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A Haar wavelet collocation approach for solving one and two‐dimensional second‐order linear and nonlinear hyperbolic telegraph equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200804
Muhammad Asif; Nadeem Haider; Qasem Al‐Mdallal; Imran KhanWe have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one‐ and two‐dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one‐ and two‐dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal

Geometric trigonometrically convexity and better approximations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200801
Mahir KadakalIn this manuscript, we introduce the concept of geometric trigonometrically convex functions. We obtain some refinements of the Hermite–Hadamard inequality for functions whose first derivative in absolute value is geometric trigonometrically convex. In addition, it has been proved that the results obtained with Hölder–İşcan and improved power‐mean integral inequalities give a better approach than Hölder

A second‐order space–time accurate scheme for nonlinear diffusion equation with general capacity term Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200731
Huan Zheng; Guangwei Yuan; Xia CuiA fully implicit finite difference (FIFD) scheme with second‐order space–time accuracy is studied for a nonlinear diffusion equation with general capacity term. A new reasoning procedure is introduced to overcome difficulties caused by the nonlinearity of the capacity term and the diffusion operator in the theoretical analysis. The existence of the FIFD solution is investigated at first which plays

Investigation of Coriolis effect on oceanic flows and its bifurcation via geophysical Korteweg–de Vries equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200721
Turgut Ak; Asit Saha; Sharanjeet Dhawan; Abdul Hamid KaraIn this work, we have investigated Coriolis effect on oceanic flows in the equatorial region with the help of geophysical Korteweg–de Vries equation (GKdVE). First, Lie symmetries and conservation laws for the GKdVE have been studied. Later, we implement finite element method for numerical simulations. Propagation of nonlinear solitary structures, their interaction and advancement of solitons can be

Optimal error analysis of Crank–Nicolson lowest‐order Galerkin‐mixed finite element method for incompressible miscible flow in porous media Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200721
Huadong Gao; Weiwei SunNumerical 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 mixed element pair is used for the Darcy velocity and pressure. The existing error estimate

The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200721
Qifeng Zhang; Lingling Liu; Jiyuan ZhangIn the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin–Bona–Mahony–Burgers (BBMB) equation. For the construction of the two‐level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O (τ 2 + h 2). For the three‐level linearized scheme

Two‐grid methods for –P1 mixed finite element approximation of general elliptic optimal control problems with low regularity Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200515
Tianliang Hou; Haitao Leng; Tian LuanIn this paper, we present a two‐grid mixed finite element scheme for distributed optimal control governed by general elliptic equations. –P 1 mixed finite elements are used for the discretization of the state and co‐state variables, whereas piecewise constant function is used to approximate the control variable. We first use a new approach to obtain the superclose property between the centroid interpolation

A fourth–order orthogonal spline collocation method for two‐dimensional Helmholtz problems with interfaces Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200717
Santosh Kumar Bhal; Palla Danumjaya; Graeme FairweatherOrthogonal spline collocation is implemented for the numerical solution of two‐dimensional Helmholtz problems with discontinuous coefficients in the unit square. A matrix decomposition algorithm is used to solve the collocation matrix system at a cost of O (N 2 log N ) on an N × N partition of the unit square. The results of numerical experiments demonstrate the efficacy of this approach, exhibiting

Exponential collocation methods based on continuous finite element approximations for efficiently solving the cubic Schrödinger equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200714
Bin Wang; Xinyuan WuIn this paper we derive and analyze new exponential collocation methods to efficiently solve the cubic Schrödinger Cauchy problem on a d ‐dimensional torus. The novel methods are formulated based on continuous time finite element approximations in a generalized function space. Energy preservation is a key feature of the cubic Schrödinger equation. It is proved that the novel methods can be of arbitrarily

Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200714
Tofigh Allahviranloo; Hussein SahihiIn this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram–Schmidt orthogonalization process

A Petrov–Galerkin RBF method for diffusion equation on the unit sphere Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200714
Mohammadreza Ahmadi Darani; Davoud MirzaeiThis paper concerns a numerical solution for the diffusion equation on the unit sphere. The given method is based on the spherical basis function approximation and the Petrov–Galerkin test discretization. The method is meshless because spherical triangulation is not required neither for approximation nor for numerical integration. This feature is achieved through the spherical basis function approximation

A local hybrid kernel meshless method for numerical solutions of two‐dimensional fractional cable equation in neuronal dynamics Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200714
Ömer OruçThis study deals with obtaining numerical solutions of two‐dimensional (2D) fractional cable equation in neuronal dynamics by using a recently introduced meshless method. In solution process at first stage, time derivatives that are appeared in the considered problem are discretized by using finite difference method. Then a meshless method based on hybridization of Gaussian and cubic kernels is developed

An observation on the uniform preconditioners for the mixed Darcy problem Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200710
Trygve Bærland; Miroslav Kuchta; Kent‐Andre Mardal; Travis ThompsonWhen solving a multiphysics problem one often decomposes a monolithic system into simpler, frequently single‐physics, subproblems. A comprehensive solution strategy may commonly be attempted, then, by means of combining strategies devised for the constituent subproblems. When decomposing the monolithic problem, however, it may be that requiring a particular scaling for one subproblem enforces an undesired

Compact block boundary value methods for semi‐linear delay‐reaction–diffusion equations with algebraic constraints Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200707
Xiaoqiang Yan; Chengjian ZhangIn the present paper, we study a class of linear approximation methods for solving semi‐linear delay‐reaction–diffusion equations with algebraic constraint (SDEACs). By combining a fourth‐order compact difference scheme with block boundary value methods (BBVMs), a class of compact block boundary value methods (CBBVMs) for SDEACs are suggested. It is proved under some suitable conditions that the CBBVMs

Numerical solutions and stability analysis for solitary waves of complex modified Korteweg–de Vries equation using Chebyshev pseudospectral methods Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200706
Avinash K. Mittal; Lokendra K. BalyanIn this research article, the authors investigate the interaction of solitary waves for complex modified Korteweg–de Vries (CMKdV) equations using Chebyshev pseudospectral methods. The proposed method is established in both time and space to approximate the solutions and to prove the stability analysis for the equations. The derivative matrices are defined at Chebyshev–Gauss–Lobbato points and the

Numerical approximation of a stochastic age‐structured population model in a polluted environment with Markovian switching Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200702
Wenrui Li; Ming Ye; Qimin Zhang; Yan LiIn this paper, a stochastic age‐structured population model with Markovian switching is investigated in a polluted environment. Both the stochastic disturbance of environment and the Markovian switching are incorporated into the model. By Itô formula and several assumptions, the boundedness in the q th moment of exact solutions of model are proved. Furthermore, making use of truncated Euler–Maruyama

Applying the three‐dimensional block‐pulse functions to solve system of Volterra–Hammerstein integral equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200702
Jiaquan Xie; Xiaoyuan Gong; Wei Shi; Ruili Li; Weici Zhao; Tao WangIn this paper, a numerical scheme is utilized to solve three‐dimensional nonlinear system of Volterra‐Hammerstein integrals equations, which is based on the three‐dimensional block‐pulse functions (3D‐BPFs) and their operational matrices. Then the primary nonlinear system is transferred into a linear system of algebraic equations by applying the approximate expression and operational matrices, which

A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200625
Muhammad Usman; Muhammad Hamid; Muhammad Saif Ullah Khalid; Rizwan Ul Haq; Moubin LiuIn this paper, we present a novel approach based on shifted Gegenbauer wavelets to attain approximate solutions of some classed of time‐fractional nonlinear problems. First, we present the approximation of a function of two variables u (x ,t ) with help of shifted Gegenbauer wavelets and then some novel operational matrices are proposed with the help of piecewise functions to investigate the positive

Novel numerical techniques for the finite moment log stable computational model for European call option Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200619
Xingyu An; Fawang Liu; Shanzhen Chen; Vo V. AnhOption pricing models are often used to describe the dynamic characteristics of prices in financial markets. Unlike the classical Black–Scholes (BS) model, the finite moment log stable (FMLS) model can explain large movements of prices during small time steps. In the FMLS, the second‐order spatial derivative of the BS model is replaced by a fractional operator of order α which generates an α ‐stable

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200619
Jiansong Zhang; Huiran Han; Hui Guo; Xiaomang ShenIn this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration

On refinements of some integral inequalities using improved power‐mean integral inequalities Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200618
Huriye KadakalIn this study, using power‐mean inequality and improved power‐mean integral inequality better approach than power‐mean inequality and an identity for differentiable functions, we get inequalities for functions whose derivatives in absolute value at certain power are convex. Numerically, it is shown that improved power‐mean integral inequality gives better approach than power‐mean inequality. Some applications

Numerical analysis of a continuous Galerkin method for damped sine‐Gordon equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200617
Zhihui Zhao; Hong LiIn this article, we discuss the numerical solution for the two‐dimensional (2‐D) damped sine‐Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher‐order accuracy in both space

Superconvergence analysis of a nonconforming finite element method for monotone semilinear elliptic optimal control problems Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200617
Hongbo Guan; Dongyang ShiA nonconforming finite element method (FEM) is proposed for optimal control problems (OCPs) governed by monotone semilinear elliptic equations. The state and adjoint state are approximated by the nonconforming elements, and the control is approximated by the orthogonal projection of the adjoint state, respectively. Some global supercloseness and superconvergence estimates are achieved by making full

A two‐grid method with backtracking for the mixed Navier–Stokes/Darcy model Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200612
Guangzhi Du; Qingtao Li; Yuhong ZhangIn this paper, we consider the effect of adding a coarse mesh correction to the two‐grid algorithm for the mixed Navier–Stokes/Darcy model. The method yields both L 2 and H 1 optimal velocity and piezometric head approximations and an L 2 optimal pressure approximation. The method involves solving one small, coupled, nonlinear coarse mesh problem, two independent subproblems (linear Navier–Stokes equation

A reduced‐order extrapolating space–time continuous finite element method for the 2D Sobolev equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200612
Jing Yang; Zhendong LuoThis paper mainly concerns with the order reduction to the coefficient vectors of the classical space–time continuous finite element (STCFE) solutions for a two‐dimensional Sobolev equation. The classical STCFE model is first constructed for the governing equation, and the theoretical results of the existence, stability, and convergence are provided for the STCFE solutions. We then employ a proper

Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200611
Ellya L. KaweckiIn this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains

The pointwise estimates of a conservative difference scheme for Burgers' equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200611
Qifeng Zhang; Xuping Wang; Zhi‐zhong SunIn this article, we are concerned with the numerical analysis of a nonlinear implicit difference scheme for Burgers' equation. A priori estimation of the analytical solution is provided in the sense of L ∞‐norm when the initial value is bounded in H 1‐norm. Conservation, boundedness, and unique solvability are proved at length. Inspired by the method of the priori estimation for the analytical solution

A least‐squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200610
Delyan Z. Kalchev; Thomas A. ManteuffelIn this paper, a least‐squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative‐norm least‐squares, is described in detail

A numerical algorithm for the nonlinear Timoshenko beam system Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200610
Jemal Peradze; Zviad KalichavaAn initial boundary value problem is considered for the dynamic beam system (1)

An L2 finite element approximation for the incompressible Navier–Stokes equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200609
Eunjung Lee; Wonjoon Choi; Heonkyu HaThis paper utilizes the Picard method and Newton's method to linearize the stationary incompressible Navier–Stokes equations and then uses an LL* approach, which is a least‐squares finite element method applied to the dual problem of the corresponding linear system. The LL* approach provides an L 2‐approximation to a given problem, which is not typically available with conventional finite element methods

Analytical and numerical approaches to nerve impulse model of fractional‐order Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200602
Mehmet Yavuz; Asıf YokusWe consider a fractional‐order nerve impulse model which is known as FitzHugh–Nagumo (F–N) model in this paper. Knowing the solutions of this model allows the management of the nerve impulses process. Especially, considering this model as fractional‐order ensures to be able to analyze in detail because of the memory effect. In this context, first, we use an analytical solution and with the aim of this

An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200523
Jiachuan Zhang; Jingshi Li; Jingzhi Li; Kai ZhangBased on a posteriori error estimator with hierarchical bases, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the elliptic problem with mixed boundary conditions. For the posteriori error estimator, we are only required to solve a linear algebraic system with diagonal entries corresponding to the degree of freedoms, which significantly reduces the computational cost. The upper

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200518
Manzoor Hussain; Sirajul HaqIn this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection–diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c*) and weight parameter (ϵ) that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method

Numerical solutions of the equal width equation by trigonometric cubic B‐spline collocation method based on Rubin–Graves type linearization Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200505
Nuri Murat Yağmurlu; Ali Sercan KarakaşIn this article, the equal width (EW) equation is going to be solved numerically. In order to show the accuracy of the presented method, six test problems namely single solitary wave, interaction of two solitary waves, interaction of three solitary waves, Maxwellian initial condition, undular bore, and soliton collision are going to be solved. For the first test problem, since it has exact solution

Construction and analysis of some nonstandard finite difference methods for the FitzHugh–Nagumo equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200430
Koffi M. Agbavon; Appanah Rao AppaduIn this work, we construct four versions of nonstandard finite difference schemes in order to solve the FitzHugh–Nagumo equation with specified initial and boundary conditions under three different regimes giving rise to three cases. The properties of the methods such as positivity and boundedness are studied. The numerical experiment chosen is quite challenging due to shock‐like profiles. The performance

Finding multiple solutions to elliptic systems with polynomial nonlinearity Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200130
Xuping Zhang; Jintao Zhang; Bo YuElliptic systems with polynomial nonlinearity usually possess multiple solutions. In order to find multiple solutions, such elliptic systems are discretized by eigenfunction expansion method (EEM). Error analysis of the discretization is presented, which is different from the error analysis of EEM for scalar elliptic equations in three aspects: first, the choice of framework for the nonlinear operator

Improved L2 and H1 error estimates for the Hessian discretization method Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200129
Devika ShylajaThe Hessian discretization method (HDM) for fourth‐order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit‐conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods (ncFEMs), finite volume methods (FVMs) and methods based on gradient recovery operators

An enhanced finite difference time domain method for two dimensional Maxwell's equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200123
Timothy Meagher; Bin Jiang; Peng JiangAn enhanced finite‐difference time‐domain (FDTD) algorithm is built to solve the transverse electric two‐dimensional Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface. The new algorithm is derived based upon the integral version of the Maxwell's equations as well as the relationship between the electric fields across

An asymptotic preserving scheme on staggered grids for the barotropic Euler system in low Mach regimes Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200123
Thierry Goudon; Julie Llobell; Sebastian MinjeaudWe present a new scheme for the simulation of the barotropic Euler equation in low Mach regimes. The method uses two main ingredients. First, the system is treated with a suitable time splitting strategy, directly inspired from the previous study that separates low and fast waves. Second, we adapt a numerical scheme where the discrete densities and velocities are stored on staggered grids, in the spirit

The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200121
Demin Liu; Linjin LiIn this paper, the blood flow problem is considered in a blood vessel, and a coupling system of Navier–Stokes equations and linear elastic equations, Navier–Lame equations, in a cylinder with cylindrical elastic shell is given as the governing equations of the problem. We provide two finite element models to simulating the three‐dimensional Navier–Stokes equations in the cylinder while the asymptotic

High order convergent modified nodal bi‐cubic spline collocation method for elliptic partial differential equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200120
Suruchi Singh; Swarn SinghA high order modified nodal bi‐cubic spline collocation method is proposed for numerical solution of second‐order elliptic partial differential equation subject to Dirichlet boundary conditions. The approximation is defined on a square mesh stencil using nine grid points. The solution of the method exists and is unique. Convergence analysis has been presented. Moreover, the superconvergent phenomena

High‐order dual‐parametric finite element methods for cavitation computation in nonlinear elasticity Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200110
Weijie Huang; Weijun Ma; Liang Wei; Zhiping LiIn this paper, we present the numerical analysis on high order dual parametric finite element methods for the cavitation computation problems in nonlinear elasticity, which leads to a meshing strategy assuring high efficiency on numerical approximations to cavity deformations. Furthermore, to cope with the high order approximation of the finite element methods, properly chosen weighted Gaussian type

Energy stability and convergence of the scalar auxiliary variable Fourier‐spectral method for the viscous Cahn–Hilliard equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200107
Nan Zheng; Xiaoli LiIn this paper, we develop two linear and unconditionally energy stable Fourier‐spectral schemes for solving viscous Cahn–Hilliard equation based on the recently scalar auxiliary variable approach. The temporal discretizations are built upon the first‐order Euler method and second‐order Crank–Nicolson method, respectively. We carry out the energy stability and error analysis rigorously. Various classical

Numerical analysis of a parabolic variational inequality system modeling biofilm growth at the porescale Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200107
Azhar Alhammali; Malgorzata PeszynskaIn this article, we consider a system of two coupled nonlinear diffusion–reaction partial differential equations (PDEs) which model the growth of biofilm and consumption of the nutrient. At the scale of interest the biofilm density is subject to a pointwise constraint, thus the biofilm PDE is framed as a parabolic variational inequality. We derive rigorous error estimates for a finite element approximation

DOF‐gathering stable generalized finite element methods for crack problems Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200106
Qinghui ZhangGeneralized or eXtended finite element methods (GFEM/XFEM) have been studied extensively for crack problems. Most of the studies were concentrated on localized enrichment schemes where nodes around the crack tip are enriched by products of singular and finite element shape functions. To attain the optimal convergence rate O(h) (h is the mesh‐size), nodes in a fixed domain containing the tip have to

Numerical method for generalized time fractional KdV‐type equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191225
Desong Kong; Yufeng Xu; Zhoushun ZhengIn this article, an efficient numerical method for linearized and nonlinear generalized time‐fractional KdV‐type equations is proposed by combining the finite difference scheme and Petrov–Galerkin spectral method. The scale and weight functions involved in generalized fractional derivative cause too much difficulty in discretization and numerical analysis. Fortunately, motivated by finite difference

A parameter‐uniform scheme for singularly perturbed partial differential equations with a time lag Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191224
Devendra Kumar; Parvin KumariA numerical scheme for a class of singularly perturbed delay parabolic partial differential equations which has wide applications in the various branches of science and engineering is suggested. The solution of these problems exhibits a parabolic boundary layer on the lateral side of the rectangular domain which continuously depends on the perturbation parameter. For the small perturbation parameter

A finite difference scheme for smooth solutions of the general Degasperis–Procesi equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191223
Jesus Noyola Rodriguez; Georgy Omel'yanovThe general Degasperis–Procesi equation (gDP) describes the evolution of the water surface in a unidirectional shallow water approximation. We propose a finite‐difference scheme for this equation that preserves some conservation and balance laws. In addition, the stability of the scheme and the convergence of numerical solutions to exact solutions for solitons are proved. Numerical experiments confirm

A modified sensitivity equation method for the Euler equations in presence of shocks Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191218
Camilla Fiorini; Christophe Chalons; Régis DuvigneauThe continuous sensitivity equation method allows to quantify how changes in the input of a partial differential equation (PDE) model affect the outputs, by solving additional PDEs obtained by differentiating the model. However, this method cannot be used directly in the framework of hyperbolic PDE systems with discontinuous solution, because it yields Dirac delta functions in the sensitivity solution

Second‐order, loosely coupled methods for fluid‐poroelastic material interaction Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191213
Oyekola Oyekole; Martina BukačThis work focuses on modeling the interaction between an incompressible, viscous fluid and a poroviscoelastic material. The fluid flow is described using the time‐dependent Stokes equations, and the poroelastic material using the Biot model. The viscoelasticity is incorporated in the equations using a linear Kelvin–Voigt model. We introduce two novel, noniterative, partitioned numerical schemes for

Fourier spectral method with an adaptive time strategy for nonlinear fractional Schrödinger equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191211
Haidong Qu; Zihang SheIn this paper, a Fourier spectral method with an adaptive time step strategy is proposed to solve the fractional nonlinear Schrödinger (FNLS) equation with periodic initial value problem. First, we prove the conservation law of the mass and the energy for the semi‐discrete Fourier spectral scheme. Second, the error estimation of the semi‐discrete scheme is given in the relevant fractional Sobolev space

Numerical algorithms for the time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191210
Hengfei Ding; Changpin LiIn this paper, high‐order numerical methods for time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations in one‐ and two‐dimensional space are constructed, where the second‐order backward fractional difference operator and the sixth‐order fractional‐compact difference operator are applied to approximate the time and space fractional derivatives, respectively. The stability and convergence of the

A discontinuous Galerkin method for Stokes equation by divergence‐free patch reconstruction Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191130
Ruo Li; Zhiyuan Sun; Zhijian YangA discontinuous Galerkin method by patch reconstruction is proposed for Stokes flows. A locally divergence‐free reconstruction space is employed as the approximation space, and the interior penalty method is adopted which imposes the normal component penalty terms to cancel out the pressure term. Consequently, the Stokes equation can be solved as an elliptic system instead of a saddle‐point problem

Splitting method for an inverse source problem in parabolic differential equations: Error analysis and applications Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191128
Simin Shekarpaz; Hossein AzariIn this work, we present a numerical method based on a splitting algorithm to find the solution of an inverse source problem with the integral condition. The source term is reconstructed by using the specified data and by employing the Lie splitting method, we decompose the equation into linear and nonlinear parts. Each subproblem is solved by the Fourier transform and then by combining the solutions

Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191126
Bhupen Deka; Papri RoyIn this paper, the weak Galerkin finite element method (WG‐FEM) is applied to a pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. A fitted WG‐FEM is proposed to approximate the voltage of the pulsed electric model across the physical media involving an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity

A hybrid high‐order formulation for a Neumann problem on polytopal meshes Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191126
Rommel Bustinza; Jonathan Munguia‐La‐CoteraIn this work, we study a hybrid high‐order (HHO) method for an elliptic diffusion problem with Neumann boundary condition. The proposed method has several features, such as: (a) the support of arbitrary approximation order polynomial at mesh elements and faces on polytopal meshes, (b) the design of a local (element‐wise) potential reconstruction operator and a local stabilization term, that weakly

A fast algorithm for a total variation based phase demodulation model Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191126
Carlos Brito‐Loeza; Ricardo Legarda‐Saenz; Anabel Martin‐GonzalezIn this paper we introduce fast numerical algorithms for the solution of the model. For each variable, background illumination, amplitude modulation and phase map, we develop a fixed point method. Then, we write all three algorithms in the same framework and analyze their convergence rates, local smoothing factors by means of local Fourier analysis and present experimental evidence of their performance

A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191121
Jintao Cui; Fuzheng Gao; Zhengjia Sun; Peng ZhuIn this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness

Numeric solution of advection–diffusion equations by a discrete time random walk scheme Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191121
Christopher N. Angstmann; Bruce I. Henry; Byron A. Jacobs; Anna V. McGannExplicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups, and discontinuities. Here we present an explicit numerical scheme for solving nonlinear advection–diffusion equations admitting shock solutions that is both easy to implement and stable

Well‐posedness and finite element approximation of time dependent generalized bioconvective flow Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20191118
Yanzhao Cao; Song Chen; Hans‐Werner van WykWe consider the finite element approximation of a time dependent generalized bioconvective flow. The underlying system of partial differential equations consists of incompressible Navier–Stokes type convection equations coupled with an equation describing the transport of micro‐organisms. The viscosity of the fluid is assumed to be a function of the concentration of the micro‐organisms. We show the