Fourth-order Finite Difference Research Articles

Propagation of laser-generated shock waves in metals: 3D axisymmetric simulations compared to experiments

This work aims at demonstrating the ability of an acoustic linear code to model the propagation of a shock wave created by a laser impact over a metallic surface. In this process, a high pressure surface level is reached using a ns laser pulse that heats the surface of the material and generates a dense plasma expansion. The pressure reaches few GPa so shock waves are generated and propagate into the bulk of the material. Currently, shock wave propagation is modeled using continuity equations and an ad hoc equation of state for the illuminated material, very limiting because it is numerically intensive. Here, we propose to model the shock wave bulk propagation using a linear acoustic code. A nonlinear surface pressure term, resulting from the laser–matter interaction, is used as a boundary condition. The applied numerical scheme is based on the Virieux scheme, including a fourth order finite difference discretization of the linearized elastomechanical equations. The role of longitudinal and transverse waves and their origins are highlighted. The importance of considering 3D geometries is pointed out. Simulations are finally confronted with experimental results obtained with the Hephaistos Laserlab facility (energy up to 14 J at 532 nm wavelength laser; pulse duration: 7 ns). Illuminations up to the optical breakdown in water are easily achieved with laser focal spots of 5 mm width. Excellent agreement between experiments and simulations is observed for several sets of experimental parameters for titanium, a material of high elastic limit, while limitations are founded for aluminum. The code is available in the MetaData.

Optimal point-wise error estimate of two conservative fourth-order compact finite difference schemes for the nonlinear Dirac equation

In this paper, we propose and analyze two conservative fourth-order compact finite difference schemes for the (1+1) dimensional nonlinear Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise forms of the proposed compact schemes into equivalent vector forms and analyze their conservative and convergence properties. We prove that the proposed schemes preserve the total mass and energy in the discrete level and the convergence rate of the schemes, without any restrictions on the grid ratio, are at the order of O(h4+τ2) in l∞-norm, where h and τ are spatial and temporal steps, respectively. The error analysis techniques include the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. The numerical experiments are carried out to confirm our theoretical analysis. The research method in this paper can be easily extended to higher order compact schemes or other types of wave equations.

Transportation of magnetite nanofluid flow and heat transfer over a rotating porous disk with Arrhenius activation energy: Fourth order Noumerov’s method

This article explains the effect of magnetite nanofluid taking into account water as the base fluid, over a rotating disk in the presence of external magnetic field. The governing boundary layer equations of laminar and incompressible flow over rotating disk have been formulated under the simultaneous influence of thermophoresis and Brownian motion. Additionally, influence of chemical reaction between species, prompted by Arrhenius activation energy is considered into the model. The Von Karman’s transformations of the Navier-Stokes equations are used to obtain the non-dimensional forms of the governing equations, and then fourth order finite difference scheme is used to find the numerical solutions using Noumerov’s discretization. This technique has strong impact on the accuracy of finite difference scheme. The physical descriptions of fluid velocity, temperature, concentration profiles and quantities of practical interest such as skin friction coefficient, Nusselt number and Sherwood number are presented with the help of graphs. One of the significant findings of this analysis include that an intensification in the thermophoresis parameter causes a downfall in the rate of heat transfer at the surface of the disk whereas another significant outcomes of the present investigation is that there is a downfall in the behaviours of the species concentration with the strengthening of Arrhenius activation energy. The fluid flow due to rotating disks has application in mechanisms of thermal transformations for nuclear propulsion devices.

Higher-Order Compact Finite Difference for Certain PDEs in Arbitrary Dimensions

In this paper, we first present the expression of a model of a fourth-order compact finite difference (CFD) scheme for the convection diffusion equation with variable convection coefficient. Then, we also obtain the fourth-order CFD schemes of the diffusion equation with variable diffusion coefficients. In addition, a fine description of the sixth-order CFD schemes is also developed for equations with constant coefficients, which is used to discuss certain partial differential equations (PDEs) with arbitrary dimensions. In this paper, various ways of numerical test calculations are prepared to evaluate performance of the fourth-order CFD and sixth-order CFD schemes, respectively, and the empirical results are proved to verify the effectiveness of the schemes in this paper.

Fast high-order method for multi-dimensional space-fractional reaction–diffusion equations with general boundary conditions

To achieve the efficient and accurate long-time integration, we propose a fast and stable high-order numerical method for solving fractional-in-space reaction–diffusion equations. The proposed method is explicit in nature and utilizes the fourth-order compact finite difference scheme and matrix transfer technique (MTT) in space with FFT-based implementation. Time integration is done through the modified fourth-order exponential time differencing Runge–Kutta scheme. The linear stability analysis and various numerical experiments including two-dimensional (2D) Fitzhugh–Nagumo, Allen–Cahn, Gierer–Meinhardt, Gray–Scott and three-dimensional (3D) Schnakenberg models are presented to demonstrate the accuracy, efficiency, and stability of the proposed method.

Fourth-order time-stepping compact finite difference method for multi-dimensional space-fractional coupled nonlinear Schrödinger equations

In this work, an efficient fourth-order time-stepping compact finite difference scheme is devised for the numerical solution of multi-dimensional space-fractional coupled nonlinear Schrodinger equations. Some existing numerical schemes for these equations lead to full and dense matrices due to the non-locality of the fractional operator. To overcome this challenge, the spatial discretization in our method is carried out by using the compact finite difference scheme and matrix transfer technique in which FFT-based computations can be utilized. This avoids storing the large matrix from discretizing the fractional operator and also significantly reduces the computational costs. The amplification symbol of this scheme is investigated by plotting its stability regions, which indicates the stability of the scheme. Numerical experiments show that this scheme preserves the conservation laws of mass and energy, and achieves the fourth-order accuracy in both space and time.

A high-order compact difference method on fitted meshes for Neumann problems of time-fractional reaction–diffusion equations with variable coefficients

This paper is concerned with numerical methods for a class of nonhomogeneous Neumann problems of time-fractional reaction–diffusion equations with variable coefficients. The solutions of this kind of problems often have weak singularity at the initial time. This makes the existing numerical methods with uniform time mesh often lose accuracy. In this paper, we propose and analyze a high-order compact finite difference method with nonuniform time mesh. The time-fractional derivative is approximated by Alikhanov’s high-order approximation on a class of fitted time meshes. For the spatial variable coefficient differential operator, a new fourth-order boundary discretization is developed under the nonhomogeneous Neumann boundary condition, and then a new fourth-order compact finite difference approximation on a space uniform mesh is obtained. Under the assumption of the weak initial singularity of solution, we prove that for the general case of the variable coefficients, the proposed method is unconditionally stable and the numerical solution converges to the solution of the problem under consideration. The convergence result also gives an optimal error estimate of the numerical solution in the discrete L2-norm, which shows that the method has the spatial fourth-order convergence, while it attains the temporal optimal second-order convergence provided a proper mesh grading parameter is employed. Numerical results that confirm the sharpness of the error analysis are presented.

Time fourth-order energy-preserving AVF finite difference method for nonlinear space-fractional wave equations

In this paper, we develop and analyze a new time fourth-order energy-preserving average vector field (AVF) finite difference method for the nonlinear fractional wave equations with Riesz space-fractional derivative. To the corresponding Hamiltonian system of the nonlinear fractional wave equations, the fourth-order weighted and shifted Lubich difference operator in space and the fourth-order AVF method in time are used to develop the time fourth-order energy-preserving AVF finite difference method for the nonlinear fractional wave equations. The energy conservation in the discrete form and unique solvability of the proposed scheme are proved and error estimates of the scheme are further proved to be order of O((Δt)4+h4) in the discrete L2- norm. Numerical experiments confirm energy conservation and high-order accuracy of the proposed scheme.

A high-order accurate finite difference scheme for the KdV equation with time-periodic boundary forcing

In this article, we present an accurate finite difference scheme for solving the KdV equation with time-periodic forcing at boundary. We first employ a variable transformation to change the time-periodic boundary to a homogeneous boundary. We then develop a three-level fourth-order accurate finite difference scheme for solving the transformed KdV problem. The stability, existence and uniqueness of the numerical solution are analyzed. Numerical examples are provided to confirm the accuracy of the scheme, and the effectiveness for time-periodic simulation.

On high-order compact schemes for the multidimensional time-fractional Schr\xf6dinger equation

In this article, some high-order compact finite difference schemes are presented and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations. The time Caputo fractional derivative is evaluated by the L1 and L1-2 approximation. The space discretization is based on the fourth-order compact finite difference method. For the one-dimensional problem, the rates of the presented schemes are of order O(tau ^{2-alpha }+h^{4}) and O(tau ^{3-alpha }+h^{4}), respectively, with the temporal step size τ and the spatial step size h, and alpha in (0,1). For the two-dimensional problem, the high-order compact alternating direction implicit method is used. Moreover, unconditional stability of the proposed schemes is discussed by using the Fourier analysis method. Numerical tests are performed to support the theoretical results, and these show the accuracy and efficiency of the proposed schemes.

Impacts of Viscous Dissipation and Brownian motion on Jeffrey Nanofluid Flow over an Unsteady Stretching Surface with Thermophoresis

The goal of this investigation is to explore the influence of viscous dissipation and Brownian motion on Jeffrey nanofluid flow over an unsteady moving surface with thermophoresis and mixed convection. Zero mass flux is also addressed at the surface such that the nanoparticles fraction of maintains itself on huge obstruction. An aiding transformation is adopted to renovate the governing equations into a set of partial differential equations which is solved using a new fourth-order finite difference continuation method and various graphical outcomes are discussed in detail with several employed parameters. The spectacular influence of pertinent constraints on velocity and thermal curves are inspected through various plots. Computational data for the heat transfer rate and skin-friction coefficient are also reported graphically. Graphical outcomes indicate that an augmentation in buoyance ratio and thermophoretic parameter leads to diminish the velocity curves and increase the temperature curves. Furthermore, it is inspected that escalating Deborah number exhibits increasing in the skin friction and salient decreasing heat transmission. Increasing magnetic strength leads to a reduction in the skin friction and enhancement in the Nusselt number, whilst a reverse reaction is manifested with mixed convection aspects.

Heat Transfer Enhancement of Magneto-Micropolar Nanofluid Over a Wedge

Current investigation characterizes the flow and heat transmission of magneto-micropolar nanofluid through a non-isothermal wedge. The base-fluid as water and micropolar nanofluid as Copper or Alumina-nanoparticles are considered. Applying the similarity transformations along with non-dimensional quantities the formulated equations of the investigation are transmuted into a system of non-linear ODEs with a collection of convenient boundary conditions. The fourth-order finite difference method (FFDM) is then applied to determine the solution of a collection of resultant equations. The outcomes obtained by FFDM have also compared with cited works. Illustrations describing influences of prominent parameters which provide physical interpretations of temperature, micro rotation and velocity fields are examined in detail with the help of graphical representations. Both the skin friction coefficient and Nusselt number are computed and exhibited through tabular forms. This investigation determined that the skin-friction coefficient and heat transport rate improved along with augmentation in the magnetic force. micropolar parameter and the nanoparticle volume fraction augmented the Both skin friction coefficient and Nusselt number.

Numerical groundwater quality assessment model using a new fourth-order scheme with Saulyev method

Landfill is a cause of environmental groundwater pollutant. Landfill operations most feasible in many medium income developing countries. The groundwater pollutant measurement is required to control the drinking water quality. The mathematical model are introduced to predict the environmental impact assessment. The predicted impact of groundwater quality in is needed if a landfill project is planned to constant in considered area. In this research, a long term groundwater quality assessment in heterogenous soil model is proposed. There are two numerical models are introduced. The traditional forward time centered space finite difference technique is used to approximate the groundwater pollutant concentration in an area around a landfill. The new fourth-order finite difference technique with Saulyev method is also employed to approximate the solution as well. The approximated solutions are compared with the ideal exact solution. Both numerical techniques give good agreement approximated solution. The proposed new fourth-order scheme with Saulyev method give better approximated solution than the traditional method. The propose numerical models can be apply to predict the groundwater quality long period of time in another type of soil-physics.

A new conservative finite difference scheme for the generalized Rosenau–KdV–RLW equation

In this paper, a new conservative fourth-order finite difference scheme is proposed for solving the generalized Rosenau–KdV–RLW equation. The solvability, convergence, and conservation of the numerical solution are discussed by the discrete energy method. The scheme is convergent of $$O(\tau ^{2}+h^{4})$$ and unconditionally stable. Several numerical experiment results show that the proposed scheme is efficient and reliable.

Energy-preserving time high-order AVF compact finite difference schemes for nonlinear wave equations with variable coefficients

In this article, we develop and analyze two energy-preserving high-order average vector field (AVF) compact finite difference schemes for solving variable coefficient nonlinear wave equations with periodic boundary conditions. Specifically, we first consider the variable coefficient nonlinear wave equation as an infinite-dimensional Hamiltonian system. Then the fourth-order compact finite difference and AVF techniques are applied to the resulting Hamiltonian system for the spatial discretization and time integration, respectively, which yield two energy-preserving high-order schemes, one is a time second-order AVF compact finite difference scheme (AVF(2)-CFD) and the other is a time fourth-order AVF compact finite difference scheme (AVF(4)-CFD). We theoretically prove that the proposed schemes satisfy energy conservations in the discrete forms and are uniquely solvable. Also, we prove the convergence of the proposed schemes and their error estimates, where the AVF(4)-CFD scheme is of fourth-order convergence in both time and space. Numerical experiments for the nonlinear wave equations with various nonlinearities are given to show their performances, which confirm the theoretical results.

A long-term numerical groundwater quality assessment model using a modified fourth-order finite difference method with Saulyev scheme

Mathematical models are required for long-term groundwater quality assessment around landfill sites. The long-term groundwater-quality prediction is used to report the environmental impact assessment of landfill site projects. In this research, a one- dimensional transient advection-diffusion equation is used to describe groundwater pollutant concentration measurement. Numerical simulations of groundwater quality measurement for long periods of time are proposed. The traditional and modified finite difference techniques are used to approximate the solutions. The traditional explicit forward time central space and the standard fourth-order finite difference methods are introduced. The standard fourth-order finite difference methods with Saulyev Scheme and a modified fourth-order finite difference methods with Saulyev Scheme are proposed. The approximated solutions are compared with the exact solutions in a specified case. The proposed modified new fourth-order finite difference technique gives accurate approximated solutions. The proposed numerical simulation can be applied to several types of soil physics.

A fourth order finite difference method for solving elliptic interface problems with the FFT acceleration

In this paper, a new Cartesian grid finite difference method is introduced based on the fourth order accurate matched interface and boundary (MIB) method and fast Fourier transform (FFT). The proposed augmented MIB method consists of two major parts for solving elliptic interface problems in two dimensions. For the interior part, in treating a smoothly curved interface, zeroth and first order jump conditions are enforced repeatedly by the MIB scheme to generate fictitious values near the interface. For the exterior part, two layers of zero-padding solutions are introduced beyond the original rectangular domain so that the FFT inversion becomes feasible. Different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations, can be imposed via the MIB scheme to generate fictitious values near boundaries. Based on fictitious values at both interfaces and boundaries, the augmented MIB method reconstructs Cartesian derivative jumps as auxiliary variables. Then, by treating such variables as unknowns, an enlarged linear system is obtained. In the Schur complement solution of such system, the FFT algorithm will not sense the solution discontinuities, so that the discrete Laplacian can be efficiently inverted. Therefore, the FFT-based augmented MIB not only achieves a fourth order of accuracy in dealing with interfaces and boundaries, but also produces an overall complexity of O(n2log⁡n) for a n×n uniform grid. Moreover, the augmented MIB scheme can provide fourth order accurate approximations to solution gradients and fluxes.

FD3D_TSN: A Fast and Simple Code for Dynamic Rupture Simulations with GPU Acceleration

AbstractWe introduce FD3D_TSN—an open-source Fortran code for 3D dynamic earthquake rupture modeling based on the staggered grid fourth-order finite-difference method employing a regular cubical spatial discretization. Slip-weakening and fast-velocity-weakening rate-and-state fault friction laws are combined with vertical planar fault geometry, orthogonal to a planar free surface. FD3D_TSN demonstrates good agreement with other methods in a range of benchmark exercises of the Southern California Earthquake Center and U.S. Geological Survey dynamic rupture code verification project. Efficient graphic processing units (GPU) acceleration using the OpenACC framework yields a factor of 10 speed-up in terms of time to solution compared to a single-core solution for current hardware (Intel i9-9900K and Nvidia RTX 2070). The software is fast and easy-to-use and suitable explicitly for data-driven applications requiring a large number of forward simulations such as dynamic source inversion or probabilistic ground-motion modeling. The code is freely available for the scientific community and may be incorporated in physics-based earthquake source imaging and seismic hazard assessment, or for teaching purposes.

Analysis of a high-order compact finite difference method for Robin problems of time-fractional sub-diffusion equations with variable coefficients

This paper is concerned with the construction and analysis of a high-order compact finite difference method for a class of time-fractional sub-diffusion equations under the Robin boundary condition. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α∈(0,1). A (3−α)th-order numerical formula (called the L2 formula here) without any sub-stepping scheme for the approximation at the first-time level is applied to the discretization of the Caputo time-fractional derivative. A new fourth-order compact finite difference operator is constructed to approximate the variable coefficient spatial differential operator under the Robin boundary condition. By developing a technique of discrete energy analysis, the unconditional stability of the proposed method and its convergence of (3−α)th-order in time and fourth-order in space are rigorously proved for the general case of variable coefficient and for all α∈(0,1). Further approximations are considered for enlarging the applicability of the method while preserving its high-order accuracy. Numerical results are provided to demonstrate the theoretical analysis results.

Performance of compact and non-compact structure preserving algorithms to traveling wave solutions modeled by the Kawahara equation

The main contribution of this article is to introduce new compact fourth-order, standard fourth-order, and standard second-order finite difference schemes for solving the Kawahara equation, the fifth-order partial derivative equation. The conservation of mass only of the numerical solution obtained by the compact fourth-order finite difference scheme is proven. However, the standard fourth-order and standard second-order finite difference schemes can preserve both mass and energy. The stability is also proven by von Neumann analysis. According to analysis for numerical experiments, the order of accuracy for each scheme and the computational efficiency of the compact scheme are presented. To validate the potential of the presented methods, we also consider long-time behavior. Finally, results obtained from the compact scheme are superior than those from the non-compact schemes.