Fourth-order Finite Difference Scheme Research Articles

A conservative fourth-order stable finite difference scheme for the generalized Rosenau–KdV equation in both 1D and 2D

In the present work, a conservative fourth-order stable finite difference scheme is proposed to solve the generalized Rosenau–KdV equation in both 1D and 2D. The existence, uniqueness, and mass and energy conservations of the numerical solution are proved using the discrete energy method. The new scheme is convergent with O ( τ 2 + h 4 ) and unconditionally stable. Numerical experiments are carried out to show that the scheme is efficient and reliable. • A new conservative fourth-order stable finite difference scheme is proposed. • The scheme is convergent with O ( τ 2 + h 4 ) and unconditionally stable. • The scheme is mass and energy conserved. • The scheme can be applied to study the solitary wave traveling in a long time.

Fourth order computational method for two parameters singularly perturbed boundary value problem using non-polynomial cubic spline

In this paper, we proposed a fourth order finite difference scheme using non-polynomial cubic spline for the solution of two parameters singularly perturbed two-point boundary value problem having dual boundary layer on a uniform mesh. In this method, the first order derivatives in the non-polynomial cubic spline finite difference scheme are replaced by the higher order finite differences to get the discretisation equation for the problem. The discretisation equation is solved by the tridiagonal solver discrete invariant imbedding. The proposed method is analysed for convergence and a fourth order rate of convergence is proved. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods.

High-Order Finite Difference Method for Helmholtz Equation in Polar Coordinates

We present a fourth-order finite difference scheme for the Helmholtz equation in polar coordinates. We employ the finite difference format in the interior of the region and derive a nine-point fourth-order scheme. Specially, ghost points outside the region are applied to obtain the approximation for the Neumann boundary condition. We obtain the matrix form of the linear system and the sparsity of the coefficient matrix is favorable for the computation of the Helmholtz equation. The feasibility and accuracy of the method are validated by two test examples which have exact solutions.

An unconditional stable compact fourth-order finite difference scheme for three dimensional Allen–Cahn equation

In this paper, we present an unconditional stable linear high-order finite difference scheme for three dimensional Allen–Cahn equation. This scheme, which is based on a backward differentiation scheme combined with a fourth-order compact finite difference formula, is second order accurate in time and fourth order accurate in space. A linearly stabilized splitting scheme is used to remove the restriction of time step. We prove the unconditional stability of our proposed method in analysis. A fast and efficient linear multigrid solver is employed to solve the resulting discrete system. We perform various numerical experiments to confirm the high-order accuracy, unconditional stability and efficiency of our proposed method. In particular, we show two applications of our proposed method: triply-periodic minimal surface and volume inpainting.

Simulation of plasma turbulence in the periphery of diverted tokamak by using the GBS code

We report on the implementation of diverted magnetic equilibria in GBS and on first simulations in this geometry. GBS is a simulation code used to evolve plasma turbulence in the tokamak periphery by solving the drift-reduced Braginskii's equations. The model equations are written in toroidal coordinates, abandoning flux coordinate systems that are not defined at the X-point. A fourth order finite difference scheme is used for the implementation of the spatial operators on poloidally and toroidally staggered grids. The GBS numerical implementation is verified through the method of manufactured solutions. The code convergence properties are tested on a relatively simple analytical X-point configuration. Finally, the diverted equilibrium from a TCV tokamak discharge is implemented in the new version of GBS. The analysis of the simulation results is focused on blob formation, radial transport, and plasma poloidal rotation mechanisms.

High order locally one-dimensional methods for solving two-dimensional parabolic equations

Based on the locally one-dimensional strategy, we propose two high order finite difference schemes for solving two-dimensional linear parabolic equations. In the first method, fourth order approximation in space and (2,2) Padé formula in time are considered. These lead to a fourth order finite difference scheme in both space and time. For the second method, we employ sixth order approximation in space and (3,3) Padé formula in time. This yields a novel sixth order scheme in both space and time. The methods are proved to be unconditionally stable, and the Sheng–Suzuki barrier is successfully avoided. Numerical experiments are given to illustrate our conclusions as well as computational effectiveness.

A new implicit energy conservative difference scheme with fourth-order accuracy for the generalized Rosenau–Kawahara-RLW equation

In the present work, a new implicit fourth-order energy conservative finite difference scheme is proposed for solving the generalized Rosenau–Kawahara-RLW equation. We first design two high-order operators to approximate the third- and fifth-order derivatives in the generalized equation, respectively. Then, the generalized Rosenau–Kawahara-RLW equation is discreted by a three-level implicit finite difference technique in time, and a fourth-order accurate in space. Furthermore, we prove that the new scheme is energy conserved, unconditionally stable, and convergent with \(O(\tau ^{2}+h^{4})\). Finally, two numerical experiments are carried out to show that the present scheme is efficient, reliable, high-order accurate, and can be used to study the solitary wave at long time.

Entropy generation of dissipative flow of carbon nanotubes in rotating frame with Darcy-Forchheimer porous medium: A numerical study

This research article explains the influence of magnetic field and volume fraction of carbon nanotubes (CNTs) on the flow over a stretching sheet in rotating frame. We analysed the behaviour of base fluid such as water with both kind of CNTs i.e. multi-wall carbon nanotubes (MWCNTs) and single-wall carbon nanotubes (SWCNTs). The nanofluid transport mechanism is obtained following Xue model. The mathematical model is developed by order of magnitude analysis. Governing partial differential equations are first converted into a system of ordinary differential equations making the use of suitable similarity transformation and then solved by fourth order finite difference scheme using Newton's linearization technique, which helps us to deal with the non-linear terms in a very smooth way. Numerical simulation for the fluid velocity and fluid temperature along with Nusselt number and skin friction coefficient are carried out for a set of values of different regulatory flow parameters that characterize the physics of the flow graphically. An outstanding correlation is perceived in tabular form of mesh density solutions of different length of domain obtained by fourth order finite difference scheme. The Biot number, Brinkmann number and applied magnetic field have vital effects on the thermal energy irreversibility in terms of Bejan number.

An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

In this paper we propose and analyze an energy stable numerical scheme for the Cahn–Hilliard equation, with second order accuracy in time and the fourth order finite difference approximation in space. In particular, the truncation error for the long stencil fourth order finite difference approximation, over a uniform numerical grid with a periodic boundary condition, is analyzed, via the help of discrete Fourier analysis instead of the standard Taylor expansion. This in turn results in a reduced regularity requirement for the test function. In the temporal approximation, we apply a second order BDF stencil, combined with a second order extrapolation formula applied to the concave diffusion term, as well as a second order artificial Douglas–Dupont regularization term, for the sake of energy stability. As a result, the unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) norm. A few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

Fourth Order Numerical Method for Singularly Perturbed Delay Differential Equations

ABSTRACT This paper presents a numerical method to solve singularly perturbed delay differential equations. The solution of this problem exhibits layer or oscillatory behaviour depending on the sign of the sum of coefficients in reaction terms. A fourth order finite difference scheme on a uniform mesh is developed. The stability and convergence of the proposed method have been established. The effect of delay parameter (small shift) on the boundary layer(s) has also been analyzed and depicted in graphs. The applicability of the proposed scheme is validated by implementing it on four model examples. To show the accuracy of the method, the results are presented in terms of maximum absolute errors.

Effect of leading-edge curvature variation on flat-plate boundary-layer receptivity

Boundary-layer receptivity is the initial stage of the laminar-turbulent transition, which is the key step to implement the prediction and control of laminar-turbulent transition in the boundary layer. Current studies show that under the action of acoustic wave or vortical disturbance, the variation of leading-edge curvature significantly affects the boundary-layer receptivity. Additionally, the free-stream turbulence is universal in nature. Therefore, direct numerical simulation is performed in this paper to study the receptivity to free-stream turbulence in the flat-plate boundary layer with an elliptic leading edge. To discretize the Navier-Stokes equation, a modified fourth-order Runge-Kutta scheme is introduced for the temporal discretization; the high-order compact finite difference scheme is utilized for the x- and y-direction spatial discretization; the Fourier transform is conducted in the z-direction. The pressure Helmholtz equation is solved by iterating a fourth-order finite difference scheme. In addition, the Jaccobi transform is introduced to convert the curvilinear coordinate system into Cartesian coordinate system. And elliptic equation technique is adopted to generate the body-fitted mesh. Then the effect of elliptic leading-edge curvature on the receptivity mechanism and the propagation speed of the excited Tollmien-Schlichting (T-S) wave packet in the flat-plate boundary layer are revealed. Subsequently, a group of multi-frequency T-S waves is extracted from the T-S wave packets by temporal fast Fourier transform. The influences of different leading-edge curvatures on the amplitudes, dispersion relations, growth rates, phases and shape functions of the excited T-S waves are analyzed in detail. Finally, the position occupied by leading-edge curvature in the boundary-layer receptivity process for the excitation of T-S wave is also confirmed. The numerical results show that the more intensive receptivity is triggered in the smaller leading-edge curvature; on the contrary, the less intensive receptivity is triggered in the greater leading-edge curvature. But in different leading-edge curvatures, the structures of the excited T-S wave packets are almost identical, and the group velocity is close to constant, which is approximate to one-third of the free-stream velocity. Similarly, the greater amplitude of the excited T-S wave can be induced with the smaller leading-edge curvature; whereas the smaller amplitude of the excited T-S wave can be induced with the greater leading-edge curvature. Moreover, the dispersion relations, growth rates, phases and shape function of the excited T-S waves in the boundary layer are found to be nearly invariable in different leading-edge curvatures. Through the above study, a further step can be made to understand the boundary-layer leading-edge receptivity and also improve the theory of the hydrodynamic stability.

A Fast Fourth-Order Method for 3D Helmholtz Equation with Neumann Boundary

We present fast fourth-order finite difference scheme for 3D Helmholtz equation with Neumann boundary condition. We employ the discrete Fourier transform operator and divide the problem into some independent subproblems. By means of the Gaussian elimination in the vertical direction, the problem is reduced into a small system on the top layer of the domain. The procedure for solving the numerical solutions is accelerated by the sparsity of Fourier operator under the space complexity of O(M3). Furthermore, the method makes it possible to solve the 3D Helmholtz equation with large grid number. The accuracy and efficiency of the method are validated by two test examples which have exact solutions.

Efficient and accurate numerical simulation of acoustic wave propagation in a 2D heterogeneous media

In this paper, a compact fourth-order finite difference scheme is derived to solve the 2D acoustic wave equation in heterogenous media. The Padé approximation is used to obtain fourth-order accuracy in both temporal and spatial dimensions, and the alternating direction implicit (ADI) technique is used to reduce the computational cost. Due to the non-constant wave velocity, the conventional ADI method is hard to implement as the algebraic manipulation cannot be used here. A novel numerical strategy is proposed in this work so that the compact scheme still maintains fourth-order accuracy in time and space. The fourth-order convergence order was firstly proved by theoretical error analysis, then was confirmed by numerical examples. It was shown that the proposed method is conditionally stable with a Courant–Friedrichs–Lewy (CFL) condition that is comparable to other existing finite difference schemes. Several numerical examples were solved to demonstrate the efficiency and accuracy of the new algorithm.

Resolution estimates for simple one-dimensional flames

The accurate resolution of flame structure is critically important to the direct numerical simulation of reacting flows. While grid-resolution estimates are readily available for cold flows, premixed flames appear to have received relatively little attention. In this paper, a premixed flame characterised by single-step chemistry at moderate-to-high Zel'dovitch numbers (β) is analysed, and its structure is used to provide estimates for sufficient grid resolution. It is found that the critical region of the flame exhibits a weak inverse Zel'dovitch number dependence, and that heuristic methods of resolution estimation based on flame thickness grow relatively less meaningful as β → ∞. Resolution estimates for second- and fourth-order finite-difference schemes are provided.

A simple finite-difference scheme for handling topography with the second-order wave equation

ABSTRACTThe presence of topography poses a challenge for seismic modeling with finite-difference codes. The representation of topography by means of an air layer or vacuum often leads to a substantial loss of numerical accuracy. A suitable modification of the finite-difference weights near the free surface can decrease that error. An existing approach requires extrapolation of interior solution values to the exterior while using the boundary condition at the free surface. However, schemes of this type occasionally become unstable and may be impossible to implement with highly irregular topography. One-dimensional extrapolation along coordinate lines results in a simple and efficient scheme. The stability of the 1D scheme is improved by ignoring the interior point nearest to the boundary during extrapolation in case its distance to the boundary is less than half a grid spacing. The generalization of the 1D scheme to more than one dimension requires a modification if the boundary intersects the finite-difference stencil on both sides of the central evaluation point and if there are not enough interior points to build the finite-difference stencil. Examples for the 2D constant-density acoustic case with a fourth-order finite-difference scheme demonstrate the method’s capability. Because the 1D assumption is not valid in two dimensions if the boundary does not follow grid lines, the formal numerical accuracy is not always obtained, but the method can handle highly irregular topography.

An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation

In this paper, we present a high-order accurate compact scheme for the phase field crystal model in two- and three-dimensional spaces. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differentiation for the time derivative term, which is second-order accurate in time. Furthermore, a nonlinearly stabilized splitting scheme is used and thus a larger time step can be allowed. Since the equations at the implicit time level are nonlinear, we introduce a Newton-type iterative method and employ a fast and efficient nonlinear multigrid solver to solve the resulting discrete system. In particular, we implement the compact scheme in the adaptive mesh refinement framework. An adaptive time step method for the phase field crystal model is also proposed. Various numerical experiments are presented and confirm the accuracy, stability, and efficiency of our proposed method.

Unsteady Magneto Hydro Dynamic Flow of a Second Order Fluid over an Oscillating Sheet with a Second Order Slip Flow Model

Unsteady slip-flow of second grade non-Newtonian electrically conducting fluid over an oscillating sheet has been considered and solved numerically. A second-order slip velocity model is used to predict the flow characteristic past the wall. With the assumption of infinite length in x-direction, velocity of the fluid can be assumed as a function of y and t, hence, with proper variable change partial governing equations are converted to ordinary differential equations, and resulting equations are solved numerically. Fourth-order finite difference scheme is used to solve the transformed governing equations. The effects of magnetic field applied on surface, slip flow parameters, frequency of oscillating, mass suction or injection and elastic second number on the velocity distribution are shown graphically and discussed. With increase of slip flow parameter, unlike that of other parameters, thickness of the fluid affected by motion of boundary will decrease. It is also realized that both injection and suction of mass on the sheet, will increase amplitude of velocity.

Development of an explicit non-staggered scheme for solving three-dimensional Maxwell’s equations

An explicit finite-difference scheme for solving the three-dimensional Maxwell’s equations in non-staggered grids is presented. We aspire to obtain time-dependent solutions of the Faraday’s and Ampère’s equations and predict the electric and magnetic fields within the discrete zero-divergence context (or Gauss’s law). The local conservation laws in Maxwell’s equations are numerically preserved using the explicit second-order accurate symplectic partitioned Runge–Kutta temporal scheme. Following the method of lines, the spatial derivative terms in the semi-discretized Faraday’s and Ampère’s equations are approximated theoretically to obtain a highly accurate numerical phase velocity. The proposed fourth-order accurate space-centered finite difference scheme minimizes the discrepancy between the exact and numerical phase velocities. This minimization process considerably reduces the dispersion and anisotropy errors normally associated with finite difference time-domain methods. The computational efficiency of getting the same level of accuracy at less computing time and the ability of preserving the symplectic property have been numerically demonstrated through several test problems.

A Compact Fourth-Order Finite Difference Scheme for the Improved Boussinesq Equation with Damping Terms

A Compact Fourth-Order Finite Difference Scheme for the Improved Boussinesq Equation with Damping Terms

Quad-plane stereoscopic PIV for fine-scale structure measurements in turbulence

The fine-scale structure in turbulence is investigated by quad-plane stereoscopic particle image velocimetry (QPSPIV). The quad-plane consists of two each of different polarizations and wavelengths, and it provides three velocity components at four independent parallel planes. Measurements have been undertaken in the developed region of a turbulent round jet with a spatial resolution sufficient to capture the small-scale structures. The advantage of the QPSPIV is presented in terms of the spectral response in the evaluation of the out-of-plane velocity gradient. The full velocity gradient tensor is computed with a fourth-order finite difference scheme in the out-of-plane direction as well as the in-plane directions. The turbulence quantities, such as the vorticity components, the energy dissipation rate and the second and third invariants of the velocity gradient tensor, are computed according to their faithful definitions. The coherent fine-scale eddies are extracted from the present QPSPIV data. The probability density functions of the diameter and the maximum azimuthal velocity of the extracted eddies exhibit their peak at approximately $$8\eta $$ and $$1.5u_k$$ , respectively, where $$\eta $$ and $$u_k$$ are the Kolmogorov length and velocity. These values agree well with the data in the literature. The phase-averaged distributions of turbulence quantities around the coherent fine-scale eddy indicate an apparent elliptic feature around the axis. Furthermore, the state of the strain rate exerting the eddy is quantified from the phase-averaged distributions of eigenvalues of the strain rate tensor and the alignment of the corresponding eigenvectors against the axis. The present study gives a solid experimental support of the coherent fine-scale structures in turbulence, and the technique can be applied to various flow fields and to the higher Reynolds number condition.