Fourth-order Accurate Finite Difference Research Articles

Positivity and bound preserving well-balanced high order compact finite difference scheme for Ripa and pollutant transport model

We construct a fourth-order accurate compact finite difference scheme that is well-balanced, positivity-preserving of water height, and bound-preserving of temperature for Ripa and concentration for pollutant transport systems. The proposed scheme preserves the still-water steady state and the positivity of water height. It also maintains concentration bounds for pollutants across nonflat bottom topographies, regardless of the presence of a pollutant source. Our approach incorporates water height and pollutant concentration constraints within the same discretization, utilizing weak monotonicity and a simple bound-preserving limiter while preserving the well-balanced property. Through extensive numerical simulations encompassing Ripa and pollutant transport models, we demonstrate the effectiveness of our method, verifying its well-balanced property, high-order accuracy, positivity-preserving, and bound-preserving capabilities.

3D time-dependent scattering about complex shapes using high order difference potentials

We compute the scattering of unsteady acoustic waves about complex three-dimensional bodies with high order accuracy. The geometry of a scattering body is defined with the help of CAD. Its surface is represented as a collection of non-overlapping patches, each parameterized independently by means of high order splines (NURBS). As a specific example, we consider a submarine-like scatterer constructed using five different patches.The acoustic wave equation on the region exterior to the scatterer is solved by first reducing it to a system of Calderon's boundary operator equations. The latter are obtained using the method of difference potentials coupled with a compact fourth order accurate finite difference scheme. When solving the boundary operator equations, we employ Huygens' principle. It allows us to work on a sliding time window of non-increasing duration rather than keep the entire temporal history of the solution at the boundary.The proposed methodology demonstrates grid-independent computational complexity at the boundary and sub-linear complexity with respect to the grid dimension. It efficiently handles complex non-conforming geometries on Cartesian grids with no penalty for either accuracy or stability due to the cut cells. Its performance does not deteriorate over arbitrarily long simulation times. The exact treatment of artificial outer boundaries is inherently built in. Finally, multiple similar problems can be solved efficiently at a low individual cost per problem. This is important when, for example, the boundary condition on the surface changes but the scattering body stays the same.

Discrete maximum principle of a high order finite difference scheme for a generalized Allen–Cahn equation

We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the $Q^2$ spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to construct a bound-preserving scheme for any passive convection with an incompressible velocity field.

Stability analysis of a high-order finite-difference scheme for the Korteweg–de Vries equation with non-homogeneous boundaries

In this paper, we analyze the stability of a high-order finite-difference scheme for the Korteweg–de Vries (KdV) equation with non-homogeneous boundaries. We first employ a variable transformation to change the non-homogeneous boundaries to homogeneous boundaries. We then develop a fourth-order accurate finite-difference scheme for solving the transformed KdV problem. The stability, convergence and solvability of the numerical solution are analyzed. Numerical examples are given to confirm the good accuracy and the effectiveness of the present method for handling the KdV equations with non-homogeneous boundaries.

Elastic Wave Propagation in Curvilinear Coordinates with Mesh Refinement Interfaces by a Fourth Order Finite Difference Method

We develop a fourth order accurate finite difference method for the three dimensional elastic wave equation in isotropic media with the piecewise smooth material property. In our model, the materia...

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.

Simulating the near-field pulse-like ground motions of the Imperial Valley, California, earthquake

Long-period pulse-like ground motions in the near-field are likely to induce severe damage to large structures due to resonance at the natural frequencies of engineered construction. To investigate the origin and characteristics of large velocity pulses from the seismic source, this study identifies 14 stations with velocity pulses from 32 strong-motion seismographs located in the near-field of the Imperial Valley MW 6.5 earthquake. The pulse-like ground motions are simulated by a kinematic approach using a fourth-order accurate finite-difference method. The results show that the slip asperity in the northern part of the fault plane contributes more to large velocity pulses than the asperity that originates in the hypocentral zone. The two-sided pulses on the fault-normal component and the one-sided pulses on the fault-parallel component are mainly regulated by the fault rupture and slip directions, respectively. Compared with the pseudo-velocity response spectrum, the displacement response spectrum better captures the resonance effect of long-period pulse-like ground motion on large structures. The location of the maximum ground motion coincides with the surface projection area of the large asperity, and the amplitude and the attenuation rate of each horizontal component vary with increasing distance from the fault. The difference in the ground motion peaks on both sides of the fault implies that the fault-normal component is more sensitive to the fault dip angle than the fault-parallel component. We reproduce the pulse-like ground motions up to 1 Hz by fitting the simulated velocity time histories, response spectrum and ground motions to observed data. Using the pulse records of the Imperial Valley earthquake, the 1994 Northridge MW 6.7 earthquake, and the 1999 Chi-Chi MW 7.6 earthquake, we compare the effects of earthquake magnitude on peak value and distribution of large velocity pulses, in which a reasonable consistency can be observed. Our simulated results of near-field pulse-like ground motions are beneficial to clarify the pulse mechanism.

A New Approach in Differential transformation method with application on MHD flow in non-Darcy medium between porous parallel plates considering hall current

The study of fluid flow in non-Darcy porous medium subjected to Hall current is very important in scientific and engineering applications such as water filters, groundwater discharge in aquifers, petroleum engineering, MHD generators and chemical engineering. Forchheimer model is needed at high flow rate, where, flow will exhibit non-linearity with respect to velocity which make Darcy law inapplicable at these conditions. In this case, the momentum equations become non-linear. The classical differential transformation method transforms the non-linear differential equations to a non-linear algebraic system which gives more than one solution and may be unstable that leads to divergence of the required solution. The novelty of present method that it is a power series solution avoiding solution multiplicity and divergence by a linearization technique that is applied on non-linear governing equations to obtain the unique and convergent solution. The uniqueness, convergence and stability of the new technique are tested by comparisons with previously available works and it is also verified by a fourth order accurate finite difference (FOFDM) solution. The effects flow parameters on the velocity and friction factor are illustrated. The values of parameters in the present work are chosen according to: the available previous results, the power of the persent method to compute over a lagre range of parameters and the distinctiveness between curves in figures.

Fourth-order accurate fractional-step IMEX schemes for the incompressible Navier–Stokes equations on moving overlapping grids

Two efficient fractional-step schemes for the incompressible Navier–Stokes equations in two and three dimensions are described. The schemes are fourth-order accurate in space and time, and are based on solving the velocity-pressure form of the equations. Both schemes employ predictor-corrector time-stepping approaches. The first is an explicit Adams-type scheme, while the second is an IMEX-BDF-type scheme in which the viscous/advective terms in the equations are treated implicitly/explicitly. The equations and boundary conditions are discretized in space using fourth-order accurate finite-difference approximations. The formulation of discrete boundary conditions for each stage of the fractional-step scheme is found to be critical to the accuracy and stability of the approach. A WENO-based scheme, called BWENO, provides upwind dissipation and ensures robustness of the schemes for problems where the solution is under-resolved on the grid (e.g. near boundary or shear layers). Complex, and possibly moving, domains are handled efficiently using composite overlapping grids. A variety of problems in two and three dimensions, some for which exact solutions are either known or manufactured, are used to verify the stability and accuracy of the new schemes.

On the monotonicity and discrete maximum principle of the finite difference implementation of $$C^0$$-$$Q^2$$ finite element method

We show that the fourth order accurate finite difference implementation of continuous finite element method with tensor product of quadratic polynomial basis is monotone thus satisfies the discrete maximum principle for solving a scalar variable coefficient equation $$-\nabla \cdot (a\nabla u)+cu=f$$ under a suitable mesh constraint.

Parabolic two-step model and accurate numerical scheme for nanoscale heat conduction induced by ultrashort-pulsed laser heating

Simulation of nanoscale heat transfer phenomena has attracted great attention, particularly the nano-scale heat conduction induced by ultrashort-pulsed laser heating. In this article, we propose a nanoscale parabolic two-step model and an accurate numerical scheme for thermal analysis of the nanoscale heat conduction induced by ultrashort-pulsed laser heating. To this end, we first introduce the Knudsen number (Kn) into the original parabolic two-step heat conduction equations and couple them with the Kn-dependent and temperature-jump boundary condition. We then develop a fourth-order accurate compact finite difference method for solving the nanoscale model. Stability and convergence of the obtained numerical scheme are analyzed theoretically. We finally test the accuracy and applicability of the nanoscale model and the obtained numerical scheme in three examples. By choosing various values of the Kn and the parameter α in the boundary condition, the simulation could be a tool for analyzing the nanoscale heat conduction induced by ultrashort-pulsed laser heating.

A fourth-order accurate finite difference method to evaluate the true transient behaviour of natural convection flow in enclosures

PurposeModelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.Design/methodology/approachFourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.FindingsError magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 103-108, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.Research limitations/implicationsThe developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.Originality/valueThe proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.

Compact High Order Accurate Schemes for the Three Dimensional Wave Equation

We construct a family of compact fourth order accurate finite difference schemes for the three dimensional scalar wave (d’Alembert) equation with constant or variable propagation speed. High order accuracy is of key importance for the numerical simulation of waves as it reduces the dispersion error (i.e., the pollution effect). The schemes that we propose are built on a stencil that has only three nodes in any coordinate direction or in time, which eliminates the need for auxiliary initial or boundary conditions. These schemes are implicit in time and conditionally stable. A particular scheme with the maximum Courant number can be chosen within the proposed class. The inversion at the upper time level is done by FFT for constant coefficients and multigrid for variable coefficients, which keeps the overall complexity of time marching comparable to that of a typical explicit scheme.

Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented.

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Scalar Convection Diffusion Equations

We show that the classical fourth order accurate compact finite difference scheme with high order strong stability preserving time discretizations for convection diffusion problems satisfies a weak...

DEVELOPMENT OF A HIGH ORDER DISCRETIZATION SCHEME FOR SOLVING FULLY NONLINEAR MAGNETOHYDRODYNAMIC EQUATIONS

We have developed fully fourth order accurate compact finite difference discretization scheme for the Navier-Stokes equations coupled with Maxwell's equations. The implementation is done in cylindrical polar geometry. Due to the full-MHD modeling of physical flow, the modeled equations are fully nonlinear coupled hydrodynamic equations which are again coupled with Maxwells equations. In our computations, we have accounted for the induced magnetic field in the flow of an electrically conducting fluid in an external magnetic field. The code is tested against available experimental and theoretical data where applicable. It is observed that a smaller grid of 64×64 is sufficient for weakly nonlinear problems and higher grids up to 512×512 are needed as the degree of nonlinearities grow in the modeled equation. In the absence of magnetic field, a discontinuity of total drag coefficient and separation length is noted for Re=73 which is in agreement with literature. When the magnetic Reynolds number Rm 1, it decreases nonlinearly, at a much faster rate. Thermal boundary layer thickness decreases as the strength of magnetic field increases and it forces the thermal convection to take place in a laminar structure as observed from thermal contour lines. Finally, using divided differences, we establish that the accuracy of the proposed numerical scheme is in fact fourth order.

Mixed Convection and Heat Transfer Studies in Non-Uniformly Heated Buoyancy Driven Cavity Flow

We analyse the mixed convection flow in a cavity flow which is driven by buoyancy generated due to a non-uniformly heated top wall which is moving uniformly. A fourth order accurate finite difference scheme is used in this study and our code is first validated against available data in the literature. The results are obtained for different sets of Reynolds number Re, Prandtl number Pr and Grashof number Gr which are in the ranges 100 - 3000, 0.0152 - 10 and 102 - 106 respectively. Here Gr is related to the Richardson number according to Ri=Gr/Re2. While increasing the Richardson number, the growth of upstream secondary eddy (USE) is observed together with a degradation of downstream secondary eddy (DSE). When mixed convection is dominant, the upstream secondary eddy and the downstream secondary eddy merge to form a large recirculation region. When the effect of Pr is studied in the forced convection regime, Ri<<1, the temperature in the central region of the cavity remains nearly a constant. However, in the mixed convection regime, the temperature in cavity undergoes non-monotonic changes. Finally, using the method of divided differences, it is shown that numerical accuracy of the derived numerical scheme used in this work is four.

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.

On the accuracy of some explicit and implicit methods for the inviscid GRLW equation subject to initial Gaussian conditions

Purpose – The purpose of this paper is to both determine the effects of the nonlinearity on the wave dynamics and assess the temporal and spatial accuracy of five finite difference methods for the solution of the inviscid generalized regularized long-wave (GRLW) equation subject to initial Gaussian conditions. Design/methodology/approach – Two implicit second- and fourth-order accurate finite difference methods and three Runge-Kutta procedures are introduced. The methods employ a new dependent variable which contains the wave amplitude and its second-order spatial derivative. Numerical experiments are reported for several temporal and spatial step sizes in order to assess their accuracy and the preservation of the first two invariants of the inviscid GRLW equation as functions of the spatial and temporal orders of accuracy, and thus determine the conditions under which grid-independent results are obtained. Findings – It has been found that the steepening of the wave increase as the nonlinearity exponent is increased and that the accuracy of the fourth-order Runge-Kutta method is comparable to that of a second-order implicit procedure for time steps smaller than 100th, and that only the fourth-order compact method is almost grid-independent if the time step is on the order of 1,000th and more than 5,000 grid points are used, because of the initial steepening of the initial profile, wave breakup and solitary wave propagation. Originality/value – This is the first study where an accuracy assessment of wave breakup of the inviscid GRLW equation subject to initial Gaussian conditions is reported.

High-Order Finite Difference Schemes for the First Derivative in Von Mises Coordinates

Third- and fourth-order accurate finite difference schemes for the first derivative of the square of the speed are developed, for both uniform and non-uniform grids, and applied in the study of a two-dimensional viscous fluid flow through an irregular domain. The von Mises transformation is used to transform the governing equations, and map the irregular domain onto a rectangular computational domain. Vorticity on the solid boundary is expressed in terms of the first partial derivative of the square of the speed of the flow in the computational domain, and the schemes are used to calculate the vorticity at the computational boundary grid points using up to five computational domain grid points. In all schemes developed, we study the effect of coordinate clustering on the computed results.