specific problems, for instance unconstrained and equality-constrained Rayleigh quotient problems, we consider the problem of finding zeros of a tangent vector field on Riemannian manifold. More precisely, we focus on the study of self-adjoint tangent vector field in this paper. By making full use of the self-adjointness property of the tangent vector field, we propose an

We develop and analyze an hp-version of the discontinuous Galerkin time-stepping method for linear Volterra integral equations with weakly singular kernels. We derive a priori error bound in the L2-norm that is fully explicit in the local time steps, the local approximation orders, and the local regularity of the exact solutions. For solutions with singular behavior near t=0 caused by the weakly singular

A non-C0 rectangular Morley element method is discussed to solve the singularly perturbed Bi-wave equation and the penalty terms are included to guarantee convergence. The quasi-uniform convergence rate of order O(h) is derived in the energy norm irrelevant to the negative powers of the real perturbation parameter δ appearing in the considered problem, which improves an existing result, here h is the

A modified SSOR-like (MSSOR-like) preconditioner is constructed for a non-Hermitian positive definite linear system with a dominant Hermitian part. The eigenvalue distribution of the MSSOR-like preconditioned matrix and the convergence property of the corresponding MSSOR-like iteration method are discussed in depth. Numerical experiments show that the MSSOR-like preconditioner can lead to a high-speed

Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. To reduce this sensitivity, the original problem may be replaced by a minimization problem with a fidelity term and a regularization term. We consider minimization problems of this kind, in which the fidelity term is the square of the ℓ2-norm of a discrepancy