Efficient Finite Difference Scheme Research Articles

Two linear energy-preserving compact finite difference schemes for coupled nonlinear wave equations

In this paper, we propose and analyze two highly efficient compact finite difference schemes for coupled nonlinear wave equations containing coupled sine-Gordon equations and coupled Klein-Gordon equations. To construct energy-preserving, high-order accurate and linear numerical methods, we first utilize the scalar auxiliary variable (SAV) approach and introduce three auxiliary functions to rewrite the original problem as a new equivalent system. Then we make use of the compact finite difference method and the Crank-Nicolson method to propose an efficient fully-discrete scheme (SAV-CFD-CN). The modified energy conservation and the convergence of the SAV-CFD-CN scheme are proved in detail, which has fourth-order convergence in space and second-order convergence in time. In order to preserve the discrete energy of original system, we further combine Lagrange multiplier approach, compact finite difference method and the Crank-Nicolson method to propose the second fully-discrete scheme (LM-CFD-CN). The proposed two schemes are high-order accurate, linear and highly efficient, only four symmetric positive definite systems with constant coefficients are required to be solved at each time level. Numerical experiments for the coupled nonlinear wave equations are given to confirm theoretical findings.

Solar radiation and lower gravitational effects on wave oscillations in heat transfer along magnetic-driven porous cone in the presence of Joule heating

Solar radiation is a very useful form of energy such as heat and current density in gas turbines, nuclear power plants and thermal energy storage. The fundamental source of heat for multiple reactions in the climate, oceans, and ecological system is solar radiation. The current radiative flow model has significant applications in nuclear power generation, radioactive reactor cooling mechanisms, heating boilers, recycling of underground radioactive materials, fabrication of glass-fiber material and thermal textiles. The significance of study to reduce excessive heating by using magnetized surface. The main goal of current research is to report the Joule heating effect on heat and magnetic flux along the magnetic-driven porous cone under solar radiations and lower gravitational region. The novelty of this work is to explore wave oscillations in heat and magnetic flux with solar radiations under lower gravitational region. The Joule heating is applied to control the thermal boundary layer along gravity-driven cone. The coupled mathematical model is changed into non-dimensional form for physical parameters. The steady and oscillatory parts are obtained by using Stokes conditions. For smooth programming in FORTRAN computing tool, the primitive variables are used. The efficient finite difference scheme is used to plot the numerical findings with Gaussian elimination technique. The impact of governing parameters involved in the problem on wave oscillations of magnetic flux and heat transmission are drafted physically and numerically. It is noted that the fluid velocity enhances with significant amplitude as lower gravity and solar radiation increases along porous cone. It is found that wave oscillations of heat and magnetic flux increases as magnetic Prandtl number enhances under lower gravitational region.

Efficient energy-preserving finite difference schemes for the Klein-Gordon-Schrödinger equations

Efficient energy-preserving finite difference schemes for the Klein-Gordon-Schrödinger equations

Efficient Upwind Finite-Difference Schemes for Wave Equations on Overset Grids

Efficient Upwind Finite-Difference Schemes for Wave Equations on Overset Grids

Fractional-step finite difference schemes for incompressible elasticity on overset grids

Efficient finite-difference schemes for the numerical solution of the time-dependent equations of incompressible linear elasticity on complex geometry are presented. The schemes are second-order accurate in space and time, and solve the equations in displacement-pressure form. The algorithms use a fractional-step approach in which the time-step of the displacements is performed separately from the solution of a Poisson problem to update the pressure. Complex geometry is treated with curvilinear overset grids. Compatibility boundary conditions and an upwind dissipation for wave equations in second-order form are included to ensure stability for overset grids, and for the difficult case of traction (free-surface) boundary conditions. A divergence damping term is added to keep the dilatation small. The stability of the schemes is studied with GKS mode analysis. Exact eigen-mode solutions are obtained for several problems involving rectangular, cylindrical and spherical configurations, and these are valuable as benchmark problems. The accuracy and stability of the approach in two and three space dimensions is illustrated by comparing the numerical solutions with the exact solutions of the benchmark problems.

Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation

In this paper, three efficient energy conserving compact finite difference schemes are developed to numerically solve the sine-Gordon equation with homogeneous Dirichlet boundary conditions. To be specific, the spatial discretization is carried out by a novel eighth-order accurate compact difference scheme in which the fast discrete sine transform can be utilized for efficient implementation. The second-order conservative Crank–Nicolson scheme is considered in the temporal direction. Then the conservative property and convergence of the first scheme in two-dimensional space are discussed. A linearized iteration based on the fast discrete sine transform technique is devised to solve the nonlinear system effectively. Since the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. Furthermore, the two other schemes are constructed based on improved scalar auxiliary variable approaches by converting the sine-Gordon equation into an equivalent new system which involves solving linear systems with constant coefficients at each time step. Finally, numerical experiments are presented to validate the correctness of the theoretical findings and demonstrate the excellent performance in long-time conservation of the schemes.

Efficient third-order BDF finite difference scheme for the generalized viscous Burgers’ equation

This article presents a third-order backward differentiation formula (BDF3) finite difference scheme for the generalized viscous Burgers’ equation. The discretization of time and space directions is accomplished by the BDF3 method and standard second-order difference formula, respectively, thereby constructing a fully-discrete scheme. For the proposed scheme, we yield the convergence of h2+τ3 by means of the energy argument and the cut-off function method. Besides, a comparison of the time convergence rate and numerical accuracy with those of recent existing work shows the effectiveness and competitiveness of our approach. A numerical experiment is carried out to verify the theoretical predictions.

Mode Propagation Analysis of Magnetically Biased Graphene Microstrips via an Efficient Finite-Difference Scheme

A full-vectorial finite-difference (FD) scheme is proposed in this work to accurately extract the propagating modes on a magnetically biased graphene microstrip. Initially, the anisotropic surface conductivity of graphene is introduced, and the appropriate eigenvalue problem is formulated starting from Maxwell’s equations. In particular, an FD approximation is utilized, while the discretization of the computation domain is based on the popular Yee cell. Then, the relationship between tangential, to the propagation direction, electromagnetic components is derived, leading to a linear eigenvalue problem. The numerical results highlight the expected difference between the propagation properties of the edge modes, thus validating the successful implementation of the featured modal solver. Moreover, it is shown that the number of unknown components is efficiently reduced due to the proper elimination of the longitudinal fields and the linearity of Maxwell’s equations.

Magneto-thermogravitational convection for hybrid nanofluid in a novel shaped enclosure

The proposed work exhibits a numerical simulation of steady thermogravitational convection of Cu–Al2O3(50%−50%) water hybrid magneto-nanofluid in a novel shaped enclosure regulated by magnetic field dependent (MFD) viscosity in presence of flush mounted variable heaters. Various cases (Case-I to Case-IV) have been considered depending on the location of the heaters on the bottom and left boundaries. Pertaining to this analysis, the streamfunction (ψ)- vorticity (ζ) form including the energy transport equation of coupled partial differential equations are evaluated with the help of an efficient higher order compact finite difference scheme. Results are exhibited in terms of streamlines, isotherms and Nusselt numbers to describe the transport phenomena obtained for different physical parametric values to this simulation. The influence of the physical parameters such as, Rayleigh number (104≤Ra≤106), Hartmann number (0≤Ha≤60), hybrid nanoparticles volume fraction (0≤ϕhnp≤0.04), orientation of magnetic field (00≤γ≤600) and magnetic number (0≤δ0≤1) on the hybrid nanofluid flow and energy transfer are analyzed. Obtained results indicate that the combinations of the heaters location and the shaped of the geometry have major impact on all the transport phenomena. Moreover, Cu–Al2O3 hybrid nanoparticles play an essential role to levitate the thermal transmission compared to other physical parameters. An addition of hybrid nanoparticles leads to an increase in the average Nusselt number up to 10.8% in Case-I, 12% in Case-II, 11.66% in Case-III and 11.65% in Case-IV. The novelty of the present work is to examine the rheological properties of hybrid nanoliquid in a novel shaped cavity subjected to magnetic field dependent viscosity (MFDV) along with flush mounted variable heaters.

An efficient operator-splitting radial basis function-generated finite difference (RBF-FD) scheme for image noise removal based on nonlinear total variation models

Image noise removal is one of the most important parts of image processing that can dramatically improve other parts of image processing performance by enhancing the quality of images in databases. Total variation models are second-order partial differential equations for image denoising. These models have some complexities, such as being multidimensional problems, non-linearity, and having large spatial and temporal domains making them challenging problems to be solved numerically. Thus, in this work, we propose the radial basis function generated finite differences (RBF-FD) method in conjunction with a suitable operator splitting technique to overcome these difficulties. This approach has some significant advantages, such as high accuracy, low computational complexity, and the sparsity of the coefficients matrices derived from it. The peak signal-to-noise ratio, structure similarity index measure, and mean-square error metrics are considered to evaluate the proposed approach’s effectiveness and accuracy compared to the other common denoising approaches.

A High Accuracy Local One-Dimensional Explicit Compact Scheme for the 2D Acoustic Wave Equation

In this paper, we develop a highly accurate and efficient finite difference scheme for solving the two-dimensional (2D) wave equation. Based on the local one-dimensional (LOD) method and Padé difference approximation, a fourth-order accuracy explicit compact difference scheme is proposed. Then, the Fourier analysis method is used to analyze the stability of the scheme, which shows that the new scheme is conditionally stable and the Courant-Friedrichs-Lewy (CFL) condition is superior to most existing methods of equivalent order of accuracy in the literature. Finally, numerical experiments demonstrate the high accuracy, stability, and efficiency of the proposed method.

An Efficient Non-standard Finite Difference Scheme for Solving Distributed Order Time Fractional Reaction–Diffusion Equation

An Efficient Non-standard Finite Difference Scheme for Solving Distributed Order Time Fractional Reaction–Diffusion Equation

Fast and efficient finite difference method for the distributed-order diffusion equation based on the staggered grids

In this article, an efficient block-centered finite difference scheme is constructed to solve the distributed-order diffusion equation with Neumann boundary condition on the rectangular grids. We use the mid-rectangle formula to approximate the integral term and a fast technique to evaluate the fractional derivative to derive a valid and efficient scheme. The stability and convergence of the efficient scheme for both pressure and velocity are proved rigorously. Finally, we carry out some numerical examples to verify our theoretical analysis and present the efficiency of our scheme.

Finite Element Method and numerical study for a super-linear reaction-diffusion problem with integral conditions

In this work, we prove the existence, uniqueness, and continuous dependence of generalized solution of a nonlinear reaction-diffusion problem with only integral terms in the boundaries, by using the finite element method.Also we have developed an efficient numerical finite difference schemes. Some numerical results are reported to show the efficiency and accuracy of the scheme.

An accurate, robust, and efficient finite difference scheme with graded meshes for the time-fractional Burgers’ equation

This work presents the L1 implicit difference scheme based on non-uniform meshes for the time-fractional Burgers equation. We find that the difficulty arising from the singularity of the exact solution at t=0 can be overcome by non-uniform meshes. We yield the unconditional stability and optimal convergence rate of h2+τmin{rα,2−α} by means of the energy method, when selecting r=(2−α)/α. Convergence rates are higher than some recently studied schemes. Numerical experiments confirm the predicted theoretical estimate.

A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system

The aim of this paper is to introduce and analyze a novel fractional chaotic system including quadratic and cubic nonlinearities. We take into account the Caputo derivative for the fractional model and study the stability of the equilibrium points by the fractional Routh–Hurwitz criteria. We also utilize an efficient nonstandard finite difference (NSFD) scheme to implement the new model and investigate its chaotic behavior in both time-domain and phase-plane. According to the obtained results, we find that the new model portrays both chaotic and nonchaotic behaviors for different values of the fractional order, so that the lowest order in which the system remains chaotic is found via the numerical simulations. Afterward, a nonidentical synchronization is applied between the presented model and the fractional Volta equations using an active control technique. The numerical simulations of the master, the slave, and the error dynamics using the NSFD scheme are plotted showing that the synchronization is achieved properly, an outcome which confirms the effectiveness of the proposed active control strategy.

High-order conservative scheme for the coupled space fractional nonlinear Schrödinger equations

In this paper, an efficient finite difference scheme is proposed for one dimension and two dimension coupled space fractional nonlinear Schrödinger equations. First, the high-order difference scheme and Crank–Nicolson scheme are used to one dimension coupled space fractional nonlinear Schrödinger equations. second, we show that the high-order conservative difference scheme satisfies the mass and energy conservation laws respectively, and convergence and unconditional stability of the scheme are also proved. Next, we give the high-order conservative scheme for two dimension coupled space fractional nonlinear Schrödinger equations. Finally, some numerical results are reported to verify our theoretical analysis.

Convergence and Stability in Maximum Norms of Linearized Fourth-Order Conservative Compact Scheme for Benjamin–Bona–Mahony–Burgers’ Equation

In the paper, a newly developed three-point fourth-order compact operator is utilized to construct an efficient compact finite difference scheme for the Benjamin–Bona–Mahony–Burgers’ (BBMB) equation. The detailed derivation is carried out based on the reduction order method together with a three-level linearized technique. The conservative invariant, boundedness and unique solvability are studied at length. The convergence is proved by the technical energy argument and induction method with the optimal convergence order $$\mathcal {O}(\tau ^2+h^4)$$ in the sense of the maximum norm. The stability under mild conditions can be achieved based on the uniform boundedness of the numerical solution. The present scheme is very efficient in practical computation since only a linear system needs to be solved at each time. The extensive numerical examples verify our theoretical results and demonstrate the scheme’s superiority when compared with state-of-the-art those in the references.

Effect of Heat Generation (Absorption) on Unsteady MHD flow over a Truncated Cone Embedded in a Stable Thermally Stratified Environment

This paper examines the effect of heat generation (absorption) on unsteady free convection flow of an incompressible electrically conducting fluid over a truncated cone embedded in a thermally stratified medium in the presence of a transverse magnetic field. The governing nonlinear partial differential equations have been solved numerically, using an efficient implicit finite difference scheme along with quasilinearization technique. The numerical results for skin-friction coefficients and velocity components in x- and y- directions, Nusselt number and temperature profiles are obtained and analyzed for different values of governing parameters. It is observed that, heat generation increases skin friction and decreases heat transfer coefficients, whereas opposite trend is observed during heat absorption. Also it is found that heat generation has a significant effect on temperature profile.

An efficient higher order compact finite difference schemes for linear hyperbolic partial differential equations

To approximate the partial derivative of Linear Hyperbolic Partial Differential Equations (LPHDE), a system of compact schemes used at non-boundary nodes. For the spatial discretization of the advection term, instead of using the upwind schemes, central difference based compact 4th order and conventional 2nd order schemes are experimented. The main aim of the numerical experiments carried out is, to assess the ability of the compact 4th order scheme in capturing the convection process, in comparison with the conventional 2nd order scheme.