High-order Finite Difference Method Research Articles

On the spurious solutions in the high-order finite difference methods for eigenvalue problems

In this paper, the origins of spurious solutions occurring in the high-order finite difference methods are studied. Based on a uniform mesh, spurious modes are found in the high-order one-sided finite difference discretizations of many eigenvalue problems. Spurious modes are classified as spectral pollution and non-spectral pollution. The latter can be partially avoided by mesh refinement, while the former persists when the mesh is refined. Through numerical studies of some prototype eigenvalue problems, such as those of the Helmholtz and beam equations, we show that perfect central differentiation schemes do not produce any spurious modes. Nevertheless, high-order central difference schemes encounter difficulties in implementing complex boundary conditions. We further show that the central difference schemes will produce spurious modes as well, if asymmetric approximation is involved in boundary treatments. In general, central difference schemes are less likely to produce spurious modes than one-sided difference schemes.

1,3,5-trinitro-1,3,5-triazine decomposition and chemisorption on Al(111) surface: First-principles molecular dynamics study

We have investigated the decomposition and chemisorption of a 1,3,5-trinitro-1,3,5-triazine (RDX) molecule on Al(111) surface using molecular dynamics simulations, in which interatomic forces are computed quantum mechanically in the framework of the density functional theory (DFT). The real-space DFT calculations are based on higher-order finite difference and norm-conserving pseudopotential methods. Strong attractive forces between oxygen and aluminum atoms break N-O and N-N bonds in the RDX and, subsequently, the dissociated oxygen atoms and NO molecules oxidize the Al surface. In addition to these Al surface-assisted decompositions, ring cleavage of the RDX molecule is also observed. These reactions occur spontaneously without potential barriers and result in the attachment of the rest of the RDX molecule to the surface. This opens up the possibility of coating Al nanoparticles with RDX molecules to avoid the detrimental effect of oxidation in high energy density material applications.

A stable and efficient hybrid scheme for viscous problems in complex geometries

In this paper, we present a stable hybrid scheme for viscous problems. The hybrid method combines the unstructured finite volume method with high-order finite difference methods on complex geometries. The coupling procedure between the two numerical methods is based on energy estimates and stable interface conditions are constructed. Numerical calculations show that the hybrid method is efficient and accurate.

Stabilities of combined radiation and Rayleigh–Bénard–Marangoni convection in an open vertical cylinder

This paper studies the combined radiation and natural convection in an open vertical cylinder. The cylinder is heated through its sidewall at a constant heat flux and cooled at the top surface via radiation. The high order finite difference method is used to simulate the fluid flow and heat transfer inside the cylinder and the internal radiation is solved using discontinuous finite element method. The two numerical methods are coupled through an iterative process. Based on the numerical simulations, a linear stability analysis is carried out to investigate how the internal radiation changes the stability of the flow.

High order numerical simulation of aeolian tones

High order finite difference methods have been constructed to be strictly stable for linear hyperbolic and parabolic problems. The difference operators have been extended from the Euler to the Navier–Stokes equations for computational aeroacoustics. Aeolian tones generated by vortex shedding from a circular cylinder have been simulated.

A stable high-order finite difference scheme for the compressible Navier–Stokes equations, far-field boundary conditions

We construct a stable high-order finite difference scheme for the compressible Navier–Stokes equations, that satisfy an energy estimate. The equations are discretized with high-order accurate finite difference methods that satisfy a Summation-By-Parts rule. The boundary conditions are imposed with penalty terms known as the Simultaneous Approximation Term technique. The main result is a stability proof for the full three-dimensional Navier–Stokes equations, including the boundary conditions. We show the theoretical third-, fourth-, and fifth-order convergence rate, for a viscous shock, where the analytic solution is known. We demonstrate the stability and discuss the non-reflecting properties of the outflow conditions for a vortex in free space. Furthermore, we compute the three-dimensional vortex shedding behind a circular cylinder in an oblique free stream for Mach number 0.5 and Reynolds number 500.

Lattice-free finite difference method for numerical solution of inverse heat conduction problem

Inverse heat conduction problem consists of finding an initial temperature distribution from the knowledge of a distribution of the temperature at the present time. Here, we assume that the associated boundary conditions are known. The heat conduction problem backward in time is a typical example of ill-posed problems in the sense that the solution exists only for regular functions of some kind describing the present temperature distribution and also the solution is unstable for the present temperature distribution function. Conventional numerical methods often suffer from instability of the problem itself when high accuracy is intended in the approximation. Our aim is to create a meshless method which is applicable to the ill-posed inverse heat conduction problem. We construct a high order finite difference method in which quadrature points do not need to have a lattice structure. In order to develop our new method we show a tool in using exponential functions in Taylor's expansion. From numerical experiments we confirmed that our method is effective for solving two-dimensional inverse heat conduction problem numerically subject to mixed boundary conditions.

A Fourier Pseudospectral Method for Some Computational Aeroacoustics Problems

A Fourier pseudospectral time-domain method is applied to wave propagation problems pertinent to computational aeroacoustics. The original algorithm of the Fourier pseudospectral time-domain method works for periodical problems without the interaction with physical boundaries. In this paper we develop a slip wall boundary condition, combined with buffer zone technique to solve some non-periodical problems. For a linear sound propagation problem whose governing equations could be transferred to ordinary differential equations in pseudospectral space, a new algorithm only requiring time stepping is developed and tested. For other wave propagation problems, the original algorithm has to be employed, and the developed slip wall boundary condition still works. The accuracy of the presented numerical algorithm is validated by benchmark problems, and the efficiency is assessed by comparing with high-order finite difference methods. It is indicated that the Fourier pseudospectral time-domain method, time stepping method, slip wall and absorbing boundary conditions combine together to form a fully-fledged computational algorithm.

Modeling complex biological flows in multi-scale systems using the APDEC framework

We have developed advanced numerical algorithms to model biological fluids in multiscale flow environments using the software framework developed under the SciDAC APDEC ISIC. The foundation of our computational effort is an approach for modeling DNA laden fluids as ‘‘bead-rod’’ polymers whose dynamics are fully coupled to an incompressible viscous solvent. The method is capable of modeling short range forces and interactions between particles using soft potentials and rigid constraints. Our methods are based on higher-order finite difference methods in complex geometry with adaptivity, leveraging algorithms and solvers in the APDEC Framework. Our Cartesian grid embedded boundary approach to incompressible viscous flow in irregular geometries has also been interfaced to a fast and accurate level-sets method within the APDEC Framework for extracting surfaces from volume renderings of medical image data and used to simulate cardio-vascular and pulmonary flows in critical anatomies.

High-order compact exponential finite difference methods for convection–diffusion type problems

A class of high-order compact (HOC) exponential finite difference (FD) methods is proposed for solving one- and two-dimensional steady-state convection–diffusion problems. The newly proposed HOC exponential FD schemes have nonoscillation property and yield high accuracy approximation solution as well as are suitable for convection-dominated problems. The O( h 4) compact exponential FD schemes developed for the one-dimensional (1D) problems produce diagonally dominant tri-diagonal system of equations which can be solved by applying the tridiagonal Thomas algorithm. For the two-dimensional (2D) problems, O( h 4 + k 4) compact exponential FD schemes are formulated on the nine-point 2D stencil and the line iterative approach with alternating direction implicit (ADI) procedure enables us to deal with diagonally dominant tridiagonal matrix equations which can be solved by application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. To validate the present HOC exponential FD methods, three linear and nonlinear problems, mostly with boundary or internal layers where sharp gradients may appear due to high Peclet or Reynolds numbers, are numerically solved. Comparisons are made between analytical solutions and numerical results for the currently proposed HOC exponential FD methods and some previously published HOC methods. The present HOC exponential FD methods produce excellent results for all test problems. It is shown that, besides including the excellent performances in computational accuracy, efficiency and stability, the present method has the advantage of better scale resolution. The method developed in this article is easy to implement and has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2D incompressible Navier–Stokes equations using the stream function-vorticity formulation.

High order finite difference methods for wave propagation in discontinuous media

High order finite difference approximations are derived for the second order wave equation with discontinuous coefficients, on rectangular geometries. The discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion. Each sub-domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method. This guarantees a conservative, strictly stable and high order accurate scheme. The analysis is verified by numerical simulations in one and two spatial dimensions.

Particle methods revisited: a class of high order finite-difference methods

We propose a new analysis of particle method with remeshing. We derive a class of high-order finite difference methods. Our analysis is completed by numerical comparisons with Lax–Wendroff schemes for the Burger equation. To cite this article: G.-H. Cottet, L. Weynans, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Thermocapillary Instabilities of Low Prandtl Number Fluid in a Laterally Heated Vertical Cylinder

Stabilities of surface-tension-driven convection in an open cylinder are investigated numerically. The cylinder is heated laterally through its sidewall and is cooled at free surface by radiation. A seeding crystal at constant temperature is in contact with the free surface. Axisymmetric base flow is solved using the high-order finite difference method. Three-dimensional perturbation is applied to the obtained base flow to determine the critical Marangoni numbers at which the axisymmetry is broken. The eigenvalue matrix equation is solved using linear fractional transformation with banded matrix structure taken into account. Critical Marangoni-Reynolds numbers are obtained at various boundary conditions.

Three-dimensional numerical simulation of thermosolutal convection in enclosures using lattice Boltzmann method

Lattice Boltzmann method of two-and three-dimensional thermosolutal convection is investigated in this paper. Lattice Boltzmann method with multiple components was applied to simulate two-dimensional thermosolutal convection flow on a square cavity. The numerical results agree well with those of high order finite difference method. It shows that the numerical scheme is efficient. Therefore the lattice Boltzmann method is extended to investigate three-dimensional thermosolutal convection flow in cubic cavity. The comparison between the proposed lattice Boltzmann method and a finite difference method shows a satisfactory agreement. The limitation of the lattice Boltzmann model in the double-diffusive convection is discussed.

A stable and efficient hybrid method for aeroacoustic sound generation and propagation

We discuss how to combine the node based unstructured finite volume method widely used to handle complex geometries and nonlinear phenomena with very efficient high order finite difference methods suitable for wave propagation dominated problems. This fully coupled numerical procedure reflects the coupled character of the sound generation and propagation problem. The coupling procedure is based on energy estimates and stability can be guaranteed. Numerical experiments using finite difference methods that shed light on the theoretical results are performed. To cite this article: J. Nordström, J. Gong, C. R. Mecanique 333 (2005).

A stable hybrid method for hyperbolic problems

A stable hybrid method for hyperbolic problems that combines the unstructured finite volume method with high-order finite difference methods has been developed. The coupling procedure is based on energy estimates and stability can be guaranteed. Numerical calculations verify that the hybrid method is efficient and accurate.

A moving collocation method for the solution of the transient convection–diffusion-reaction problems

An adaptive collocation method is introduced for simulating the short time diffusion–convection-reaction problem. The method is based on dividing the solution domain into active and inactive zones in such a way that the collocation points remain concentrated in regions of solution variation. The numerical results show that the proposed method is efficient in simulating the sharp profile at short time for the convective-dominant case. The adaptive scheme performance is found compatible with the high-order finite difference method, the QUICK method in terms of the CPU time and average numerical errors.

Accuracy requirements for the calculation of gravitational waveforms from coalescing compact binaries in numerical relativity

I discuss the accuracy requirements on numerical relativity calculations of inspiraling compact object binaries whose extracted gravitational waveforms are to be used as templates for matched filtering signal extraction and physical parameter estimation in modern interferometric gravitational wave detectors. Using a post-Newtonian point particle model for the pre-merger phase of the binary inspiral, I calculate the maximum allowable errors for the mass and relative velocity and positions of the binary during numerical simulations of the binary inspiral. These maximum allowable errors are compared to the errors of state-of-the-art numerical simulations of multiple-orbit binary neutron star calculations in full general relativity, and are found to be smaller by several orders of magnitude. A post-Newtonian model for the error of these numerical simulations suggests that adaptive mesh refinement coupled with second order accurate finite difference codes will {\it not} be able to robustly obtain the accuracy required for reliable gravitational wave extraction on Terabyte-scale computers. I conclude that higher order methods (higher order finite difference methods and/or spectral methods) combined with adaptive mesh refinement and/or multipatch technology will be needed for robustly accurate gravitational wave extraction from numerical relativity calculations of binary coalescence scenarios.

Multi-Azimuth Three-Component Surface Seismic Modeling in Cracked Monoclinic Media

The snapshots of the seismic wave propagation in monoclinic media are simulated using high-order staggered-grid finite-difference method. The monoclinic medium is a typical anisotropic model in fractured reservoir which includes two sets of non-orthorhombic vertical fractures in isotropic background and with different fracture infill. The modeling results show that the velocities of wave propagation in anisotropic models vary greatly in different directions, and the property of the fracture infill has a strong effect on velocity anisotropy. Then the axial rotation method is used in the modeling of multi-azimuth three-component surface seismic data in layered monoclinic media. The results illustrate that seismic velocities vary in directions not only with the polar angle, but also with the azimuth angle. Furthermore, the numerical modeling results may provide a theoretical basis for the inversion of anisotropic parameters and the fracture parameters by using the surface multi-azimuth seismic attributes.

A quasi-spectral method for Cauchy problem of 2/D Laplace equation on an annulus

Real numbers are usually represented in the computer as a finite number of digits hexa-decimal floating point numbers. Accordingly the numerical analysis is often suffered from rounding errors. The rounding errors particularly deteriorate the precision of numerical solution in inverse and ill-posed problems. We attempt to use a multi-precision arithmetic for reducing the rounding error evil. The use of the multi-precision arithmetic system is by the courtesy of Dr Fujiwara of Kyoto University. In this paper we try to show effectiveness of the multi-precision arithmetic by taking two typical examples; the Cauchy problem of the Laplace equation in two dimensions and the shape identification problem by inverse scattering in three dimensions. It is concluded from a few numerical examples that the multi-precision arithmetic works well on the resolution of those numerical solutions, as it is combined with the high order finite difference method for the Cauchy problem and with the eigenfunction expansion method for the inverse scattering problem.