Search

The Klein–Gordon–Schrödinger equations describe a classical model of interaction of nucleon field with meson field in physics, how to design the energy conservative and stable schemes is an important issue. This paper aims to develop a linearized energy‐preserve, unconditionally stable and efficient scheme for Klein–Gordon–Schrödinger equations. Some auxiliary variables are utilized to circumvent the imaginary functions of Klein–Gordon–Schrödinger equations, and transform the original system into its real formulation. Based on the invariant energy quadratization approach, an equivalent system is deduced by introducing a Lagrange multiplier. Then the efficient and unconditionally stable scheme is designed to discretize the deduced equivalent system. A numerical analysis of the proposed scheme is presented to illustrate its uniquely solvability and convergence. Numerical examples are provided to validate accuracy, energy and mass conservation laws, and stability of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider leap‐frog finite element methods with EQ1rot$$ {\mathrm{EQ}}_1^{\mathrm{rot}} $$ element for the nonlinear Schrödinger equation with wave operator. We propose that both the continuous and discrete systems can keep mass and energy conservation. In addition, we focus on the unconditional superconvergence analysis of the numerical scheme, the key of which is the time‐space error splitting technique. The spatial error is derived τ$$ \tau $$ independently with order O(h2+hτ)$$ O\left({h}^2+ h\tau \right) $$ in H1$$ {H}^1 $$‐norm, where h$$ h $$ and τ$$ \tau $$ denote the space and time step size. Then the unconditional optimal L2$$ {L}^2 $$ error and superclose result with order O(h2+τ2)$$ O\left({h}^2+{\tau}^2\right) $$ are deduced, and the unconditional optimal H1$$ {H}^1 $$ error is obtained with order O(h+τ2)$$ O\left(h+{\tau}^2\right) $$ by using interpolation theory. The final unconditional superconvergence result with order O(h2+τ2)$$ O\left({h}^2+{\tau}^2\right) $$ is derived by the interpolation postprocessing technique. Furthermore, we apply the proposed leap‐frog finite element methods to solve the logarithmic Schrödinger equation with wave operator by introducing a regularized system with a small regularization parameter 0<ϵ≪1$$ 0<\epsilon \ll 1 $$. The detailed theoretical conclusions, including the mass and energy conservation laws of the continuous regularized and discrete regularized systems and the convergence and superconvergence results, are presented as prolongation of the previous work. At last, some numerical experiments are given to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical computation and analysis for nonlinear Sobolev equations with distributed delay. We present linearized compact difference methods for solving one‐ and two‐dimensional cases, respectively. The solvability, convergence and stability of the methods are derived and it is shown under some suitable conditions that the methods have second‐order accuracy in time and fourth‐order accuracy in space. Moreover, with several numerical examples, the theoretical accuracy and computational effectiveness of the proposed methods are confirmed. [ABSTRACT FROM AUTHOR]

We 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 solution, we obtain numerical solutions by using two numerical schemes. Then, we demonstrate the walking wave‐type solutions of the stated problem. These solutions include complex trigonometric functions, complex hyperbolic functions, and algebraic functions. In addition, the linear stability analysis is performed and the absolute error is occurred by comparing the numerical results with the analytical result. All of the results are depicted by tables and figures. This paper not only points out the exact and numerical solutions of the model but also compares the differences and the similarities of the stated solution methods. Therefore, the results of this paper are important and useful for either neuroscientists and physicists or mathematicians and engineers. [ABSTRACT FROM AUTHOR]

This paper discusses the numerical solutions of the nonlinear Klein–Gordon–Zakharov model by the use of five recent numerical schemes (Adomian decomposition, El‐kalla, cubic B‐spline, extended cubic B‐spline, and exponential cubic B‐spline). The object of this numerical analysis is to demonstrate the accuracy of the analytical solutions obtained using the generalized Khater system and also to display the accuracy of the five numerical schemes mentioned above. This mathematical model explains the development in plasma physics of efficient Langmuir turbulence. We use computational solutions to assess the original and boundary conditions which require the numerical schemes to be implemented. Many distinct sketches are given to demonstrate the exactness of the numerical solutions measured. [ABSTRACT FROM AUTHOR]

The present paper is devoted to study the first order of accuracy absolute stable difference scheme for the approximate solution of the space identification problem for the elliptic‐telegraph equation in Hilbert spaces with the self‐adjoint positive definite operator. The main theorem on the stability of the difference scheme is established. In applications, theorems on the stability of difference schemes for two types of the space identification problems for multidimensional elliptic‐telegraph partial differential equations are proved. Numerical analysis is provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P0, P0, RT0). The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper is primarily concern with the formulation and analysis of a reliable numerical method based on the novel alternating direction implicit finite difference scheme for the solution of the fractional reaction–diffusion system. In the work, the integer first‐order derivative in time is replaced with the Caputo fractional derivative operator. As a case study, the dynamics of predator–prey model is considered. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction–diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve both self‐diffusion and cross‐diffusion problems in two‐dimensions. We observed in the experimental results a range of spatiotemporal and chaotic structures that are related to Turing pattern. It was also discovered in the simulations that cross‐diffusive case gives rise to spatial patterns faster than the diffusive case. Apart from chaotic spiral‐like structures obtained in this work, it should also be mentioned that Turing patterns such as stationary spots and stripes are obtainable, depending on the initial and parameters choices. [ABSTRACT FROM AUTHOR]

In 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 L2 and H1 optimal velocity and piezometric head approximations and an L2 optimal pressure approximation. The method involves solving one small, coupled, nonlinear coarse mesh problem, two independent subproblems (linear Navier–Stokes equation and Darcy equation) on the fine mesh, and a correction problem on the coarse mesh. Theoretical analysis and numerical tests are done to indicate the significance of this method. [ABSTRACT FROM AUTHOR]

In 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. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp‐DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates. [ABSTRACT FROM AUTHOR]

In 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 numerical quadrature is applied to the singular part of the elastic energy. Our numerical experiments show that the high order dual parametric finite element methods work well when coupled with properly designed weighted Gaussian type numerical quadratures for the singular part of the elastic energy, and the convergence rates of the numerical cavity solutions are shown to be significantly improved as expected. [ABSTRACT FROM AUTHOR]

This paper presents a numerical solution of the coupled system of the time‐dependent Stokes and fully dynamic Biot equations. The numerical scheme is based on standard inf‐sup stable finite elements in space and the Backward Euler scheme in time. We establish stability of the scheme and derive error estimates for the fully discrete coupled scheme. To handle realistic parameters which may cause nonphysical oscillations in the pore fluid pressure, a heuristic stabilization technique is considered. Numerical errors and convergence rates for smooth problems as well as tests on realistic material parameters are presented. [ABSTRACT FROM AUTHOR]

In this paper, we will investigate a two grid finite element discretization method for the semi‐linear hyperbolic integro‐differential equations by piecewise continuous finite element method. In order to deal with the semi‐linearity of the model, we use the two grid technique and derive that once the coarse and fine mesh sizes H, h satisfy the relation h = H2 for the two‐step two grid discretization method, the two grid method achieves the same convergence accuracy as the ordinary finite element method. Both theoretical analysis and numerical experiments are given to verify the results. [ABSTRACT FROM AUTHOR]

A nonlinear coupled mathematical system of two‐phase seepage flow displacement is discussed in this paper including an elliptic equation for the pressure and a convection‐dominated diffusion equation for the saturation. In fact, the boundary of an underground region where the fluid flows through is nonstationary. So a moving boundary should be considered. The saturation equation is convection‐dominated, therefore the method of upwind finite difference is introduced for the accurate computation. The upwind approximation could eliminate numerical oscillation and strong stability is shown. Since the computational work of saturation is larger than the pressure, the authors apply a parallel method, decomposing the whole domain into several nonoverlapping subdomains, to simplify the computation. A domain decomposition method coupled with upwind differences is presented for the saturation. The pressure equation is discretized by a five‐point center finite difference method. By using a transformation and defining new inner products and norms, error estimates in l2 norm is discussed. Finally, two experimental tests are given to illustrate the efficiency and accuracy of the parallel algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we derive a theoretical analysis of nonsymmetric interior penalty discontinuous Galerkin methods for solving the Cahn–Hilliard equation. We prove unconditional unique solvability of the discrete system and derive stability bounds with a generalized chemical energy density. Convergence of the method is obtained by optimal a priori error estimates. Our analysis is valid for both symmetric and nonsymmetric versions of the discontinuous Galerkin formulation. [ABSTRACT FROM AUTHOR]

In this paper, we numerically investigate the BBM‐Burgers equation with a nonlocal viscous term u t + u x − β u t x x + ν π ∂ ∂ t ∫ 0 t u ( s ) t − s d s + γ u u x = α u x x , where 1 π ∂ ∂ t ∫ 0 t u ( s ) t − s d s is the Riemann‐Liouville half derivative. In particular, we implement different numerical schemes to approximate the solution and its asymptotical behavior. Also, we compare our numerical results with those given in for similar models. [ABSTRACT FROM AUTHOR]

In this paper, we develop a fully implicit scheme on staggered grids to solve the Maxwell's equations when Drude metamaterial is involved. Unconditional stability and optimal error estimate of the scheme are proved. Numerical results are provided to support the theoretical analysis, and used to demonstrate the applicability of the scheme to simulate the complicated backward wave propagation phenomenon occurring in metamaterials. [ABSTRACT FROM AUTHOR]

The numerical solution for the time fractional advection‐diffusion problem in one‐dimension with the initial‐boundary condition is proposed in this paper by B‐spline finite volume element method. The fractional derivative is Caputo in the proposed scheme. The stability of the proposed numerical method is studied, and the numerical results presented support the theoretical results. [ABSTRACT FROM AUTHOR]

In the last decade, theoretical and applied studies were done in order to provide a suitable definition of fractional derivative, which meets all the requirement of a derivative in its primary sense. It was concluded by some eminent researchers that the Riemann‐Liouville version was the most suitable. However, many numerical approximation of fractional derivative were done with Caputo version. This paper addresses the numerical approximation of fractional differentiation based on the Riemann‐Liouville definition, from power‐law kernel to generalized Mittag‐Leffler‐law via exponential‐decay‐law. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical treatment of a fourth‐order fractional diffusion‐wave problem. Our proposed method includes the use of parametric quintic spline in the spatial dimension and the weighted shifted Grünwald‐Letnikov approximation of fractional integral. The solvability, stability, and convergence of the numerical scheme are rigorously proved. It is shown that the theoretical convergence order improves those of earlier work. Simulation is further carried out to demonstrate the numerical efficiency of the proposed scheme and to compare with other methods. [ABSTRACT FROM AUTHOR]

Because of the finiteness of the life span and boundedness of the physical space, the more reasonable or physical choice is the tempered power‐law instead of pure power‐law for the CTRW model in characterizing the waiting time and jump length of the motion of particles. This paper focuses on providing the variational formulation and efficient implementation for solving the corresponding deterministic/macroscopic models, including the space tempered fractional equation and time tempered fractional equation. The convergence, numerical stability, and a series of variational equalities are theoretically proved. And the theoretical results are confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]

In this continuing paper of (Zhu and Qiu, J Comput Phys 318 (2016), 110-121), a new fifth order finite difference weighted essentially non-oscillatory (WENO) scheme is designed to approximate the viscosity numerical solution of the Hamilton-Jacobi equations. This new WENO scheme uses the same numbers of spatial nodes as the classical fifth order WENO scheme which is proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126-2143), and could get less absolute truncation errors and obtain the same order of accuracy in smooth region simultaneously avoiding spurious oscillations nearby discontinuities. Such new WENO scheme is a convex combination of a fourth degree accurate polynomial and two linear polynomials in aWENOtype fashion in the spatial reconstruction procedures. The linear weights of three polynomials are artificially set to be any random positive constants with a minor restriction and the new nonlinear weights are proposed for the sake of keeping the accuracy of the scheme in smooth region, avoiding spurious oscillations and keeping sharp discontinuous transitions in nonsmooth region simultaneously. The main advantages of such new WENO scheme comparing with the classical WENO scheme proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126-2143) are its efficiency, robustness and easy implementation to higher dimensions. Extensive numerical tests are performed to illustrate the capability of the new fifth WENO scheme. [ABSTRACT FROM AUTHOR]