High-order Finite Difference Method Research Articles

High order finite difference methods on non-uniform meshes for space fractional operators

In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform meshes. The nonlocal property of space fractional operator makes it difficult to design the finite difference scheme on non-uniform meshes. This paper provides a basic strategy to derive the first and high order discretization schemes on non-uniform meshes for fractional operators. And the obtained first and second schemes on non-uniform meshes are used to solve space fractional diffusion equations. The error estimates and stability analysis are detailedly performed; and extensive numerical experiments confirm the theoretical analysis or verify the convergence orders.

Development of the SSPH Method for Real-Space Electronic Structure Calculations

We have applied symmetric smoothed particle hydrodynamics (SSPH) to real-space electronic structure calculations. The electronic structure of a simple atom such as hydrogen, helium and lithium is calculated. The results of SSPH are compared to that of the higher-order finite difference method (FD), and are in good agreement with that of FD. Our results indicate that SSPH can be applicable to the electronic structure calculations that require high accuracy.

On Nonlocal Computation of Eigenfrequencies of Beams Using Finite Difference and Finite Element Methods

In this paper, we show that two numerical methods, specifically the finite difference method and the finite element method applied to continuous beam dynamics problems, can be asymptotically investigated by some kind of enriched continuum approach (gradient elasticity or nonlocal elasticity). The analysis is restricted to the vibrations of elastic beams, and more specifically the computation of the natural frequencies for each numerical method. The analogy between the finite numerical approaches and the equivalent enriched continuum is demonstrated, using a continualization procedure, which converts the discrete numerical problem into a continuous one. It is shown that the finite element problem can be transformed into a system of finite difference equations. The convergence rate of the finite numerical approaches is quantified by an equivalent Rayleigh's quotient. We also present some exact analytical solutions for a first-order finite difference method, a higher-order finite difference method or a cubic Hermitian finite element, valid for arbitrary number of nodes or segments. The comparison between the exact numerical solution and the approximated nonlocal approaches shows the efficiency of the continualization methodology. These analogies between enriched continuum and finite numerical schemes provide a new attractive framework for potential applications of enriched continua in computational mechanics.

High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

We consider high order finite difference methods for two-dimensional fractional differential equations with temporal Caputo and spatial Riemann-Liouville derivatives in this paper. We propose a scheme and show that it converges with second order in time and fourth order in space. The accuracy of our proposed method can be improved by Richardson extrapolation. Approximate solution is obtained by the generalized minimal residual (GMRES) method. A preconditioner is proposed to improve the efficiency for the implementation of the GMRES method.

Numerical investigation of binary fluid convection with a weak negative separation ratio in finite containers

By using a high-order compact finite difference method to solve the full hydrodynamic field equations, convection in binary fluid mixtures with a weak negative separation ratio of −0.1 in rectangular containers heated from below is numerically investigated. We consider the problem with the Prandtl number Pr ranging from 0.01 to 10 and the Lewis number Le from 0.0005 to 1. Several convective structures such as traveling wave, localized traveling wave, and undulation traveling wave convection as well as stationary overturning convection (SOC) are obtained. For the separation ratio considered, localized traveling wave state exists in a range of Rayleigh numbers spanning the critical point (the critical Rayleigh number at the onset of convection), and their length of the convective region is uniquely selected for a given parameter set. A bifurcation diagram of solution is drawn and the transitions between various traveling waves and the steady states on their upper branches are discussed. The effects of the fluid parameters and the aspect ratio of the container on the onset of convection and their saturated structures are studied in detail. Finally, several types of initial temperature fields are used to start simulations and five different stable SOC states with different mean wavenumbers are found. The corresponding heat and mass transfer properties of these stable SOC states are also investigated.

Efficient high-order finite difference methods for nonlinear Klein–Gordon equations. I: Variants of the phi-four model and the form-I of the nonlinear Klein–Gordon equation

In this paper, the problem of solving some nonlinear Klein-Gordon equations (KGEs) is considered. Here, we derive different fourth- and sixth-order explicit and implicit algorithms to solve the phi-four equation and the form-I of the nonlinear Klein-Gordon equation. Stability and consistency of the proposed schemes are studied under certain conditions. Numerical results are presented and then compared with others obtained from some methods already existing in the scientific literature to explain the efficiency of the new algorithms. It is also shown that similar schemes can be proposed to solve many classes of nonlinear KGEs.

Handling state constraints and economics in feedback control of transport-reaction processes

Transport-reaction processes, which are typically described by parabolic partial differential equations (PDEs), play an important role within the chemical process industries. Therefore, it is important to develop feedback control techniques that operate transport-reaction processes in an economically optimal fashion in the presence of constraints in the process states and manipulated inputs. Economic model predictive control (EMPC) is a predictive control scheme that combines process economics and feedback control into an integrated framework with the potential of improving the closed-loop process economic performance compared to traditional control methodologies. In this work, we focus on systems of nonlinear parabolic PDEs and propose a novel EMPC design integrating adaptive proper orthogonal decomposition (APOD) method with a high-order finite-difference method to handle state constraints. The computational efficiency and constraint handling properties of this design are evaluated using a tubular reactor example modeled by two nonlinear parabolic PDEs.

Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

In this paper, we revisit the capability of numerical approaches such as finite difference methods and finite element methods, in approximating exact one-dimensional continuous eigenvalue problems (such as lateral vibrations of a string, the axial or the torsional vibrations of a bar, and the buckling of elastic columns). The numerical methods analysed in this paper are converted into difference equations. Following a continualization procedure or the method of differential approximation, the difference operators are then expanded in differential operators via Taylor expansion or Pade approximants. Analogies between the finite numerical approaches and some equivalent enriched continuum are shown, using this continualization procedure. The finite difference methods (first-order or higher-order finite difference methods) are shown to behave as integral-based nonlocal media (or stress gradient media), while the finite element method is found to behave as gradient elasticity media (or strain gradient media). The length scale identification of each equivalent enriched continuum strongly depends on the order of the numerical method considered. For the finite difference methods, the length scale identification of the equivalent nonlocal medium depends on the static versus dynamic analysis, whereas this length scale appears to be independent of inertia effects for the finite element method. Some comparisons between the exact discrete eigenvalue problems and the approximated continuous ones show the efficiency of the continualization procedure. An equivalent enriched Rayleigh quotient can be defined for each numerical method: the integral-based nonlocal method gives a lower bound solution to the exact eigenvalue multiplier, whereas the gradient elasticity method furnishes an upper bound solution.

Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains

A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme.We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.

High-order compact operator splitting method for three-dimensional fractional equation with subdiffusion

In this paper, a high-order compact finite difference method is proposed to solve the three-dimensional (3D) time fractional convection–diffusion equation with subdiffusion (0<α<1). After a transform of the original problem, a difference scheme which is combined the Padé approximation for the space derivatives with the classical backward differentiation formula for time fractional derivative is presented. The new scheme is fourth-order accurate in space and (2-α)-order accurate in time. To increase the efficiency and stability of numerical solutions, the alternating direction implicit (ADI) operator splitting approach is employed. The stability analysis shows that this method is unconditionally stable. Numerical experiments are carried out to support the theoretical results.

Simulation of Earthquake Rupture Dynamics in Complex Geometries Using Coupled Finite Difference and Finite Volume Methods

AbstractWe couple a node-centered finite volume method to a high order finite difference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite difference method is used, enhancing efficiency. Both the finite volume and finite difference methods are in summation-by-parts form. Interface conditions coupling the numerical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneous-approximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the particle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug.

High Order Finite Difference Methods with Subcell Resolution for Stiff Multispecies Discontinuity Capturing

AbstractIn this paper, we extend the high order finite-difference method with subcell resolution (SR) in [34] for two-species stiff one-reaction models to multispecies and multireaction inviscid chemical reactive flows, which are significantly more difficult because of the multiple scales generated by different reactions. For reaction problems, when the reaction time scale is very small, the reaction zone scale is also small and the governing equations become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present SR method for reactive Euler system is a fractional step method. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with certain computed flow variables in the shock region modified by the Harten subcell resolution idea. Several numerical examples of multispecies and multireaction reactive flows are performed in both one and two dimensions. Studies demonstrate that the SR method can capture the correct propagation speed of discontinuities in very coarse meshes.

An effective method to suppress numerical dispersion in 2D acoustic and elastic modelling using a high-order Padé approximation

In this paper, we propose a method that can effectively reduce the numerical dispersion for solving the acoustic and elastic-wave equations. The method is a fourth-order Pade approximation scheme, in which the time difference operator is a rational function and a block tridiagonal system needs to be solved at each step. On the one hand, to efficiently solve a large linear system of equations we propose an explicit method for this implicit algrithom. On the other hand, to approximate the high-order spatial derivatives we use an eighth-order stereo-modelling method using wavefield displacements and their gradients simultaneously. For this new method, we investigate some mathematical properties including the stability, errors and the numerical dispersion relationship for 1D and 2D cases. We also present numerical results computed by the Pade approximation and compare them with the eighth-order Lax–Wendroff correction method and the eighth-order staggered-grid method. Numerical results show that the high-order Pade approximation scheme can effectively suppress the numerical dispersion caused by discretizing the wave equations when coarse spatial grids are used or models have strong velocity contrasts between adjacent grids. In contrast to other high-order finite-difference methods, the new method takes substantially less computational time and requires less memory because large spatial and time increments can be used. Thus the high-order Pade approximation method can potentially be used to solve large-scale wave propagation problems and seismic tomography based on the wave equations.

Comparison of Finite Difference Schemes for the Wave Equation Based on Dispersion

Finite difference techniques are widely used for the numerical simulation of time-dependent partial differential equations. In order to get better accuracy at low computational cost, researchers have attempted to develop higher order methods by improving other lower order methods. However, these types of methods usually suffer from a high degree of numerical dispersion. In this paper, we review three higher order finite difference methods, higher order compact (HOC), compact Padé based (CPD) and non-compact Padé based (NCPD) schemes for the acoustic wave equation. We present the stability analysis of the three schemes and derive dispersion characteristics for these schemes. The effects of Courant Friedrichs Lewy (CFL) number, propagation angle and number of cells per wavelength on dispersion are studied.

A Novel Efficient Nonstandard High-Order Finite-Difference Time-Domain Method Based on Dispersion Relation Analysis

A novel approach to get new high-order nonstandard finite-difference time-domain methods is proposed. By using different coefficients in electric field and magnetic field update equations, schemes up to order (2, 8) are derived. These new methods maintain comparable accuracy to the conventional high-order finite-difference time-domain counterparts while a reduction in computational cost is achieved. The dispersion and anisotropic behaviors of these new methods are investigated. Some numerical examples are studied to examine the accuracy and efficiency of the new method.

A compact finite difference method for solving a class of time fractional convection-subdiffusion equations

A high-order compact finite difference method is proposed for solving a class of time fractional convection-subdiffusion equations. The convection coefficient in the equation may be spatially variable, and the time fractional derivative is in the Caputo’s sense with the order \(\alpha \) (\(0<\alpha <1\)). After a transformation of the original equation, the spatial derivative is discretized by a fourth-order compact finite difference method and the time fractional derivative is approximated by a \((2-\alpha )\)-order implicit scheme. The local truncation error and the solvability of the method are discussed in detail. A rigorous theoretical analysis of the stability and convergence is carried out using the discrete energy method, and the optimal error estimates in the discrete \(H^{1}\), \(L^{2}\) and \(L^{\infty }\) norms are obtained. Applications using several model problems give numerical results that demonstrate the effectiveness and the accuracy of this new method.

Numerical study of particle behavior in laminar axisymmetric opposed-jet flows

The particle behavior in laminar axisymmetric opposed-jet flows is investigated numerically using a high order finite difference method and Lagrangian particle tracking. The deterministic hard-sphere collision model is used to deal with the inter-particle collisions, and the feedback of the particles on the carrier phase is neglected in the present work. In the scope of particle volume fractions and particle Stokes numbers studied, the particle collision rates in the impinging zone can agree with the prediction of cylinder formula (Saffman & Turner, 1956). The results found that inter-particle collisions have significant influence on particle distribution, velocities and residence time. When particle volume fraction at nozzle outlet is smaller than O(10−4), inter-particle collisions make little difference on the particle concentration profile, which is primarily decided by the maximum penetration distance (MPD) of a particular particle. Particle will prefer to concentrate at the location of MPD when the MPD is within the flow domain, otherwise particles will evenly distribute along the axis of two nozzles. When particle volume fraction is higher than O(10−3), the effect of inter-particle collisions become significant, which can impede particles reaching the MPD so that particles will prefer to concentrate at the impinging plane. Inter-particle collisions can restrain the penetration process of particles, and shorten the particle residence time at the same time. Due to massive collisions, particle mean velocity profile has the similar shape with the profile of gas velocity in the impinging zone, and slip velocity between particles and gas will be decreased.

Nonuniform-time-step explicit Runge–Kutta scheme for high-order finite difference method

Explicit Runge–Kutta method has been widely used for time-accurate simulations in aeroacoustics and aerodynamics, partly because of its strong stability properties. However, when dealing with problems involving irregular geometries or multi-scale phenomena where the grid is often refined in localized regions, a single global time step size dictated by the smallest grid size based on the explicit stability condition would inevitably result in excessive computational consumption. A local time stepping strategy that adopts time step size to the local stability requirement in different grid blocks is an effective way to reduce the cost of time integration and to increase the overall computational efficiency. In the present paper, a non-uniform time step (NUTS) explicit Runge–Kutta scheme, previously introduced in the framework of discontinuous Galerkin method, which makes the local time stepping strategy applicable to the explicit Runge–Kutta family without any interpolation and extrapolation in time, is further extended to high-order finite difference methods. The stability of non-uniform time step (NUTS) scheme in combination with high order difference schemes is studied. Numerical experiments are carried out using 1D and 2D acoustic problems for validation of the proposed approach. Computational cost reduction by the proposed algorithm is also discussed.

Large eddy simulation of turbulent spray combustion

The two-phase filtered mass density function (FMDF) method is employed for large eddy simulation (LES) of high speed evaporating and combusting n-heptane sprays using simple (global) and complex (skeletal) chemical kinetic mechanisms. The resolved fluid velocity and pressure fields are obtained by solving the filtered compressible Navier–Stokes equations with high-order Eulerian finite difference methods. The liquid spray and gas scalar (temperature and species mass fractions) fields are both obtained by Lagrangian stochastic models. The chemistry calculation is accelerated by incorporating the parallel in situ adaptive tabulation (ISAT) method. There are two-way interactions among Eulerian and Lagrangian fields. Simulations of evaporating sprays with and without combustion indicate that the two-phase LES/FMDF results are consistent and compare well with the available experimental data at different gas temperatures and oxygen concentrations. The spray controlled flame tends to move away from a diffusion flame structure toward a premixed one as the oxygen concentration decreases and/or the ambient gas temperature increases because of changes in spray-induced turbulence and mixing. The LES/FMDF results for ignition delay show more sensitivity to the chemical kinetic model at lower gas temperatures due to slower reaction and stronger turbulence–chemistry interactions. The liftoff length is less sensitive to the kinetics.

Switch dynamics for stochastic model of genetic toggle switch

Recently, more and more biological experiments have indicated that noise plays an important role in bistable systems, such as the case of the bimodal population distribution in the genetic toggle switch. In this paper, we further verify that noises in degradation rates can indeed induce switching in the genetic toggle switch. Meanwhile, we apply the theory of mean first passage time (MFPT) in high dimensional system to the above stochastic model. According to our assumption, the high order finite difference method is used to compute the MFPT (that the average time switching from one steady state to the other) and we find that the relationship between the MFPT and noise intensity is negative correlation. The result is also verified through another numerical simulation method.