Finite-dimensional models for equations of mathematical physics
Finite dimensional models for obtaining finite difference methods with same properties
FINITE DIFFERENCE METHOD↗
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Efficiency trade-offs of steady-state methods using FEM and FDM
The efficiency characteristics of finite element and finite difference approximations for the steady-state solution of the Navier-Stokes equations are examined. The finite element method discussed is a standard Galerkin formulation of the incompressible, steady-state Navier-Stokes equations. The finite difference formulation uses simple centered differences that are O(delta x-squared). Operation counts indicate that a rapidly converging Newton-Raphson-Kantorovitch iteration scheme is generally preferable over a Picard method. A split NOS Picard iterative algorithm for the finite difference method was most efficient.
Gartling, D. K.↗
Some Aspects of Essentially Nonoscillatory (ENO) Formulations for the Euler Equations, Part 3
An essentially nonoscillatory (ENO) formulation is described for hyperbolic systems of conservation laws. ENO approaches are based on smart interpolation to avoid spurious numerical oscillations. ENO schemes are a superset of Total Variation Diminishing (TVD) schemes. In the recent past, TVD formulations were used to construct shock capturing finite difference methods. At extremum points of the solution, TVD schemes automatically reduce to being first-order accurate discretizations locally, while away from extrema they can be constructed to be of higher order accuracy. The new framework helps construct essentially non-oscillatory finite difference methods without recourse to local reductions of accuracy to first order. Thus arbitrarily high orders of accuracy can be obtained. The basic general ideas of the new approach can be specialized in several ways and one specific implementation is described based on: (1) the integral form of the conservation laws; (2) reconstruction based on the primitive functions; (3) extension to multiple dimensions in a tensor product fashion; and (4) Runge-Kutta time integration. The resulting method is fourth-order accurate in time and space and is applicable to uniform Cartesian grids. The construction of such schemes for scalar equations and systems in one and two space dimensions is described along with several examples which illustrate interesting aspects of the new approach.
Chakravarthy, Sukumar R.↗
Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids
A new, high-order, conservative, and efficient discontinuous spectral finite difference (SD) method for conservation laws on unstructured grids is developed. The concept of discontinuous and high-order local representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG) and the Spectral Volume (SV) methods, but while these methods are based on the integrated forms of the equations, the new method is based on the differential form to attain a simpler formulation and higher efficiency. Conventional unstructured finite-difference and finite-volume methods require data reconstruction based on the least-squares formulation using neighboring point or cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every point or cell at each time step, or to store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In addition, the finite-difference method does not satisfy the integral conservation in general. By contrast, the DG and SV methods employ a local, universal reconstruction of a given order of accuracy in each cell in terms of internally defined conservative unknowns. Since the solution is discontinuous across cell boundaries, a Riemann solver is necessary to evaluate boundary flux terms and maintain conservation. In the DG method, a Galerkin finite-element method is employed to update the nodal unknowns within each cell. This requires the inversion of a mass matrix, and the use of quadratures of twice the order of accuracy of the reconstruction to evaluate the surface integrals and additional volume integrals for nonlinear flux functions. In the SV method, the integral conservation law is used to update volume averages over subcells defined by a geometrically similar partition of each grid cell. As the order of accuracy increases, the partitioning for 3D requires the introduction of a large number of parameters, whose optimization to achieve convergence becomes increasingly more difficult. Also, the number of interior facets required to subdivide non-planar faces, and the additional increase in the number of quadrature points for each facet, increases the computational cost greatly.
Liu, Yen↗
Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics
Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.
Yee, H. C.↗
Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I - The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics
Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.
Yee, H. C.↗
Issac, Jason Cherian ses in transonic flow
Flutter analysis of a two degree of freedom airfoil in compressible flow is performed using a state-space representation of the unsteady aerodynamic behavior. Indicial response functions are used to represent the normal force and moment response of the airfoil. The structural equations of motion of the airfoil with bending and torsional degrees of freedom are coupled to the unsteady air loads and the aeroelastic system so modelled is solved as an eigenvalue problem to determine the stability. The aeroelastic equations are also directly integrated with respect to time and the time-domain results compared with the results from the eigenanalysis. A good agreement is obtained. The derivatives of the flutter speed obtained from the eigenanalysis are calculated with respect to the mass and stiffness parameters by both analytical and finite-difference methods for various transonic Mach numbers. The experience gained from the two degree of freedom model is applied to study the sensitivity of the flutter response of a wing with respect to various shape parameters. The parameters being considered are as follows: (1) aspect ratio; (2) surface area of the wing; (3) taper ratio; and (4) sweep. The wing deflections are represented by Chebyshev polynomials. The compressible aerodynamic state-space model used for the airfoil section is extended to represent the unsteady aerodynamic forces on a generally laminated tapered skewed wing. The aeroelastic equations are solved as an eigenvalue problem to determine the flutter speed of the wing. The derivatives of the flutter speed with respect to the shape parameters are calculated by both analytical and finite difference methods.
Issac, Jason Cherion↗
The nonlinear transient response of thin, rectangular elastic plates
The results of experiments and theoretical analysis of the nonlinear, transient response of thin, rectangular, elastic, simply supported plates are presented. In the experiments, a plane wave tube with a special pulse generator at one end and a thin glass plate mounted at the other end was used to subject the glass plate to pressure pulses of different magnitudes. Whole-field measurements of the response of the plate were obtained by the reflected moire grid technique. The largest ratio of center deflection to thickness of the plate during the experiments was 5.6. The theoretical response of the glass plate to the pressure pulses was computed by solving the von Karman equations governing large deflections in elastic plates by two methods - (1) the finite-difference method, and (2) Galerkin's method. Good agreement was found between theoretically predicted and experimentally measured response.
Rajagopal, G.↗
Supersonic turbulent boundary-layer flows with tangential slot injection.
An implicit finite-difference method has been developed for the solution of the compressible boundary-layer equations. This method is applied to tangential slot injection into supersonic turbulent boundary-layer flows. In addition, the effects induced by the interaction between the boundary-layer displacement thickness and the external pressure field are considered. Three different eddy viscosity models have been used to specify the turbulent momentum exchange. One model depends on the species concentration profile, and the species conservation equation has been included in the system of governing partial differential equations. For air injected into air, the freestream and injected gases are treated as separate species which have common fluid properties. Calculations were made and results were compared with experimental data at freestream Mach numbers of 2.4 and 6.0 and with results of another finite-difference method. Good agreement was obtained for the reduction of wall skin friction with slot injection.
Miner, E. W.↗
Symmetric and nonsymmetric buckling of finitely deformed eccentrically stiffened shells of revolution.
Finite difference method stability analysis of deformed eccentrically stiffened shells of revolution, accounting for finite prebuckling rotations
FINITE DIFFERENCE METHOD↗
Symmetric and nonsymmetric buckling of finitely deformed eccentrically stiffened shells of revolution.
Finite difference method stability analysis of deformed eccentrically stiffened shells of revolution, accounting for finite prebuckling rotations
FINITE DIFFERENCE METHOD↗
On the application of pseudo-spectral FFT technique to non-periodic problems
The reduction-to-periodicity method using the pseudo-spectral Fast Fourier Transform (FFT) technique is applied to the solution of nonperiodic problems including the two-dimensional Navier-Stokes equations. The accuracy of the method is demonstrated by calculating derivatives of given functions, one- and two-dimensional convective-diffusive problems, and by comparing the relative errors due to the FFT method with seocnd order Finite Difference Methods (FDM). Finally, the two-dimensional Navier-Stokes equations are solved by a fractional step procedure using both the FFT and the FDM methods for the driven cavity flow and the backward facing step problems. Comparisons of these solutions provide a realistic assessment of the FFT method indicating its range of applicability.
Biringen, S.↗
A zonal approach to V/STOL vehicle aerodynamics
A zonal method based on an iterative coupling between an inviscid panel method and two viscous finite difference methods is applied to solve for the region of jets issued from generic V/STOL models in order to assess the influence of the jet upon V/STOL configurations. The low-order panel method, VSAERO, is used to solve for the entire flow region, which includes the V/STOL model and jet boundary. ARC3D, a well known compressible, thin layer Navier-Stokes (NS) code developed at NASA-Ames, and APPL, a parabolized NS code, are used to solve for the jet region subject to the boundary condition provided by the VSAERO calculation. Overlapping boundaries are used for the purpose of coupling. Results are presented with the available experimental data for the jet and V/STOL configurations.
Yoo, Sungyul↗
On the Daubechies-based wavelet differentiation matrix
The differentiation matrix for a Daubechies-based wavelet basis is constructed and superconvergence is proven. That is, it will be proven that under the assumption of periodic boundary conditions that the differentiation matrix is accurate of order 2M, even though the approximation subspace can represent exactly only polynomials up to degree M-1, where M is the number of vanishing moments of the associated wavelet. It is illustrated that Daubechies-based wavelet methods are equivalent to finite difference methods with grid refinement in regions of the domain where small-scale structure is present.
Jameson, Leland↗
On the dynamics of approximating schemes for dissipative nonlinear equations
Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.
Jones, Donald A.↗
Solving Fluid Structure Interaction Problems with an Immersed Boundary Method
An immersed boundary method for the compressible Navier-Stokes equations can be used for moving boundary problems as well as fully coupled fluid-structure interaction is presented. The underlying Cartesian immersed boundary method of the Launch Ascent and Vehicle Aerodynamics (LAVA) framework, based on the locally stabilized immersed boundary method previously presented by the authors, is extended to account for unsteady boundary motion and coupled to linear and geometrically nonlinear structural finite element solvers. The approach is validated for moving boundary problems with prescribed body motion and fully coupled fluid structure interaction problems. Keywords: Immersed Boundary Method, Higher-Order Finite Difference Method, Fluid Structure Interaction.
Structure Interactions↗
Numerical techniques for the solution of symmetric positive linear differential equations
Finite difference method for solving symmetric positive linear differential equations
LINEAR EQUATION↗
A new method for analysis of transonic flow
Finite difference method for transonic flow solution in axial stator
Katsanis, T.↗