High-order Finite Difference Method Research Articles

Dissolution enhancement and mathematical modeling of removal of residual trichloroethene in sands by ozonation during flushing with micro-nano-bubble solution

Soil flushing using micro-nano-sized bubbles (MNB) in water as the flushing solution was tested in laboratory sand columns for the cleanup of residual trichloroethene (TCE) non-aqueous-phase-liquid (NAPL). Experiments considering flushing with MNB as well as ozone MNB (OZMNB) in water to treat soils contaminated with residual TCE liquid were conducted to examine effects of ozone on dissolution enhancement. The degrees of residual TCE saturation in soils, ranging from 0.44% to 7.6%, were tested. During flushings, aqueous TCE concentrations at the column exit were monitored and TCE masses remained in the columns after flushing were determined. Experimental results between runs with MNB and OZMNB in water revealed that dissolution enhancement was dependent on residual saturation conditions, and the maximum enhancement was around 9%. Governing equations consisting of three coupled partial differential equations (PDEs) were developed to model the system, and high-order finite difference (HOFD) method was employed to solve these PDEs. From mathematical modeling of reactive mass transfer under low residual saturation conditions (0.44% and 1.9%), experimental data were simulated and important controlling mechanisms were identified. It was concluded that a specific parameter pertinent to NAPL-water interfacial area in the Sherwood number had to be modified to satisfactorily describe the dissolution of TCE in the presence of MNB in water.

Suitability of artificial viscosity discontinuous Galerkin method for compressible turbulence

A discontinuous Galerkin method based on an artificial viscosity model is investigated in the context of the simulation of compressible turbulence. The effects of artificial viscosity on shock capturing ability, broadband accuracy and under-resolved instability are examined combined with various orders and mesh resolutions. For shock-dominated flows, the superior accuracy of high order methods in terms of discontinuity resolution are well retained compared with lower ones. For under-resolved simulations, the artificial viscosity model is able to enhance stability of the eighth order discontinuous Galerkin method despite of detrimental influence for accuracy. For multi-scale flows, the artificial viscosity model demonstrates biased numerical dissipation towards higher wavenumbers. Capability in terms of boundary layer flows and hybrid meshes is also demonstrated. It is concluded that the fourth order artificial viscosity discontinuous Galerkin method is comparable to typical high order finite difference methods in the literature in terms of accuracy for identical number of degrees of freedom, while the eighth order is significantly better unless the under-resolved instability issue is raised. Furthermore, the artificial viscosity discontinuous Galerkin method is shown to provide appropriate numerical dissipation as compensation for turbulent kinetic energy decaying on moderately coarse meshes, indicating good potentiality for implicit large eddy simulation.

Direct Numerical Simulation of Turbine Cascade Flow with Heat Transfer

Abstract Two- and three-dimensional direct numerical simulation (DNS) of turbine cascade flow at low Reynolds number with heat transfer are performed using high-order finite difference method. Two-dimensional laminar computation which is used to construct the initial flow of three-dimensional DNS fails to predict Stanton number on the second half of suction side where the flow is turbulent in experiment. In three-dimensional DNS, transition is triggered by periodic blow-and-suction disturbances. Numerical experiments show that phase randomness of the disturbance is not necessary to trigger the transition which can be induced by disturbances with fixed phases. In a range of time fundamental frequency of disturbance, when increasing the frequency, transition moves downstream. When time fundamental frequency is big enough, transition disappears. With fixed space phases, time phases and selected time fundamental frequency, time averaged pressure and Stanton number distributions of three-dimensional DNS coincide with the experimental datum. Averaged velocity and temperature, Root-Mean-squares (RMS) of velocity pulse，temperature pulse, Reynolds shear stress and heat flux are extracted from the DNS database. All statistics agree well with experimental and theoretical results which verify the accuracy of present database.

A Simple Bound-Preserving Sweeping Technique for Conservative Numerical Approximations

In this paper, we propose a simple bound-preserving sweeping procedure for conservative numerical approximations. Conservative schemes are of importance in many applications, yet for high order methods, the numerical solutions do not necessarily satisfy maximum principle. This paper constructs a simple sweeping algorithm to enforce the bound of the solutions. It has a very general framework acting as a postprocessing step accommodating many point-based or cell average-based discretizations. The method is proven to preserve the bounds of the numerical solution while conserving a prescribed quantity designated as a weighted average of values from all points. The technique is demonstrated to work well with a spectral method, high order finite difference and finite volume methods for scalar conservation laws and incompressible flows. Extensive numerical tests in 1D and 2D are provided to verify the accuracy of the sweeping procedure.

A method for the computational modeling of the physics of heart murmurs

A computational method for direct simulation of the generation and propagation of blood flow induced sounds is proposed. This computational hemoacoustic method is based on the immersed boundary approach and employs high-order finite difference methods to resolve wave propagation and scattering accurately. The current method employs a two-step, one-way coupled approach for the sound generation and its propagation through the tissue. The blood flow is simulated by solving the incompressible Navier–Stokes equations using the sharp-interface immersed boundary method, and the equations corresponding to the generation and propagation of the three-dimensional elastic wave corresponding to the murmur are resolved with a high-order, immersed boundary based, finite-difference methods in the time-domain. The proposed method is applied to a model problem of aortic stenosis murmur and the simulation results are verified and validated by comparing with known solutions as well as experimental measurements. The murmur propagation in a realistic model of a human thorax is also simulated by using the computational method. The roles of hemodynamics and elastic wave propagation on the murmur are discussed based on the simulation results.

Spectra of electron pair under harmonic and Debye potential

Two electron systems confined by harmonic potential is known as harmonium. Such a system has been studied for many reasons in the literature. In this work we study harmonium under Debye potential. We use higher order finite difference method for the solution of Schrodinger equation. Complete energy spectrum of harmonium and harmonium under Debye potential is studied. Debye screening length shows considerable effect on the energy levels and the radial matrix elements. The results are analysed in the light of existing results and the comparison with available results shows remarkable agreement.

Effect of Nonuniform Grids on High-Order Finite Difference Method

AbstractThe finite difference (FD) method is popular in the computational fluid dynamics and widely used in various flow simulations. Most of the FD schemes are developed on the uniform Cartesian grids; however, the use of nonuniform or curvilinear grids is inevitable for adapting to the complex configurations and the coordinate transformation is usually adopted. Therefore the question that whether the characteristics of the numerical schemes evaluated on the uniform grids can be preserved on the nonuniform grids arises, which is seldom discussed. Based on the one-dimensional wave equation, this paper systematically studies the characteristics of the high-order FD schemes on nonuniform grids, including the order of accuracy, resolution characteristics and the numerical stability. Especially, the Fourier analysis involving the metrics is presented for the first time and the relation between the resolution of numerical schemes and the stretching ratio of grids is discussed. Analysis shows that for smooth varying grids, these characteristics can be generally preserved after the coordinate transformation. Numerical tests also validate our conclusions.

Coupling Requirements for Multiphysics Problems Posed on Two Domains

We consider two hyperbolic systems in first order form of different size posed on two domains. Our ambition is to derive general conditions for when the two systems can and cannot be coupled. The adjoint equations are derived and well-posedness of the primal and dual problems is discussed. By applying the energy method, interface conditions for the primal and dual problems are derived such that the continuous problems are well posed. The equations are discretized using a high order finite difference method in summation-by-parts form and the interface conditions are imposed weakly in a stable way, using penalty formulations. It is shown that one specific choice of penalty matrices leads to a dual consistent scheme. By considering an example, it is shown that the correct physical coupling conditions are contained in the set of well posed coupling conditions. It is also shown that dual consistency leads to superconverging functionals and reduced stiffness.

Numerical simulation of ultrasonic wave transmission experiments in rocks of shale gas reservoirs

Shale gas reservoirs have risen in importance in China’s new power source exploration and development program. The investigation of the propagation of ultrasonic waves in shale forms the basis for the full waveform application of acoustic logging data to the exploration of shale gas. Using acoustic wave theory, initial conditions, vibration source conditions, and stability conditions are developed in combination with experimental background of ultrasonic wave transmission. With improved boundary conditions, we performed numerical simulations of the ultrasound transmission experiments in shale using the high-order staggered-grid finite difference method (second-order in the time domain and fourth-order in the space domain). With programs developed within MatLab, the results obtained from numerical simulations agree well with experimental results based on physical models. In addition, using snapshots of the wave field that give a microscopic perspective, the propagation laws for ultrasonic waves can be analyzed. Using this method, human error is avoided, transmission experiments costs can be reduced and efficiency improved. This method extends the scope of experimental investigations regarding the transmission of ultrasonic waves in a shale gas reservoir with increasing stratification, and thus has great theoretical value and practical significance.

Properties-preserving high order numerical methods for a kinetic eikonal equation

For the BGK (Bhatnagar–Gross–Krook) equation in the large scale hyperbolic limit, the density of particles can be transformed as the Hopf–Cole transformation, where the phase function converges uniformly to the viscosity solution of an effective Hamilton–Jacobi equation, referred to as the kinetic eikonal equation. In this work, we present efficient high order finite difference methods for numerically solving the kinetic eikonal equation. The methods are based on monotone schemes such as the Godunov scheme. High order weighted essentially non-oscillatory techniques and Runge–Kutta procedures are used to obtain high order accuracy in both space and time. The effective Hamiltonian is determined implicitly by a nonlinear equation given as integrals with respect to the velocity variable. Newton's method is applied to solve the nonlinear equation, where integrals with respect to the velocity variable are evaluated either by a Gauss quadrature formula or as expansions with respect to moments of the Maxwellian. The methods are designed such that several key properties such as the positivity of the viscosity solution and the positivity of the effective Hamiltonian are preserved. Numerical experiments are presented to demonstrate the effectiveness of the methods.

A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection–diffusion equations

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.

An efficient time-splitting compact finite difference method for Gross–Pitaevskii equation

We propose an efficient time-splitting compact finite difference method for Gross–Pitaevskii equation (GPE). In our method, we solve the GPE in time with time-splitting technique and in space by the compact finite difference method. To find the numerical solution of the resulting discretized system in one-dimension (1D), two-dimensions (2D) and three-dimensions (3D), we apply the fast discrete Sine transform in 1D, 2D and 3D respectively and get an efficient solver for the discretized system in 1D, 2D and 3D, respectively. Our numerical algorithm at every time step does not need linear-algebraic-equations-solver, whose computation cost will be much higher when the spatial dimension increases. The method also has the merit that it is unconditionally stable and conservative. Moreover the method can achieve spectral-like accuracy in space when high-order compact finite difference method is applied. Extensive numerical tests for the GPE in 1D, 2D and 3D are presented to demonstrate the power and accuracy of the proposed numerical method.

General rectangular grid based time-space domain high-order finite-difference methods for modeling scalar wave propagation

We develop the general rectangular grid discretization based time-space domain high-order staggered-grid finite-difference (SGFD) methods for modeling three-dimension (3D) scalar wave propagation. The proposed two high-order SGFD schemes can achieve the arbitrary even-order accuracy in space, and the fourth- and sixth-order accuracies in time, respectively. We derive the analytical expression of the high-order FD coefficients based on a general rectangular grid discretization with different grid spacing in all axial directions. The general rectangular grid discretization makes our time-space domain SGFD schemes more flexible than the existing ones developed on the cubic grid with the same grid spacing in the axial directions. Theoretical analysis indicates that our time-space domain SGFD schemes have a better stability and a higher accuracy than the traditional temporal second-order SGFD scheme. Our time-space domain SGFD schemes allow larger time steps than the traditional SGFD scheme for attaining a similar accuracy, and thus are more efficient. Numerical example further confirms the superior accuracy, stability and efficiency of our time-space domain SGFD schemes.

Eighth-Order Compact Finite Difference Scheme for 1D Heat Conduction Equation

The purpose of this paper is to develop a high-order compact finite difference method for solving one-dimensional (1D) heat conduction equation with Dirichlet and Neumann boundary conditions, respectively. A parameter is used for the direct implementation of Dirichlet and Neumann boundary conditions. The introduced parameter adjusts the position of the neighboring nodes very next to the boundary. In the case of Dirichlet boundary condition, we developed eighth-order compact finite difference method for the entire domain and fourth-order accurate proposal is presented for the Neumann boundary conditions. In the case of Dirichlet boundary conditions, the introduced parameter behaves like a free parameter and could take any value from its defined domain but for the Neumann boundary condition we obtained a particular value of the parameter. In both proposed compact finite difference methods, the order of accuracy is the same for all nodes. The time discretization is performed by using Crank-Nicholson finite difference method. The unconditional convergence of the proposed methods is presented. Finally, a set of 1D heat conduction equations is solved to show the validity and accuracy of our proposed methods.

Preferential concentration of heavy particles in compressible isotropic turbulence

Numerical simulations of particle-laden compressible isotropic turbulence with Taylor Reynolds number Reλ ∼ 100 are conducted by using a high-order turbulence solver, which is based on high-order compact finite difference method in the whole flow domain and localized artificial diffusivities for discontinuities. For simplicity, only one-way coupling (i.e., the influence of fluid on particles) between the carrier flow and particles is considered. The focus is on the study of the preferential concentration of heavy particles in dissipative scale of turbulence and the underlying mechanisms. Firstly, the effect of Stokes number (St) on the particle distribution in flow of Mach 1.01 (referred to as high-Mach-number case in this study) is investigated as a necessary supplementation for the previous studies in incompressible and weakly compressible flows. It turns out that heavy particles with Stokes number close to unity exhibit the strongest preferential concentration, which is in agreement with the observation in incompressible flow. All types of heavy particles have a tendency to accumulate in high-density regions of the background flow. While all kinds of particles dominantly collect in low-vorticity regions, intermediate and large particles (St = 1 and St = 5) are also found to collect in high-vorticity regions behind the randomly formed shocklets. Secondly, the impact of turbulent Mach number (Mt) (or the compressibility) of the carrier flow on the spatial distribution of the particles with St = 1 is discussed using the simulated compressible flows with Mt being 0.22, 0.68, and 1.01, respectively. In low-Mach-number flow, particles tend to concentrate in regions of low vorticity due to the centrifuge effect of vortices and particle concentration decreases monotonically with the increasing vorticity magnitude. As Mach number increases, the degree of particle clustering is slightly weakened in low-vorticity regions but is enhanced in high-vorticity regions, which only account for a small fraction of the flow domain. This observation as well as the anomalous correlation between the fluid density and vorticity can be ascribed to the appearance of randomly distributed shocklets in high-Mach-number turbulence, which enhance the vorticity and density (pressure) immediately behind them. Finally, the effects of the forcing and cooling schemes on the properties of flow and particles are also discussed.

Vlasov simulations of electron-ion collision effects on damping of electron plasma waves

Collisional effects can play an essential role in the dynamics of plasma waves by setting a minimum damping rate and by interfering with wave-particle resonances. Kinetic simulations of the effects of electron-ion pitch angle scattering on Electron Plasma Waves (EPWs) are presented here. In particular, the effects of such collisions on the frequency and damping of small-amplitude EPWs for a range of collision rates and wave phase velocities are computed and compared with theory. Both the Vlasov simulations and linear kinetic theory find the direct contribution of electron-ion collisions to wave damping significantly reduced from that obtained through linearized fluid theory. To our knowledge, this simple result has not been published before. Simulations have been carried out using a grid-based (Vlasov) approach, based on a high-order conservative finite difference method for discretizing the Fokker-Planck equation describing the evolution of the electron distribution function. Details of the implementation of the collision operator within this framework are presented. Such a grid-based approach, which is not subject to numerical noise, is of particular interest for the accurate measurements of the wave damping rates.

Asymptotic behavior and interpretation of virtual states: The effects of confinement and of basis sets.

A real-space high order finite difference method is used to analyze the effect of spherical domain size on the Hartree-Fock (and density functional theory) virtual eigenstates. We show the domain size dependence of both positive and negative virtual eigenvalues of the Hartree-Fock equations for small molecules. We demonstrate that positive states behave like a particle in spherical well and show how they approach zero. For the negative eigenstates, we show that large domains are needed to get the correct eigenvalues. We compare our results to those of Gaussian basis sets and draw some conclusions for real-space, basis-sets, and plane-waves calculations.

Enhanced instability of supersonic boundary layer using passive acoustic feedback

Tripping of boundary-layers from laminar to turbulent flow, which may be needed in specific practical applications, requires introducing high amplitude disturbances into the boundary layers without large drag penalties. As a possible improvement on fixed trip devices, a technique based on feedback instability is demonstrated in the present contribution for a Mach 1.5 boundary layer. The compressible Navier-Stokes equations are solved directly using a high-order (fifth-order in space and third-order in time) finite difference method. The geometries tested include wall-mounted rectangular blocks and off-surface cylinders. For a particular combination of an upstream cylinder and downstream block, a feedback loop is observed to provide more robust self-amplification of disturbances than a cylinder alone, whereas blocks by themselves only enhance a convective instability. A Fourier analysis and an impulse response simulation are used to confirm the presence of an acoustic feedback loop for the cylinder-block configuration. A parametric study of the size of the cylinder, the height, and the location of the rectangular block is also provided. It is observed that the feedback strength is not monotonically related to the height and location of the rectangular block.

High Order Finite Difference Methods for the Wave Equation with Non-conforming Grid Interfaces

We use high order finite difference methods to solve the wave equation in the second order form. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. The focus of this work is on the numerical treatment of non-conforming grid interfaces. The interface conditions are imposed weakly by the simultaneous approximation term technique in combination with interface operators, which move the discrete solutions between the grids on the interface. In particular, we consider interpolation operators and projection operators. A norm-compatibility condition, which leads to stability for first order hyperbolic systems, does not suffice for second order wave equations. An extra constraint on the interface operators must be satisfied to derive an energy estimate for stability. We carry out eigenvalue analyses to investigate the additional constraint and how it is related to stability, and find that the projection operators have better stability properties than the interpolation operators. In addition, a truncation error analysis is performed to study the convergence property of the numerical schemes. In the numerical experiments, the stability and accuracy properties of the numerical schemes are further explored, and the practical usefulness of non-conforming grid interfaces is presented and discussed in two efficiency studies.

Error Analysis of a High Order Method for Time-Fractional Diffusion Equations

In this paper, we consider a numerical method for the time-fractional diffusion equation. The method uses a high order finite difference method to approximate the fractional derivative in time, resulting in a time stepping scheme for the underlying equation. Then the resulting equation is discretized in space by using a spectral method based on the Legendre polynomials. The main body of this paper is devoted to carry out a rigorous analysis for the stability and convergence of the time stepping scheme. As a by-product and direct extension of our previous work, an error estimate for the spatial discretization is also provided. The key contribution of the paper is the proof of the ($3-\alpha$)-order convergence of the time scheme, where $\alpha$ is the order of the time-fractional derivative. Then the theoretical result is validated by a number of numerical tests. To the best of our knowledge, this is the first proof for the stability of the ($3-\alpha$)-order scheme for the time-fractional diffusion equation.