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The main purpose of this paper is to introduce the gradient discretisation method (GDM) to a system of reaction diffusion equations subject to non-homogeneous Dirichlet boundary conditions. Then, the authors show that the GDM provides a comprehensive convergence analysis of several numerical methods for the considered model. The convergence is established without non-physical regularity assumptions on the solutions.

In this paper, the authors use the GDM to discretise a system of reaction diffusion equations with non-homogeneous Dirichlet boundary conditions.

The authors provide a generic convergence analysis of a system of reaction diffusion equations. The authors introduce a specific example of numerical scheme that fits in the gradient discretisation method. The authors conduct a numerical test to measure the efficiency of the proposed method.

This work provides a unified convergence analysis of several numerical methods for a system of reaction diffusion equations. The generic convergence is proved under the classical assumptions on the solutions.

The purpose of this study is to investigate the magneto-hydrodynamics boundary layer Falkner–Skan flow over a flat plate numerically by using the Runge–Kutta method featuring shooting technique and analytically via a new modified analytical technique called improved generalized Adomian decomposition method (improved-GDM).

It is well established that the generalized decomposition method (GDM) (Yong-Chang et al., 2008), which uses a new kind of decomposition strategy for the nonlinear function, has proved its efficiency and superiority when compared to the standard ADM method. In this investigation, based on the idea of improved-ADM method developed by Lina and Song (Song and Wang, 2013), the authors proposed a new analytical algorithm of computation named improved-GDM. Thereafter, the proposed algorithm is tested by solving the nonlinear problem of the hydro-magnetic boundary layer flow over a flat plate.

The proposed improved generalized decomposition method (I-GDM) introduces a convergence-control parameter “ω’’ into the GDM, which accelerates the convergence of solution and reduces considerably the computation time. In fact, the key of this method is mainly based on the best selection of the convergence-control parameter ω.

The paper presents a new efficient algorithm of computation that can be considered as an alternative for solving the nonlinear initial boundary layer value problems. Obtained results show clearly the accuracy of the proposed method.

We consider, as discretization in space, the nonconforming mesh developed in SUSHI (Scheme Using Stabilization and Hybrid Interfaces) developed in Eymard et al. (2010) for a semi-linear heat equation. The time discretization is performed using a uniform mesh. We are concerned with a nonlinear scheme that has been studied in Bradji (2016) in the context of the general framework GDM (Gradient Discretization Method) (Droniou et al., 2018) which includes SUSHI. We provide sufficient conditions on the size of the spatial mesh and the time step which allow to prove a W1,∞(L2)-error estimate. This error estimate can be viewed as an improvement for the W1,2(L2)-error estimate proved in Bradji (2016). The W1,∞(L2)-error estimate we want to prove in this note was stated without proof in Bradji (2016, Remark 7.2, Page 1302). Its proof is based on a comparison with an appropriately chosen auxiliary finite volume scheme along with the derivation of some new estimates on its solution.

The main aim of the current paper is to find a numerical plan for hydraulic fracturing problem with application in extracting natural gases and oil.

First, time discretization is accomplished via Crank-Nicolson and semi-implicit techniques. At the second step, a high-order finite element method using quadratic triangular elements is proposed to derive the spatial discretization. The efficiency and time consuming of both obtained schemes will be investigated. In addition to the popular uniform mesh refinement strategy, an adaptive mesh refinement strategy will be employed to reduce computational costs.

Numerical results show a good agreement between the two schemes as well as the efficiency of the employed techniques to capture acceptable patterns of the model. In central single-crack mode, the experimental results demonstrate that maximal values of displacements in x- and y- directions are 0.1 and 0.08, respectively. They occur around both ends of the line and sides directly next to the line where pressure takes impact. Moreover, the pressure of injected fluid almost gained its initial value, i.e. 3,000 inside and close to the notch. Further, the results for non-central single-crack mode and bifurcated crack mode are depicted. In central single-crack mode and square computational area with a uniform mesh, computational times corresponding to the numerical schemes based on the high order finite element method for spatial discretization and Crank-Nicolson as well as semi-implicit techniques for temporal discretizations are 207.19s and 97.47s, respectively, with 2,048 elements, final time T = 0.2 and time step size τ = 0.01. Also, the simulations effectively illustrate a further decrease in computational time when the method is equipped with an adaptive mesh refinement strategy. The computational cost is reduced to 4.23s when the governed model is solved with the numerical scheme based on the adaptive high order finite element method and semi-implicit technique for spatial and temporal discretizations, respectively. Similarly, in other samples, the reduction of computational cost has been shown.

This is the first time that the high-order finite element method is employed to solve the model investigated in the current paper.

This paper deals with the inverse scattering problem of reconstructing the material properties of perfectly conducting or dielectric cylindrical objects. The material properties are reconstructed from measured far‐field scattering data provided by the Electromagnetics Technology Division, AFRL/SNH, 31 Grenier Street, Hanscom AFB, MA 01731‐3010. The measured data have to be calibrated for use in our reconstruction algorithm. The inverse scattering problem formulated as unconstrained nonlinear optimization problem is numerically solved using an iterative scheme with a variable calibration factor which will be determined during the optimization process. Numerical examples show the successful application of the method to the measured data.