Fourth-order In Space Research Articles

A high-order L2 type difference scheme for the time-fractional diffusion equation

The present paper is devoted to constructing L2 type difference analog of the Caputo fractional derivative. The fundamental features of this difference operator are studied and it is used to construct difference schemes generating approximations of the second and fourth order in space and the (3−α)th-order in time for the time fractional diffusion equation with variable coefficients. Difference schemes were also constructed for the variable-order diffusion equation and the generalized fractional-order diffusion equation of the Sobolev type. Stability of the schemes under consideration as well as their convergence with the rate equal to the order of the approximation error are proven. The received results are supported by the numerical computations performed for some test problems.

HOC–ADI schemes for two-dimensional Ginzburg–Landau equation in superconductivity

In this paper, we apply the high-order compact scheme coupled with alternating direction implicit (HOC–ADI) method to solve the two-dimensional Ginzburg–Landau (2D GL) equation. Five HOC–ADI schemes with second order accuracy in time and fourth order in space are proposed for 2D GL equation. Scheme I and Scheme II are both nonlinear, which need nonlinear iteration.To overcome this shortcoming, Scheme III and Scheme IV are proposed in order to avoid nonlinear iteration. With the three-level ADI scheme and the method of extrapolation, we obtain a linearized scheme V. Some numerical experiments are shown to testify and compare the superiority of the new numerical schemes.

High‑Order Bicompact Schemes for Numerical Modeling of Multispecies Multi-Reaction Gas Flows

Euler equations for multidimensional inviscid gas flows with multispecies and multi‑reaction are considered. By using the Marchuk–Strang splitting method, an implicit numerical scheme for this system is constructed. Its convection part is computed using the bicompact scheme SDIRK3B4 of fourth order in space and third order in time, while its chemical part is computed using the L-stable Runge–Kutta method of second order. The SDIRK3B4 scheme is compared with the WENO5/SR scheme in the case of one- and two-dimensional flows with detonation waves. It is shown that the SDIRK3B4 scheme has the same actual accuracy as the WENO5/SR, but the former needs fewer time steps by a factor of 20–40 and does not require any special algorithms to suppress the nonphysical breakdown of detonation waves on relatively coarse meshes.

FAST COMPACT DIFFERENCE SCHEME FOR THE FOURTH-ORDER TIME MULTI-TERM FRACTIONAL SUB-DIFFUSION EQUATIONS WITH THE FIRST DIRICHLET BOUNDARY

<abstract> In this paper, a fast compact difference scheme is proposed for the initial-boundary value problem of fourth-order time multi-term fractional sub-diffusion equations with the first Dirichlet boundary conditions. Using the method of order reduction, the original problem can be converted to an equivalent lower-order system. Then at some super-convergence points, the multi-term Caputo derivatives are fast evaluated based on the sum-of-exponentials (SOE) approximation for the kernel functions appeared in Caputo fractional derivatives. The difficulty caused by the first Dirichlet boundary conditions is carefully handled. The energy method is used to illustrate the unconditional stability and convergence of the proposed fast compact scheme. The convergence accuracy is second-order in time and fourth-order in space if the solution has enough regularity. Compared with the direct scheme without the acceleration in time direction, the CPU time of the current fast scheme is largely reduced, which is shown by numerical examples. </abstract>

AN A PRIORI ERROR ANALYSIS OF A STRAIN GRADIENT MODEL USING &lt;i&gt;C&lt;/i&gt;&lt;sup&gt;0&lt;/sup&gt; INTERIOR PENALTY METHODS

In this work we study, from the numerical point of view, a strain gradient model. It can be written as a linear fourth-order in space and second-order in time partial differential equation which leads to a parabolic variational equation in terms of the velocity field. Then, a fully discrete approximation is provided by using the implicit Euler scheme to discretize the time derivatives and the so-called $ C^0 $ interior penalty method for the spatial approximation. A priori error estimates are obtained, and from them it follows the convergence of the approximations (under suitable regularity conditions). Finally, some two-dimensional numerical simulations are shown to demonstrate the numerical behaviour.

A Unifying Numerical Framework for the “Small-Slope” Based Core-Annular Flow Instability Models

The snap-off is an instability phenomenon that takes place during the immiscible two-phase flow in porous media due to competing forces acting on the fluid phases and at the interface between them. Different theoretical approaches have been proposed for the development of mathematical models that describe the dynamics of a fluid/fluid interface in order to analyze the snap-off mechanism. The models studied here are based on the “small-slope” approach and were derived from the mass conservation and other governing equations of two-phase flow at pore scale in circular capillaries for pure and complex interfaces. The models consist of evolution equations; highly nonlinear partial differential equations of fourth order in space and first order in time. Although the structure of the models for each type of interface is similar, different numerical techniques have been employed to solve them. Here, we propose a unifying numerical framework to solve the group of such models. Such a framework is based on the Fourier pseudo-spectral differentiation method which uses the Fast Fourier Transform (FFT) and the inverse FFT (IFFT) algorithms. We compared the solutions obtained with this method to the results reported in the literature in order to validate our framework. In general, acceptable agreements were obtained in the dynamics of the snap-off.

On three-dimensional stable long-wavelength convection in the presence of Dirichlet thermal boundary conditions

It is a well-known fact that the onset of Rayleigh–Benard convection occurs via a long-wavelength instability when the horizontal boundaries are thermally insulated. The aim of this paper is to quantify the exact dimensions of a cylinder of rectangular cross-section wherein stable three-dimensional Rayleigh–Benard convection sets in via a long-wavelength instability from the motionless state at the same value of the critical Rayleigh number as the corresponding horizontally unbounded problem when the bounding horizontal walls have infinite thermal conductance. Hence, we consider three-dimensional Rayleigh–Benard convection in a cell of infinite extent in the x-direction, confined between two vertical walls located at $$y= \pm H$$ and horizontal boundaries located at $$z=0$$ and $$z=d$$ . Our analysis predicts the existence of the sought stable state for experimental velocity boundary conditions at the vertical walls provided the aspect ratio $$\delta = H/d$$ takes a certain value. In the limit $$H \rightarrow \infty $$ , we retrieve the stability characteristics of the horizontally unbounded problem. As expected, the analysis predicts two counter-rotating rolls aligned along the y-direction of period $$2 \pi /\delta $$ equal to the period of the roll in the y-direction of the corresponding unbounded problem. A long-scale asymptotic analysis leads to the derivation of an evolution partial differential equation (PDE) that is fourth order in space and contains a single bifurcation parameter. The PDE, valid for a specific value of $$\delta $$ , is analyzed analytically and numerically as function of the bifurcation parameter and for a variety of velocity boundary conditions at the vertical walls to seek the stable steady-state solutions. The same analysis is also extended to the case of convection in a fluid-saturated porous medium.

Numerical study of bright-bright-dark soliton dynamics in the mixed coupled nonlinear Schrödinger system

An efficient, accurate numerical method is developed to study the mixed (bright-bright-dark) solitons in a three component mixed coupled nonlinear Schrödinger system, arising in nonlinear optics. For this purpose, a compact finite difference method involving different boundary conditions for the bright and dark solitons is employed to derive an implicit nonlinear numerical scheme of fourth order in space and second order in time. Particularly, the scheme is shown to be unconditionally stable by von Neumann stability method. Also, the exact soliton solutions and the conserved quantities are used to assess the proposed method. Propagation dynamics as well as the standard elastic and the fascinating energy-sharing collisions of multicomponent mixed solitons are studied in this framework. Significantly, the energy-sharing collision scenario of solitons is found to be robust against small perturbations.

Adaptive ADI Numerical Analysis of 2D Quenching-Type Reaction: Diffusion Equation with Convection Term

An adaptive high-order difference solution about a 2D nonlinear degenerate singular reaction-diffusion equation with a convection term is initially proposed in the paper. After the first and the second central difference operator approximating the first-order and the second-order spatial derivative, respectively, the higher-order spatial derivatives are discretized by applying the Taylor series rule and the temporal derivative is discretized by using the Crank–Nicolson (CN) difference scheme. An alternating direction implicit (ADI) scheme with a nonuniform grid is built in this way. Meanwhile, accuracy analysis declares the second order in time and the fourth order in space under certain conditions. Sequentially, the high-order scheme is performed on an adaptive mesh to demonstrate quenching behaviors of the singular parabolic equation and analyse the influence of combustion chamber size on quenching. The paper displays rationally that the proposed scheme is practicable for solving the 2D quenching-type problem.

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.

Technologies for supporting high-order geodesic mesh frameworks for computational astrophysics and space sciences

Many important problems in astrophysics, space physics, and geophysics involve flows of (possibly ionized) gases in the vicinity of a spherical object, such as a star or planet. The geometry of such a system naturally favors numerical schemes based on a spherical mesh. Despite its orthogonality property, the polar (latitude-longitude) mesh is ill suited for computation because of the singularity on the polar axis, leading to a highly non-uniform distribution of zone sizes. The consequences are (a) loss of accuracy due to large variations in zone aspect ratios, and (b) poor computational efficiency from a severe limitations on the time stepping. Geodesic meshes, based on a central projection using a Platonic solid as a template, solve the anisotropy problem, but increase the complexity of the resulting computer code. We describe a new finite volume implementation of Euler and MHD systems of equations on a triangular geodesic mesh (TGM) that is accurate up to fourth order in space and time and conserves the divergence of magnetic field to machine precision. The paper discusses in detail the generation of a TGM, the domain decomposition techniques, three-dimensional conservative reconstruction, and time stepping.

A fast temporal second-order compact ADI difference scheme for the 2D multi-term fractional wave equation

In this paper, a fast temporal second-order compact ADI scheme is proposed for the 2D time multi-term fractional wave equation. At the super-convergence point, the multi-term Caputo derivative is approximated by combining the order reduction technique with the sum-of-exponential approximation to the kernel function appeared in Caputo derivative. The difference scheme can be solved by the recursion, which reduces the storage and computational cost significantly. The obtained scheme is uniquely solvable. The unconditional convergence and stability of the scheme in the discrete H1-norm are proved by the discrete energy method and the convergence accuracy is second-order in time and fourth-order in space. Numerical example illustrates the efficiency of the scheme.

A discrete Schwarz waveform relaxation method of higher order for singularly perturbed parabolic reaction–diffusion problems

We consider singularly perturbed parabolic reaction–diffusion problems of scalar and vector types. We construct a discrete Schwarz waveform relaxation method of higher order for their numerical solution. The method is shown to be parameter-uniformly convergent having almost fourth order in space and first order in time. Further, interesting result proven is much faster convergence of iterative process for small perturbation parameter. Numerical results demonstrating the efficiency of the proposed method and validating the theoretically proven convergence results are given.

Numerical Computation and Stability Analysis for the Fractional Subdiffusions with Spatial Variable Coefficients

In this paper, we propose an efficient compact finite difference method for a class of time-fractional subdiffusion equations with spatially variable coefficients. Based on the L2-1σ approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, an efficient compact finite difference method is developed. The local truncation error and the solvability of the developed method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Numerical examples are provided to demonstrate the accuracy and the theoretical results.

An Accurate Three-Dimensional FDTD(2,4) Method on Face-Centered Cubic Grids With Low Numerical Dispersion

In this letter, a three-dimensional finite-difference time-domain (FDTD) method that is second-order in time and fourth-order in space on a face-centered cubic (FCC) grid, termed as FCC-FDTD(2,4), is proposed to solve the time-domain Maxwell's equations. The proposed method shows a much lower numerical dispersion error compared with the errors of other FDTD schemes, such as the FCC-FDTD(2,2) method, the FDTD(2,2) method, and the FDTD(2,4) method. This improvement in the numerical dispersion results from the FCC grid and the fourth-order scheme in space used in the proposed FCC-FDTD(2,4) method. Then, the Courant-Friedrichs-Lewy condition is given through spectral analysis, and an analytical numerical dispersion relation is obtained and comprehensively studied through numerical results with the FCC-FDTD(2,2) method, the FDTD(2,2) method, and the FDTD(2,4) method. Compared with the other three FDTD methods, the proposed method has much smaller numerical dispersion errors. Finally, the accuracy of the proposed method is further validated with other numerical examples.

An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

3D Numerical Simulation of Elastic Wave Propagation in Discrete Fracture Network Rocks

Fractures play an important role in controlling rock block stability and the hydraulic properties of fractured rock formations. Understanding elastic wave propagation in fractured media can result in significant advances for the geophysical prediction of fracture parameters from seismic data. However, most natural fracture characteristics, such as fracture length, aperture, angle and location are random; therefore, fracture models must be built discretely and follow some stochastic principles. We construct stochastic models of fractured rock samples using a random fracture network rather than a single fracture. Three-dimensional (3D) wave field computation is a computationally complex problem. Here, the 3D fourth-order in space, second-order in time, displacement-stress staggered-grid finite-difference scheme is used for accurate simulations. Our numerical examples demonstrate the effects of varying fracture number, aperture, and length distribution of the fracture network on the seismic response. The wave field scattering caused by the contrast between fractures and background media is one of the key features, and the resulting scattering is more obvious for S-waves than for P-waves. Such an approach can be applied to any fracture network model that provides a link between fracture parameters and seismic attributes.

Efficient numerical treatment of nonlinearities in the advection–diffusion–reaction equations

Purpose The purpose of this study is to propose a non-classical method to obtain efficient and accurate numerical solutions of the advection–diffusion–reaction equations. Design/methodology/approach Unlike conventional numerical methods, this study proposes a numerical scheme using outer Newton iteration applied to a time-dependent PDE. The linearized time dependent PDE is discretized by trapezoidal rule, which is second order in time, and by spline-based finite difference method of fourth order in space. Findings Using the proposed technique, even when relatively large time step sizes are used in computations, the efficiency of the proposed procedure is very clear for the numerical examples in comparison with the existing classical methods. Originality/value This study, unlike these classical methods, proposes an alternative approach based on linearizing the nonlinear problem at first, and then discretizing it by an appropriate scheme. This technique helps to avoid considering the convergence issues of Newton iteration applied to nonlinear algebraic system containing many unknowns at each time step if an implicit method is used in time discretization. The linearized PDE can be solved by implicit time integrator, which enables the use of large time step size.

Устойчивость и сходимость разностных схем для уравнения диффузии дискретно-распределенного порядка с обобщенными функциями памяти

Методом энергетических неравенств получена априорная оценка решения первой краевой задачи для уравнения диффузии дискретно-распределенного порядка с обобщенными функциями памяти. Построен разностный аналог дробной производной дискретно-распределенного порядка с обобщенными функциями памяти (аналог формулы L1). Исследованы основные свойства этого разностного оператора и на его основе построены разностные схемы второго и четвертого порядков аппроксимации по пространственной переменной и дробного порядка $ 2{-}\alpha_0 $ по временной переменной. Доказана устойчивость предложенных разностных схем, а также их сходимость в сеточной $ L_2 $-норме со скоростью, равной порядку погрешности аппроксимации. Достоверность полученных результатов подтверждают численные расчеты, проведенные для тестовых примеров.

Efficient compact finite difference method for variable coefficient fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions in conservative form

In this paper, a high-order compact finite difference scheme is derived for the time fractional sub-diffusion equation with nonhomogeneous Neumann boundary conditions in conservative form. Based on the L2- $$1_{\sigma }$$ approximation formula of the time fractional derivative, a high-order efficient compact finite difference method is developed. The unconditional stability of the resulting scheme and its convergence of second order in time and fourth order in space are rigourously proved by a discrete energy analysis method. The theoretical results are illustrated by examples.