Spatial Finite Difference Research Articles

Time Explicit Schemes and Spatial Finite Differences Splittings

In this article, we conjugate time marching schemes with Finite Differences splittings into low and high modes in order to build fully explicit methods with enhanced temporal stability for the numerical solutions of PDEs. The main idea is to apply explicit schemes with less restrictive stability conditions to the linear term of the high modes equation, in order that the allowed time step for the temporal integration is only determined by the low modes. These conjugated schemes were developed in l10r for the spectral case and here we adapt them to the Finite Differences splittings provided by Incremental Unknowns, which steems from the Inertial Manifolds theory. We illustrate their improved capabilities with numerical solutions of Burgers equations, with uniform and nonuniform meshes, in dimensions one and two, when using modified Forward–Euler and Adams–Bashforth schemes. The resulting schemes use time steps of the same order of those used by semi-implicit schemes with comparable accuracy and reduced computational costs.

Simulation of the Planar Free Surface with Near-Surface Lateral Discontinuities in the Finite-Difference Modeling of Seismic Motion

Kristek et al. (2002) developed a technique for simulating the planar free surface in the 3D fourth-order staggered-grid finite-difference (FD) modeling of seismic motion. The technique is based on (1) explicit application of zero values of the stress-tensor components at the free surface and (2) adjusted FD approximations (AFDAs) to vertical derivatives at and near the free surface. The technique was shown to be more accurate and efficient than the standard stress-imaging technique in 1D models. In this study, we tested accuracy of the AFDA technique in media with lateral material discontinuities reaching the free surface. We compared the FD synthetics with synthetics calculated by the standard finite-element (FE) method because the FE method naturally and sufficiently accurately satisfies the boundary conditions at the free surface and the traction interface continuity conditions at internal material discontinuities. The comparison showed a very good level of accuracy of the AFDA technique. We also demonstrated the very good sensitivity of our FD modeling to different positions of the same physical model in the spatial FD grid.

Finite-Difference Seismic Simulation Combining Discontinuous Grids with Locally Variable Timesteps

A locally variable timestep scheme that matches with discontinuous grids in the finite-difference method is developed for the efficient simulation of seismic-wave propagation. The first-order velocity-stress formulations are used to obtain the spatial derivatives using finite-difference operators on a staggered grid. In the case of a media interface with high velocity contrast, the computational domain consists of two regions with different grid spacings. Each region roughly covers the medium of the lower or higher wave propagation velocity. There is a small overlap of the two regions, called the transitional zone, within the higher velocity medium. A grid three times coarser in the high-velocity region compared with the grid in the low-velocity region is used to avoid spatial oversampling. Temporal steps corresponding to the spatial sampling ratio between both regions are determined based on local stability criteria. The wave field in the margin of the region with the smaller timestep is linearly interpolated in time using the values calculated in the region with the larger one within the transitional zone. Since the temporal interpolation is a 1D operation performed separately from the spatial interpolation strategy employed to connect two regions with different grid spacings, the proposed scheme is not restricted to 2D or 3D problems with a specific order of accuracy of the spatial finite-difference approximation. The use of the locally variable timestep scheme with discontinuous grids results in remarkable savings of computation time and reductions in memory requirements, with the efficiency depending on the simulation model.

Non-Reflecting Boundary Conditions for Maxwell?s Equations

A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation.

Spatial discretization of partial differential equations with integrals

We consider the problem of constructing spatial finite-difference approximations on an arbitrary fixed grid which preserve any number of integrals of the partial differential equation and preserve some of its symmetries. A basis for the space of such finite-difference operators is constructed; most cases of interest involve a single such basis element. (The 'Arakawa' Jacobian is such an element, as are discretizations satisfying 'summation by parts' identities.) We show how the grid, its symmetries, and the differential operator interact to affect the complexity of the finite difference.

A multigrid solver for two-dimensional stochastic diffusion equations

Steady and unsteady diffusion equations, with stochastic diffusivity coefficient and forcing term, are modeled in two dimensions by means of stochastic spectral representations. Problem data and solution variables are expanded using the Polynomial Chaos system. The approach leads to a set of coupled problems for the stochastic modes. Spatial finite-difference discretization of these coupled problems results in a large system of equations, whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time. To accelerate the convergence of the iterative technique, a multigrid method, based on spatial coarsening, is implemented. Numerical experiments show good scaling properties of the method, both with respect to the number of spatial grid points and the stochastic resolution level.

Analysis and simulation for a system of chemical reaction equations with a vortex formulation

This work presents analytical and numerical results for the two dimensional molecular mixing and chemical reaction processes using a vortex formulation. The particular model studied here is the single-step, irreversible, exothermic Arrhenius type reaction c A + c B → c P , in an incompressible fluid under Neumann boundary conditions. We establish the existence of a unique classical solution considering the weak coupling of the reactants via advection by a independently developing velocity, and we prove the existence and uniqueness of the solution via a semigroup formulation and the maximum principle. For the numerical solution a second order accurate spatial finite difference scheme is used with a second order accurate Runge–Kutta method for the time step. The behaviour of the concentrations c A and c B , the temperature, the reaction rate, and product concentration c P , are obtained over a wide range of Reynolds and Damköhler numbers.

Estimation of optimal parameters for a surface hydrology model

The estimation of infiltration, roughness, and rainfall intensity parameters in a surface flow model is studied. The problem is formulated as an optimal control problem in which the parameters to be estimated are viewed as controls in the surface flow model and a fit-to-data functional is viewed as a control functional. The control system is obtained as the approximation of the distributed model by means of finite elements in the spatial domain and finite differences in the time domain. The leap-frog and implicit differences are used for the time differencing approximations. The resulting minimization problem is solved by a descent method, The adjoint equations are introduced in the context of a general method to calculate efficiently the derivatives of the functional. The adjoint equations are developed for the continuous as well as the finite difference equations. A numerical example is presented. This example illustrates the effect of the accuracy of prior rainfall intensity information on the estimates of roughness and infiltration parameters.

An optimal spatial finite-difference operator which reduces truncation error to a minimum

Highly accurate spatial discretization is essentially required to perform numerical climate and weather prediction. The difference between the differential and the finite-difference operator is however a primitive error source in the numerics. This paper presents an optimization of centered finite-difference operator based on the principle of constrained cost function, which can reduce the truncation error to minimum. In the optimization point of view, such optimal operator is in fact an attempt to minimize spatial truncation er-rors in atmospheric modeling, in a simple way and indeed a quite innovative way to implement Variational Continuous Assimilation (VCA) technique.

A Pseudospectral Time-Domain Algorithm for Calculating the BandStructure of a Two-Dimensional Photonic Crystal

A pseudospectral time-domain (PSTD) method is developed forcalculating the band structure of a two-dimensional photonic crystal.Maxwell's equations are rewritten in terms of period fields by using the Blochtheorem. Instead of spatial finite differences, the fast Fourier transform isused to calculate the spatial derivatives. To reach a similaraccuracy, fewer sample points are required in the present PSTDmethod as compared to the conventional finite-differencetime-domain methods. Our numerical simulation shows that the present PSTDmethod is an efficient and accurate method for calculating the bandstructure of a photonic crystal.

STABILITY ISSUES OF CONCENTRIC PIPES CONTAINING STEADY AND PULSATILE FLOWS

In this paper, we study both the static and dynamic instabilities of submerged and inclined concentric pipes conveying fluid. The governing equation for the inner tubular beam is derived under small deformation assumptions. We obtain the discretized dynamical equations using spatial finite-difference schemes. In the case of steady flow, both buckling and flutter instabilities are investigated. In the case of pulsatile flow, we compute the eigenvalues of the monodromy matrix derived from the discretized linear system with periodic coefficients, and deduce the dynamical stability information. In addition, for a special case, in which the concentric pipes have the same length, we compare the dynamic stability results with the corresponding solutions obtained with the Bolotin method.

Application of the generalized- α method to the time integration of the cable dynamics equations

The box method is a popular finite difference scheme for solving the cable dynamics equations in ocean engineering applications. In this paper, we analyze the shortcomings of the temporal integration defined by the box method and propose a new procedure that retains the box method's spatial discretization, but employs the generalized- α method for temporal discretization. The stability and accuracy of the generalized- α method, as applied to the cable dynamics equations, are developed. This analysis procedure demonstrates the formulation of the semi-discrete governing equations using spatial finite differences and the applicability of amplification matrix stability analysis to finite difference problems. An example is presented that demonstrates the robustness of the new approach.

Three-Dimensional Model of Navier-Stokes Equations for Water Waves

A three-dimensional numerical model based on the complete Navier-Stokes equations is developed and presented in this paper. The model can be used for the problem of propagation of fully nonlinear water waves. The Navier-Stokes equations are first transformed from an irregular calculation domain to a regular one using sigma coordinates. The projection method is used to separate advection and diffusion terms from the pressure terms in Navier-Stokes equations. MacCormack's explicit scheme is used for the advection and diffusion terms, and it has second-order accuracy in both space and time. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. The resulting hydrodynamic pressure equation is solved by a multigrid method. A staggered mesh and central spatial finite-difference scheme are used. The model is tested against the experimental data of Luth et al., and the comparison shows that higher harmonics can be modeled well. Comparison of the model solutions with the elliptic shoal case confirms that the present model works well for wave refraction and diffraction with strong wave focusing.

Adaptive High-Order Finite-Difference Method for Nonlinear Wave Problems

We discuss a scheme for the numerical solution of one-dimensional initial value problems exhibiting strongly localized solutions or finite-time singularities. To accurately and efficiently model such phenomena we present a full space-time adaptive scheme, based on a variable order spatial finite-difference scheme and a 4th order temporal integration with adaptively chosen time step. A wavelet analysis is utilized at regular intervals to adaptively select the order and the grid in accordance with the local behavior of the solution. Through several examples, taken from gasdynamics and nonlinear optics, we illustrate the performance of the scheme, the use of which results in several orders of magnitude reduction in the required degrees of freedom to solve a problem to a particular fidelity.

Application of Three-Dimensional Hydrodynamic Model for Lake Okeechobee

A 3D hydrodynamic and heat transport model was developed for Lake Okeechobee. Continuity, momentum, and temperature transport equations were solved. Dynamically coupled transport equations for turbulent kinetic energy and turbulent scale also were solved. The numerical scheme used spatial finite differencing and a three-time-level, external-internal mode splitting procedure. A 28-day calibration was conducted, using measured bathymetry, rainfall, relative humidity, total solar radiation, wind velocity, inflow, and outflow data. During the calibration period, little rainfall occurred, and lake water levels receded. Water surface elevation, horizontal velocities, and temperature were computed. Agreement between observed and simulated values was based on graphical comparisons, minimizing mean absolute and root-mean-square errors, and spectral analysis. Comparisons showed that the model reproduced general observed trends and short-term fluctuations. The model's heat transport and turbulence closure schemes behaved as expected with regard to water column stratification and mixing. Simulation accuracy may potentially be improved by adding wind-wave and vegetation resistance algorithms to the model.

NUMERICAL ANALYSIS OF ROTATIONALLY SYMMETRIC SHELLS BY THE MODAL SUPERPOSITION METHOD

The system of differential equations governing the analysis of rotationally symmetric shells under time-dependent or static surface loadings is formulated with the transverse, meridional, and circumferential displacements as the dependent variables. The thickness of the shell may vary, and four homogeneous boundary conditions may be prescribed at each boundary edge of the shell. The governing differential equations for each Fourier harmonic are obtained by use of Fourier series in the circumferential direction of the shell. Influence coefficients at each of the node points along the shell meridian are obtained for each Fourier component by employing ordinary finite difference representations for the meridional co-ordinate derivatives. With these influence coefficients, a set of homogeneous flexibility equations governing the free vibration characteristics of the shell is obtained and solved for the frequencies and mode shapes for each Fourier component. The solutions under time-dependent or static surface loadings and due to initial displacements and velocities are then obtained by expanding the solutions in terms of the modes of free vibration of the shell for each Fourier component. The solution for the total shell response is obtained by Fourier series summation. Solutions for typical shells have been found to be in excellent agreement with solutions by the method of temporal and spatial finite differences. Solutions for a parabolic shell are presented as an example. The solution is also presented for a published cylindrical shell example and is seen to be in excellent agreement with the published finite element method results.

FINITE DIFFERENCE ANALYSIS OF ROTATIONALLY SYMMETRIC SHELLS USING VARIABLE NODE POINT SPACINGS

The system of equations for the analysis of rotationally symmetric shells under time-dependent or static surface loadings has been formulated with the transverse, meridional, and circumferential displacements as the dependent variables. All loading functions must be continuous. The thickness h of the shell may vary along the meridian. Four of the eight natural boundary conditions will be prescribed as time-dependent boundary conditions at each boundary edge of the shell. Surface loadings and inertia forces in the three displacement directions of the shell have been considered. Fourier series are used in the circumferential direction of the shell. Solutions for each Fourier component are found by replacing all derivatives by their finite difference equivalents and solving the resulting system of algebraic equations at successive increments of the time variable. The complete system of equations is solved implicitly for the first time increment, while explicit relations are used to determine the three primary displacements within the boundary edges of the shell for the second and succeeding time increments. The remaining unspecified primary variables are then determined by separate implicit solutions at each boundary for the second and succeeding time increments. Subsequently, all remaining primary and secondary variables are found explicitly. A variable node point spacing may be specified over the full range of the spatial finite difference mesh. A numerical stability criterion which will ensure stable solutions for selected time increments and spatial meshes has been derived and found to agree with results for typical example solutions. Numerical solutions obtained with the computer program which accompanies this development have been found to be stable and in agreement for a wide range of practical values of both spatial and time increments for typical shells and loadings. Static and dynamic solutions for a parabolic shell with fixed boundary conditions at one edge and free boundary conditions at the other edge have been obtained using both constant and variable node point spacings and presented as examples.

Spatially periodic structures in electron swarms and the Franck-Hertz experiment

The seminal experiment of Franck and Hertz, which helped lay the foundations of quantum and atomic physics, is investigated through solution of the Boltzmann equation, using both eigenfunction expansion methods and spatial finite-difference techniques. We consider electrons in both a model gas and in mercury, the gas originally used by Franck and Hertz, and focus upon the effects of both inelastic and elastic collisions. It is pointed out that the periodic spatial structures encountered in the Franck-Hertz experiment have a physical origin similar to the oscillatory phenomena observed by Fletcher and others previously in low-pressure, low-current discharges, and by Winkler et al, and other contemporary authors in simulations of low-temperature plasmas.

A frequency accuraterth order spatial derivative finite difference approximation

A method for the specification and design of finite difference spatial derivative approximations of general order r is presented. The method uses a difference polynomial with undetermined coefficients. Spatial frequency domain-based criteria, which include phase velocity, group velocity, and dissipation requirements at a priori selected spatial frequencies, are used to find the appropriate coefficient values. The method is formulated as an optimal design problem but is pursued heuristically. The general derivative approximation and the design method are suitable for use in more general design problems involving finite difference schemes for linear and nonlinear partial differential equations. © 1999 John Wiley & Sons, Inc. * Numer Methods Partial Differential Eq 15: 569–580, 1999

Analytical and computational approaches for some fluid–structure interaction analyses

In this paper, we present an analytical model to study the interaction between a slender structure and the surrounding incompressible viscous flows. The governing equations for the suspended structure are derived under small deformation assumptions and with the consideration of fluid pressures and frictional forces. We obtain the discretized equations using the standard linearization procedure and spatial finite difference schemes. It is shown that for both laminar and turbulent flow conditions, the results derived from analytical approaches match the solutions of the ADINA fluid-structure analysis. Of course when geometries of fluid domains are complex, it is necessary to use a general purpose computational software. We also point out that for specific problems, such as the dynamic stability analysis of the structure surrounded by pulsatile flows, there is still a need for analytical approaches.