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Preview: Numerical Methods for Partial Differential Equations

Numerical Methods for Partial Differential Equations

Wiley Online Library : Numerical Methods for Partial Differential Equations

Published: 2018-03-01T00:00:00-05:00


Virtual Elements for the Reissner-Mindlin plate problem


In this work, we present a virtual element method for the approximation of the plate bending problem in the Reissner-Mindlin formulation. The proposed method follows the MITC approach of the FEM context. We construct a family of VEM spaces with arbitrary degree of accuracy that satisfies the conditions of the MITC philosophy. We perform some numerical tests which allow us to assess the convergence and the robustness of the method.

Stability of a finite volume element method for the time-fractional advection-diffusion equation


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.

Numerical solution of systems of partial integral differential equations with application to pricing options


We introduce and analyze a strongly stable numerical method designed to yield good performance under challenging conditions of irregular or mismatched initial data for solving systems of coupled partial integral differential equations (PIDEs). Spatial derivatives are approximated using second order central difference approximations by treating the mixed derivative terms in a special way. The integral operators are approximated using one and two–dimensional trapezoidal rule on an equidistant grid. Computational complexity of the method for solving large systems of PIDEs is discussed. A detailed treatment for the consistency, stability, and convergence of the proposed method is provided. Two asset American option under regime–switching with jump–diffusion model when solved using a penalty term, leads to a system of two dimensional PIDEs with mixed derivatives. This model involves double probability density function which brings more challenges to the numerical solution in already a complicated partial integral differential equation. The complexity of the dense jump probability generator, the nonlinear penalty term and the regime–switching terms are treated efficiently, while maintaining the stability and convergence of the method. The impact of the jump intensity and other parameters is shown in the graphs. Numerical experiments are performed to demonstrated efficiency, accuracy, and reliability of the proposed approach.

A compact adaptive approach for degenerate singular reaction-diffusion equations


This article concerns a compact adaptive method for the numerical solution of nonlinear degenerate singular reaction-diffusion equations. The partial differential equation problems exhibit strong quenching blow-up type singularities, and are critical in numerous applications ranging from optimized internal combustion designs to oil pipeline decay predictions. Adaptive schemes of fourth order in space and second order in time are acquired and discussed. Nonuniform spatial and temporal grids are utilized through suitable adaptations. Rigorous analysis is given for the numerical stability, and computational experiments are performed to illustrate our conclusions.

Three methods for two-sided bounds of eigenvalues—A comparison


We compare three finite element-based methods designed for two-sided bounds of eigenvalues of symmetric elliptic second order operators. The first method is known as the Lehmann–Goerisch method. The second method is based on Crouzeix–Raviart nonconforming finite element method. The third one is a combination of generalized Weinstein and Kato bounds with complementarity-based estimators. We concisely describe these methods and use them to solve three numerical examples. We compare their accuracy, computational performance, and generality in both the lowest and higher order case.

Error estimates of a semidiscrete finite element method for fractional stochastic diffusion-wave equations


In this paper, we consider the Galerkin finite element method for solving the fractional stochastic diffusion-wave equations driven by multiplicative noise, which can be used to describe the propagation of mechanical waves in viscoelastic media with random effects. The optimal strong convergence error estimates with respect to the semidiscrete finite element approximation in space are established. Finally, a numerical example is presented to verify the reliability of the theoretical study.

A Newton linearized compact finite difference scheme for one class of Sobolev equations


In this article, a Newton linearized compact finite difference scheme is proposed to numerically solve a class of Sobolev equations. The unique solvability, convergence, and stability of the proposed scheme are proved. It is shown that the proposed method is of order 2 in temporal direction and order 4 in spatial direction. Moreover, compare to the classical extrapolated Crank-Nicolson method or the second-order multistep implicit–explicit methods, the proposed scheme is easier to be implemented as it only requires one starting value. Finally, numerical experiments on one and two-dimensional problems are presented to illustrate our theoretical results.

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers


In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well-posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H1 and L2 norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.

Asymptotic Analysis and Optimal Error estimates for Benjamin-Bona-Mahony-Burgers' Type Equations


In this article, stabilization result for the Benjamin-Bona-Mahony-Burgers' (BBM-B) equation, that is, convergence of unsteady solution to steady state solution is established under the assumption that a linearized steady state eigenvalue problem has a minimal positive eigenvalue. Based on appropriate conditions on the forcing function, exponential decay estimates in L ∞ ( H j ) , j = 0 , 1 , 2 , and W 1 , ∞ ( L 2 ) -norms are derived, which are valid uniformly with respect to the coefficient of dispersion as it tends to zero. It is, further, observed that the decay rate for the BBM-B equation is smaller than that of the decay rate for the Burgers equation. Then, a semidiscrete Galerkin method for spatial direction keeping time variable continuous is considered and stabilization results are discussed for the semidiscrete problem. Moreover, optimal error estimates in L ∞ ( H j ) ,   j = 0 , 1 -norms preserving exponential decay property are established using the steady state error estimates. For a complete discrete scheme, a backward Euler method is applied for the time discretization and stabilization results are again proved for the fully discrete problem. Subsequently, numerical experiments are conducted, which verify our theoretical results. The article is finally concluded with a brief discussion on an extension to a multidimensional nonlinear Sobolev equation with Burgers' type nonlinearity.

High-order energy-preserving schemes for the improved Boussinesq equation


This article proposes a class of high-order energy-preserving schemes for the improved Boussinesq equation. To derive the energy-preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite-dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy-preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second-order, fourth-order, and sixth-order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break-up.

Convergence rate of collocation method based on wavelet for nonlinear weakly singular partial integro-differential equation arising from viscoelasticity


The main aim of this research article is to propose and analyze a Legendre wavelet collocation method (LWCM) for the nonlinear weakly singular partial integro-differential equation (SPIDE) arising from viscoelasticity subject to the given initial and boundary conditions. This problem can be found in the mathematical modeling of physical phenomena involving viscoelastic forces. Operational matrix of integration of Legendre wavelets along with collocation method are utilized to reduce the original SPIDE into the nonlinear system of algebraic equations. Some numerical results are presented to simplify applications of operational matrix formulation and reduce the computational cost. Convergence analysis, numerical stability and rate of convergence (C-order) of the proposed method are also investigated by considering a test function. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the L 2 and L ∞ norms. Finally, we compare the proposed LWCM with well-known Crank-Nicolson and Crandall's methods (for instance, see Table 4).

Wavelet Galerkin schemes for higher order time dependent partial differential equations


Fourth-order derivatives appearing in different linear and nonlinear transient higher order multidimensional equations modeling several physical problems have always posed computational challenges to widely prevailing numerical approaches such as FDM, FEM, and so forth. In this study, we address the issue effectively using the special features of Daubechies wavelets such as orthogonality, compact support, arbitrary regularity, high-order vanishing moments, and good localization. An efficient compression strategy is proposed to reduce the computational cost significantly. Implicitly stable backward Euler scheme is used for time marching the evolving solution. A priori error estimates have been derived to prove the convergence of the numerical scheme. The proposed approach is successfully tested on few linear and nonlinear multidimensional PDEs.

On the Euler implicit/explicit iterative scheme for the stationary Oldroyd fluid


In this article, we consider the stationary Oldroyd fluid equations from the large time behavior research of the nonstationary equations. Thus, to obtain its numerical solution, we first solve the nonstationary Oldroyd fluid equations via the Euler implicit/explicit finite element method with the integral term discretized by the right-hand rectangle rule, then increase the total time (i.e., number of time steps) to approximate the solution of the original stationary equations. Under a new uniqueness condition (A2), we prove the exponential stability of the solution pair { u ¯ , p ¯ } for the stationary equations and the almost unconditional stability of the numerical method. Furthermore, we also obtain the uniform optimal H 1 and L 2 error estimates in time integral 0 ≤ t < + ∞ . Finally, several numerical experiments are provided to verify our theoretical results.

A symmetric integrated radial basis function method for solving differential equations


In this article, integrated radial basis functions (IRBFs) are used for Hermite interpolation in the solution of differential equations, resulting in a new meshless symmetric RBF method. Both global and local approximation-based schemes are derived. For the latter, the focus is on the construction of compact approximation stencils, where a sparse system matrix and a high-order accuracy can be achieved together. Cartesian-grid-based stencils are possible for problems defined on nonrectangular domains. Furthermore, the effects of the RBF width on the solution accuracy for a given grid size are fully explored with a reasonable computational cost. The proposed schemes are numerically verified in some elliptic boundary-value problems governed by the Poisson and convection-diffusion equations. High levels of the solution accuracy are obtained using relatively coarse discretisations.

A reliable algorithm for the approximate solution of the nonlinear Lane-Emden type equations arising in astrophysics


In this paper, we present a reliable algorithm to obtain the approximate solution of the nonlinear Lane-Emden type equations arising in astrophysics. The suggested algorithm is based upon the operational matrix of integration for Jacobi polynomials and the collocation method. Convergence analysis and numerical stability of the suggested method are provided. Numerical results for several interesting nonlinear cases of the Lane-Emden type equations such as the standard Lane-Emden equation, the white-dwarf equation, and the isothermal gas spheres equation, as well as Richardson's theory of thermionic current are discussed. These numerical results are shown in the form of tables and figures for the particular cases of Jacobi polynomials such as the Legendre polynomial (P1), the Chebyshev polynomials of the second kind (P2), the Chebyshev polynomials of the third kind (P3), the Chebyshev polynomials of the fourth kind (P4), and the Gegenbauer (or ultraspherical) polynomials (P5). Numerical results are also compared with those that were derived earlier by applying some well-known and recently developed numerical methods and it is observed that our numerical results are more accurate. The maximum absolute errors and the root mean square errors are calculated by using P1, P2, P3, P4, and P5 for comparison purposes.

A backward Euler alternating direction implicit difference scheme for the three-dimensional fractional evolution equation


A backward Euler alternating direction implicit (ADI) difference scheme is formulated and analyzed for the three-dimensional fractional evolution equation. In our method, the Riemann-Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, an ADI technique is adopted to reduce the multidimensional problem to a series of one-dimensional problems. A fully discrete difference scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. Numerical experiments are reported to demonstrate the efficiency of our scheme.

Solutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space


The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this research article is to present results on the numerical simulation for time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space that were found in the transonic flows. Those resulting mathematical models are solved using the reproducing kernel algorithm which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations, and error bounds under some hypotheses which provide the theoretical basis of the proposed algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the algorithm are discussed through academic validations.

A computational method based on Hermite wavelets for two-dimensional Sobolev and regularized long wave equations in fluids


This paper deals with the numerical solutions of two-dimensional Sobolev and regularized long wave equations which commonly emerge in the flow of fluids or used to explain motion of wave in media. The proposed computational method in this paper is based on Hermite wavelets. We first discretize time derivatives in the considered equations by finite difference approaches then we use Hermite wavelets for discretization of space variables. By doing so computing the numerical solutions of Sobolev and regularized long wave equations is reduced to computing the solution of an algebraic system of equations whose solution gives Hermite wavelet coefficients. Then with these wavelet coefficients numerical solutions can be computed successively. The main objective of this paper is to indicate that Hermite wavelets based computational method is proper and efficient for two-dimensional Sobolev and regularized long wave equations. We consider five test problems and calculate L2 and L∞ error norms for comparison of results of the current paper with the exact results and with the results of earlier studies based on such as finite difference, finite element and meshless methods. The obtained results verify the feasibility and efficiency of the proposed method.

Spectral technique for solving variable-order fractional Volterra integro-differential equations


This article, presented a shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method which is introduced for solving variable-order fractional Volterra integro-differential equation (VO-FVIDEs) subject to initial or nonlocal conditions. Based on shifted Legendre Gauss-Lobatto (SL-GL) quadrature, we treat with integral term in the aforementioned problems. Via the current approach, we convert such problem into a system of algebraic equations. After that we obtain the spectral solution directly for the proposed problem. The high accuracy of the method was proved by several illustrative examples.

The fictitious domain method with L2-penalty for the Stokes problem with the Dirichlet boundary condition


We consider the fictitious domain method with L2-penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order O ( ϵ 1 4 ) in H1-norm for the velocity and in L2-norm for the pressure, where ϵ is the penalty parameter. The L2-norm error estimate for the velocity is upgraded to O ( ϵ ) . Moreover, we derive the a priori estimates depending on ϵ for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate O ( h + ϵ 1 4 ) in H1-norm for the velocity and in L2-norm for the pressure, where h denotes the discretization parameter. For the velocity in L2-norm, the convergence rate is improved to O ( h + ϵ 1 2 ) . The theoretical results are verified by the numerical experiments.

Numerical analysis of a finite volume scheme for a seawater intrusion model with cross-diffusion in an unconfined aquifer


We consider a degenerate parabolic system modeling the flow of fresh and saltwater in a porous medium in the context of seawater intrusion. We propose and analyze a finite volume scheme based on two-point flux approximation with upwind mobilities. The scheme preserves at the discrete level the main features of the continuous problem, namely the nonnegativity of the solutions, the decay of the energy and the control of the entropy and its dissipation. Based on these nonlinear stability results, we show that the scheme converges toward a weak solution to the problem. Numerical results are provided to illustrate the behavior of the model and of the scheme.

Heat and mass transfer in unsteady MHD slip flow of Casson fluid over a moving wedge embedded in a porous medium in the presence of chemical reaction: Numerical Solutions using Keller-Box Method


The hydromagnetic mixed convection flow of Casson fluid due to moving wedge embedded in a porous medium in the presence of chemical reaction and viscous dissipation is investigated. The Joule heating due to a magnetic field and porous medium heating is also considered. Similarity transformations are utilized to convert nonlinear partial differential equations into nonlinear ordinary differential equations. The resulting equations are solved numerically via Keller-box method. The numerical results are achieved for limiting cases and are revealed in close agreement with those of the results available in the literature. It is noticed that unsteadiness parameter thinning the velocity boundary layer while opposite to this was found in the thermal and concentration boundary layers. The dimensionless temperature is observed to be enhanced with increment in Eckert number.

On solutions to the second-order partial differential equations by two accurate methods


In this article, we investigate the reproducing kernel method and the difference schemes method for solving the second-order partial differential equations. Numerical results have been shown to prove the efficiency of the methods. Results prove that the methods are very effective.

A triangular spectral method for the Stokes eigenvalue problem by the stream function formulation


An efficient spectral method is developed in this paper for the two-dimensional Stokes eigenvalues on arbitrary triangle. By using the spectral theory of compact operator and approximate property of orthogonal polynomial, we give the error estimate of the approximate eigenvalues and eigenfunctions. In addition, we also present some numerical results to show the validity of our algorithm and the correctness of the theoretical results.

Superconvergence of Ritz-Galerkin finite element approximations for second order elliptic problems


In this paper, the author derives an O ( h 4 ) -superconvergence for the piecewise linear Ritz-Galerkin finite element approximations for the second-order elliptic equation − ∇ · ( A ∇ u ) = f equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor A and the usual shape functions on each element, called A -equilateral assumption in this paper. Several examples are presented for the coefficient tensor A and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.

Numerical simulation of time fractional Cable equations and convergence analysis


In this article, the numerical solution of time fractional Cable equation is considered. We convert the time fractional Cable equations into equivalent integral equations with singular kernel, then propose a spectral collection method in both time and space discretizations with a spectral expansion of Lagrange interpolation polynomial for this equation. The convergence of the method is rigorously established. Numerical tests are carried out to confirm the theoretical results.

Dynamical study of two predators and one prey system with fractional Fourier transform method


In this work, we investigate both the analytical and numerical studies of the dynamical model comprising of three species systems. We analyze the linear stability of stationary solutions in the one-dimensional multisystem modeling the interactions of two predators and one prey species. The stability analysis has a lot of implications for understanding the various spatiotemporal and chaotic behaviors of the species in the spatial domain. The analysis results presented have established the possibility of the three-interacting species to coexist harmoniously, this feat is achieved by combining the local and global analyses to determine the global dynamics of the system. In the presence of a fractional diffusion term, we introduced a fractional Fourier transform for solving the system modeled by fractional partial differential equations. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator with exponential accuracy and a completely straightforward extension to high spatial dimensions. The scheme is described in detail and justified by a number of computational experiments.

New two step Laplace Adam-Bashforth method for integer a noninteger order partial differential equations


This article presents a novel method that allows to generalize the use of the Adam-Bashforth to Partial Differential Equations with local and nonlocal operator. The Method derives a two step Adam-Bashforth numerical scheme in Laplace space and the solution is taken back into the real space via inverse Laplace transform. The method yields a powerful numerical algorithm for fractional order derivative where the usually very difficult to manage summation in the numerical scheme disappears. Error Analysis of the method is also presented. Applications of the method and numerical simulations are presented on a wave–equation-like, and on a fractional order diffusion equation.

Numerical solution for high-order linear complex differential equations with variable coefficients


In this paper, we have obtained the numerical solutions of complex differential equations with variable coefficients by using the Legendre Polynomials and we have performed it on two test problems. Then, we applied with different technical of error analysis to the test problems. When we compared exact solutions and numerical solutions of tables and graphs, we realized that our method is reliable, practical, and functional.

Chaos in a nonlinear Bloch system with Atangana–Baleanu fractional derivatives


In this article, a nonlinear model of the Bloch equation to include both fractional derivatives with variable-order, constant-order, and time delays was considered. The fractional derivative with the generalized Mittag-Leffler function as kernel is introduced due to the nonlocality of the dynamical system. To find a numerical solution of the delay variable-order model, a predictor corrector method had been developed to solve this system. The existence and uniqueness of the numerical scheme was discussed in detail. For the constant-order, we presented the existence and uniqueness of a positive set of the solutions for the new model and the Adams–Moulton rule was considered to solved numerically the fractional equations. The behavior of the fractional commensurate order nonlinear delay-dependent Bloch system with total order less than 3, which exhibits chaos and transient chaos, was presented. In addition, it is found that the presence of fractional variable-order in the nonlinear Bloch system exhibit more complicated dynamics can improve the stability of the solutions.

Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions


Recently, many new applications in engineering and science are governed by a series of time-fractional partial integrodifferential equations, which will lead to new challenges for numerical simulation. In this article, we propose and analyze an efficient iterative algorithm for the numerical solutions of such equations subject to initial and Dirichlet boundary conditions. The algorithm provide appropriate representation of the solutions in infinite series formula with accurately computable structures. By interrupting the n -term of exact solutions, numerical solutions of linear and nonlinear time-fractional equations of nonhomogeneous function type are studied from mathematical viewpoint. Convergence analysis, error estimations, and error bounds under some hypotheses which provide the theoretical basis of the proposed algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such integrodifferential equations.

An Investigation on Reliable Analytical and Numerical Methods for the Riesz Fractional Nonlinear Schrödinger Equation inQuantum Mechanics


In the article, a nonlinear Schrödinger equation with the Riesz fractional derivative has been considered. This equation has been solved by two reliable methods to investigate the accuracy of the solutions. In the implicit finite difference numerical scheme, the fractional centered difference is utilized to approximate the Riesz fractional derivative. Also, a novel modified optimal homotopy asymptotic method with Fourier transform (MOHAM-FT) has been proposed to compute the approximate solution of Riesz fractional nonlinear Schrödinger equation(RFNLSE). Further the numerical solutions of RFNLSE obtained by proposed implicit finite difference method, have been compared with that obtained by MOHAM-FT to exhibit the effectiveness of the suggested methods. Finally, the obtained solutions have been presented graphically to justify the efficiency of the methods.

Solving the burgers' and regularized long wave equations using the new perturbation iteration technique


In this study, an efficient framework is provided to handle nonlinear partial differential equations by implementing perturbation iteration method. This method is recovered and amended to solve the Burgers' and regularized long wave equations. Comparing our new solutions with the exact solutions reveals that this technique is extremely accurate and effective in solving nonlinear models. Convergence analysis and error estimate are also supplied using some critical theorems.

Engine oil based generalized brinkman-type nano-liquid with molybdenum disulphide nanoparticles of spherical shape: Atangana-Baleanu fractional model


The impact of magnetic field on Engine Oil based generalized Brinkman-type nanofluid over an oscillating vertical plate embedded in a porous medium is studied. Molybdenum Disulphide (MoS2) nanoparticles of spherical shape are suspended in Engine Oil, taken as conventional base fluid. Effect of thermal radiation in energy equation is also considered. A generalized model of Brinkman-type fluid is considered with newly introduced fractional derivatives known as Atangana-Baleanu Derivative (ABD) in the presence of heat transfer due to convection. Exact solution of the problem is determined by means of the Laplace transform. Expressions for velocity and temperature are obtained in terms of Mittag-Leffler and General Wright function. The effects of various pertinent parameters on velocity are portrayed and discussed graphically. Numerical results of rate of heat transfer are computed in tabular form. Which showed that increasing values of volume fraction and Prandtl number increase rate of heat transfer.

The Hunter-Saxton Equation: A Numerical Approach Using Collocation Method


In this study, we are going to present an overview on the Hunter-Saxton equation which is a famous equation modelling waves in a massive director field of a nematic liquid crystal. The collocation finite element method is based on quintic B-spline basis for obtaining numerical solutions of the equation. Using this method, after discretization, solution of the equation expressed as linear combination of shape functions and B-spline basis. So, Hunter-Saxton equation converted to nonlinear ordinary differential equation system. With the aid of the error norms L 2 and L ∞ , some comparisons are presented between numeric and exact solutions for different step sizes. As a result, the authors observed that the method is a powerful, suitable and reliable numerical method for solving various kind of partial differential equations.

Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu


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.

Issue Information


Two-grid method for two-dimensional nonlinear Schrödinger equation by finite element method


A conservative two-grid finite element scheme is presented for the two-dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse-grid solution as the initial guess. Moreover, error estimates are conducted for the two-grid method. It is shown that the coarse space can be extremely coarse, with no loss in the order of accuracy, and still achieve the asymptotically optimal approximation as long as the mesh sizes satisfy H = O ( h 1 2 ) in the two-grid method. The numerical results show that this method is very effective.

A posteriori error analysis for solving the Navier-Stokes problem and convection-diffusion equation


In this article, we consider the finite element discretization of the Navier-Stokes problem coupled with convection-diffusion equations where both the viscosity and the diffusion coefficients depend on the temperature. Existence and uniqueness of a solution are established. We prove a posteriori error estimates.

Long-time stability and asymptotic analysis of the IFE method for the multilayer porous wall model


In this article, we study the long-time stability and asymptotic behavior of the immersed finite element (IFE) method for the multilayer porous wall model for the drug-eluting stents. First, with the IFE method for the spatial descretization, and the implicit Euler scheme for the temporal discretization, respectively, we deduce the global stability of fully discrete solution. Then, we investigate the asymptotic behavior of the discrete scheme which reveals that the multilayer porous wall model converges to the corresponding elliptic equation if f ( x , t ) approaches to a steady-state f ¯ ( x ) in both L 1 ( 0 , t ; L 2 ( Ω ) ) and L ∞ ( 0 , t ; L 2 ( Ω ) ) norms as t + ∞ . Finally, some numerical experiments are given to verify the theoretical predictions.

A scattering-based algorithm for wave propagation in one dimension


We present an explicit numerical scheme to solve the variable coefficient wave equation in one space dimension with minimal restrictions on the coefficient and initial data.

New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves


In this article, some high-order accurate difference schemes of dispersive shallow water waves with Rosenau-KdV-RLW-equation are presented. The corresponding conservative quantities are discussed. Existence of the numerical solution has been shown. A priori estimates, convergence, uniqueness, and stability of the difference schemes are proved. The convergence order is O ( h 4 + k 2 ) in the uniform norm without any restrictions on the mesh sizes. At last numerical results are given to support the theoretical analysis.

Spurious solutions for the advection-diffusion equation using wide stencils for approximating the second derivative


A one-dimensional steady-state advection-diffusion problem using summation-by-parts operators is investigated. For approximating the second derivative, a wide stencil is used, which simplifies implementation and stability proofs. However, it also introduces spurious, oscillating, modes for all mesh sizes. We prove that the size of the spurious modes are equal to the size of the truncation error for a stable approximation and hence disappears with the convergence rate. The theoretical results are verified with numerical experiments.

Maximum norm stability and error estimates for the evolving surface finite element method


We show convergence in the natural L ∞ and W 1 , ∞ norm for a semidiscretization with linear finite elements of a linear parabolic partial differential equations on evolving surfaces. To prove this, we show error estimates for a Ritz map, error estimates for the material derivative of a Ritz map and a weak discrete maximum principle.

Orthogonal spline collocation scheme for multiterm fractional convection-diffusion equation with variable coefficients


The orthogonal spline collocation (OSC) technique is an efficient way to solve a wide variety of problems that are modeled by ordinary and partial differential equations. In this article, by using OSC method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of two-dimensional multiterm fractional convection-diffusion reaction equation with variable coefficients. The optimal estimates in Hj (j = 0, 1, 2) norms at each time step are derived. Also, L ∞ estimate in space is provided. At last, we provide some numerical results to verify the accuracy and efficiency of the proposed algorithm.

A stable boundary elements method for magnetohydrodynamic channel flows at high Hartmann numbers


The article is devoted to extension of boundary element method (BEM) for solving coupled equations in velocity and induced magnetic field for time dependent magnetohydrodynamic (MHD) flows through a rectangular pipe. The BEM is equipped with finite difference approach to solve MHD problem at high Hartmann numbers up to 106. In fact, the finite difference approach is used to approximate partial derivatives of unknown functions at boundary points respect to outward normal vector. It yields a numerical method with no singular boundary integrals. Besides, a new approach is suggested in this article where transforms 2D singular BEM's integrals to 1D nonsingular ones. The new approach reduces computational cost, significantly. Note that the stability of the numerical scheme is proved mathematically when computational domain is discretized uniformly and Hartmann number is 40 times bigger than length of boundary elements. Numerical examples show behavior of velocity and induced magnetic field across the sections.

High-order implicit staggered-grid finite differences methods for the acoustic wave equation


Motivated by the idea that staggered-grid methods give a greater stability and give energy conservation, this article presents a new family of high-order implicit staggered-grid finite difference methods with any order of accuracy to approximate partial differential equations involving second-order derivatives. In particular, we numerically analyze our new methods for the solution of the one-dimensional acoustic wave equation. The implicit formulation is based on the plane wave theory and the Taylor series expansion and only involves the solution of tridiagonal matrix equations resulting in an attractive method with higher order of accuracy but nearly the same computation cost as those of explicit formulation. The order of accuracy of the proposal staggered formulas are similar to the methods with conventional grids for a ( 2 M + 2 ) -point operator: the explicit formula is ( 2 M ) th-order and the implicit formula is ( 2 M + 2 ) th-order; however, the results demonstrate that new staggered methods are superior in terms of stability properties to the classical methods in the context of solving wave equations.

Application of finite difference method of lines on the heat equation


In this article, we apply the method of lines (MOL) for solving the heat equation. The use of MOL yields a system of first–order differential equations with initial value. The solution of this system could be obtained in the form of exponential matrix function. Two approaches could be applied on this problem. The first approach is approximation of the exponential matrix by Taylor expansion, Padé and limit approximations. Using this approach leads to create various explicit and implicit finite difference methods with different stability region and order of accuracy up to six for space and superlinear convergence for time variables. Also, the second approach is a direct method which computes the exponential matrix by applying its eigenvalues and eigenvectors analytically. The direct approach has been applied on one, two and three-dimensional heat equations with Dirichlet, Neumann, Robin and periodic boundary conditions.

The method of mixed volume element-characteristic mixed volume element and its numerical analysis for three-dimensional slightly compressible two-phase displacement


Numerical simulation of oil-water two-phase displacement is a fundamental problem in energy mathematics. The mathematical model for the compressible case is defined by a nonlinear system of two partial differential equations: (1) a parabolic equation for pressure and (2) a convection-diffusion equation for saturation. The pressure appears within the saturation equation, and the Darcy velocity controls the saturation. The flow equation is solved by the conservative mixed volume element method. The order of the accuracy is improved by the Darcy velocity. The conservative mixed volume element with characteristics is applied to compute the saturation, that is, the diffusion is discretized by the mixed volume element and convection is computed by the method of characteristics. The method of characteristics has strong computational stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation. Small time truncation error and accuracy are obtained through this method. The mixed volume element simulates diffusion, saturation, and the adjoint vector function simultaneously. By using the theory and technique of a priori estimates of differential equations, convergence of the optimal second order in l 2 norm is obtained. Numerical examples are provided to show the effectiveness and viability of this method. This method provides a powerful tool for solving challenging benchmark problems.

A two-grid stabilized mixed finite element method for Darcy-Forchheimer model


A two-grid stabilized mixed finite element method based on pressure projection stabilization is proposed for the two-dimensional Darcy-Forchheimer model. We use the derivative of a smooth function, ε 2 + u 2 , to approximate the derivative of | u | in constructing the two-grid algorithm. The two-grid method consists of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. There are a substantial reduction in computational cost. We prove the existence and uniqueness of solution of the discrete schemes on the coarse grid and the fine grid and obtain error estimates for the two-grid algorithm. Finally, some numerical experiments are carried out to verify the accuracy and efficiency of the method.

Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation


In this article, an efficient algorithm for the evaluation of the Caputo fractional derivative and the superconvergence property of fully discrete finite element approximation for the time fractional subdiffusion equation are considered. First, the space semidiscrete finite element approximation scheme for the constant coefficient problem is derived and supercloseness result is proved. The time discretization is based on the L1-type formula, whereas the space discretization is done using, the fully discrete scheme is developed. Under some regularity assumptions, the superconvergence estimate is proposed and analyzed. Then, extension to the case of variable coefficients is also discussed. To reduce the computational cost, the fast evaluation scheme of the Caputo fractional derivative to solve the fractional diffusion equations is designed. Finally, numerical experiments are presented to support the theoretical results.

Efficient hybrid method for solving special type of nonlinear partial differential equations


In this article, an efficient hybrid method has been developed for solving some special type of nonlinear partial differential equations. Hybrid method is based on tanh–coth method, quasilinearization technique and Haar wavelet method. Nonlinear partial differential equations have been converted into a nonlinear ordinary differential equation by choosing some suitable variable transformations. Quasilinearization technique is used to linearize the nonlinear ordinary differential equation and then the Haar wavelet method is applied to linearized ordinary differential equation. A tanh–coth method has been used to obtain the exact solutions of nonlinear ordinary differential equations. It is easier to handle nonlinear ordinary differential equations in comparison to nonlinear partial differential equations. A distinct feature of the proposed method is their simple applicability in a variety of two- and three-dimensional nonlinear partial differential equations. Numerical examples show better accuracy of the proposed method as compared with the methods described in past. Error analysis and stability of the proposed method have been discussed.

High-order Padé and singly diagonally Runge-Kutta schemes for linear ODEs, application to wave propagation problems


In this article, we address the problem of constructing high-order implicit time schemes for wave equations. We consider two classes of one-step A-stable schemes adapted to linear Ordinary Differential Equation (ODE). The first class, which is not dissipative is based on the diagonal Padé approximant of exponential function. For this class, the obtained schemes have the same stability function as Gauss Runge-Kutta (Gauss RK) schemes. They have the advantage to involve the solution of smaller linear systems at each time step compared to Gauss RK. The second class of schemes are constructed such that they require the inversion of a unique linear system several times at each time step like the Singly Diagonally Runge-Kutta (SDIRK) schemes. While the first class of schemes is constructed for an arbitrary order of accuracy, the second-class schemes is given up to order 12. The performance assessment we provide shows a very good level of accuracy for both classes of schemes, and the great interest of considering high-order time schemes that are faster. The diagonal Padé schemes seem to be more accurate and more robust.

Superconvergence analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation


The purpose of this article is to apply E Q 1 rot nonconforming finite element(FE) to solve a generalized nonlinear Schrödinger equation. First, a new important property of E Q 1 rot nonconforming FE (see (2.3) of Lemma 2 below) is proved by use of BHX lemma and the integral identities techniques. Second, a linearized Crank-Nicolson fully discrete scheme is constructed and the superclose error estimate of order O ( h 2 + τ 2 ) for original variable u in broken H1-norm is also derived by using the properties of E Q 1 rot element and the splitting argument for nonlinear terms, while previous works always only obtain convergent error estimates with this element. Furthermore, the global superconvergence is arrived at by the interpolated postprocessing technique. Finally, two numerical experiments are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step.