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At least 19 records

Multilevel Error Estimation and Adaptive h-Refinement for Cartesian Meshes with Embedded Boundaries

This paper presents the development of a mesh adaptation module for a multilevel Cartesian solver. While the module allows mesh refinement to be driven by a variety of different refinement parameters, a central feature in its design is the incorporation of a multilevel error estimator based upon direct estimates of the local truncation error using tau-extrapolation. This error indicator exploits the fact that in regions of uniform Cartesian mesh, the spatial operator is exactly the same on the fine and coarse grids, and local truncation error estimates can be constructed by evaluating the residual on the coarse grid of the restricted solution from the fine grid. A new strategy for adaptive h-refinement is also developed to prevent errors in smooth regions of the flow from being masked by shocks and other discontinuous features. For certain classes of error histograms, this strategy is optimal for achieving equidistribution of the refinement parameters on hierarchical meshes, and therefore ensures grid converged solutions will be achieved for appropriately chosen refinement parameters. The robustness and accuracy of the adaptation module is demonstrated using both simple model problems and complex three dimensional examples using meshes with from 10(exp 6), to 10(exp 7) cells.

Aftosmis, M. J.↗

Classical eighth- and lower-order Runge-Kutta-Nystroem formulas with a new stepsize control procedure for special second-order differential equations

New Runge-Kutta-Nystrom formulas of the eighth, seventh, sixth, and fifth order are derived for the special second-order (vector) differential equation x = f (t,x). In contrast to Runge-Kutta-Nystrom formulas of an earlier NASA report, these formulas provide a stepsize control procedure based on the leading term of the local truncation error in x. This new procedure is more accurate than the earlier Runge-Kutta-Nystrom procedure (with stepsize control based on the leading term of the local truncation error in x) when integrating close to singularities. Two central orbits are presented as examples. For these orbits, the accuracy and speed of the formulas of this report are compared with those of Runge-Kutta-Nystrom and Runge-Kutta formulas of earlier NASA reports.

Fehlberg, E.↗

Error induced by coordinate systems

It is pointed out that the choice of a curvilinear coordinate system can have a substantial effect on the error in the numerical solution of a partial differential equation. The truncation error is dependent not only on the higher order derivatives of the solution and the local grid spacing, but also on the rate-of-change of the grid spacing and on the departure of the grid from orthogonality. In connection with the present investigation, an analysis is conducted of the local truncation error in the approximation of first and second order derivatives on a curvilinear grid. Attention is given to a number of examples which illustrate the two fundamental sources of truncation error in the numerical solution of partial differential equations on curvilinear coordinate systems. The first is the grid spacing and changes in grid spacing which is measured by the first and second order derivatives of the functions defining the coordinate system. The second source is the higher order derivatives of the solution itself.

Mastin, C. W.↗

Error norms for the adaptive solution of the Navier-Stokes equations

The adaptive solution of the Navier-Stokes equations depends upon the successful interaction of three key elements: (1) the ability to flexibly select grid length scales in composite grids, (2) the ability to efficiently control residual error in composite grids, and (3) the ability to define reliable, convenient error norms to guide the grid adjustment and optimize the residual levels relative to the local truncation errors. An initial investigation was conducted to explore how to approach developing these key elements. Conventional error assessment methods were defined and defect and deferred correction methods were surveyed. The one dimensional potential equation was used as a multigrid test bed to investigate how to achieve successful interaction of these three key elements.

Forester, C. K.↗

Adaptive grid methods for RLV environment assessment and nozzle analysis

Rapid access to highly accurate data about complex configurations is needed for multi-disciplinary optimization and design. In order to efficiently meet these requirements a closer coupling between the analysis algorithms and the discretization process is needed. In some cases, such as free surface, temporally varying geometries, and fluid structure interaction, the need is unavoidable. In other cases the need is to rapidly generate and modify high quality grids. Techniques such as unstructured and/or solution-adaptive methods can be used to speed the grid generation process and to automatically cluster mesh points in regions of interest. Global features of the flow can be significantly affected by isolated regions of inadequately resolved flow. These regions may not exhibit high gradients and can be difficult to detect. Thus excessive resolution in certain regions does not necessarily increase the accuracy of the overall solution. Several approaches have been employed for both structured and unstructured grid adaption. The most widely used involve grid point redistribution, local grid point enrichment/derefinement or local modification of the actual flow solver. However, the success of any one of these methods ultimately depends on the feature detection algorithm used to determine solution domain regions which require a fine mesh for their accurate representation. Typically, weight functions are constructed to mimic the local truncation error and may require substantial user input. Most problems of engineering interest involve multi-block grids and widely disparate length scales. Hence, it is desirable that the adaptive grid feature detection algorithm be developed to recognize flow structures of different type as well as differing intensity, and adequately address scaling and normalization across blocks. These weight functions can then be used to construct blending functions for algebraic redistribution, interpolation functions for unstructured grid generation, forcing functions to attract/repel points in an elliptic system, or to trigger local refinement, based upon application of an equidistribution principle. The popularity of solution-adaptive techniques is growing in tandem with unstructured methods. The difficultly of precisely controlling mesh densities and orientations with current unstructured grid generation systems has driven the use of solution-adaptive meshing. Use of derivatives of density or pressure are widely used for construction of such weight functions, and have been proven very successful for inviscid flows with shocks. However, less success has been realized for flowfields with viscous layers, vortices or shocks of disparate strength. It is difficult to maintain the appropriate mesh point spacing in the various regions which require a fine spacing for adequate resolution. Mesh points often migrate from important regions due to refinement of dominant features. An example of this is the well know tendency of adaptive methods to increase the resolution of shocks in the flowfield around airfoils, but in the incorrect location due to inadequate resolution of the stagnation region. This problem has been the motivation for this research.

Thornburg, Hugh J.↗

Generation and application of the equations of condition for high order Runge-Kutta methods

This thesis develops the equations of condition necessary for determining the coefficients for Runge-Kutta methods used in the solution of ordinary differential equations. The equations of condition are developed for Runge-Kutta methods of order four through order nine. Once developed, these equations are used in a comparison of the local truncation errors for several sets of Runge-Kutta coefficients for methods of order three up through methods of order eight.

Haley, D. C.↗

Classical seventh-, sixth-, and fifth-order Runge-Kutta-Nystrom formulas with stepsize control for general second-order differential equations

Runge-Kutta-Nystrom formulas of the seventh, sixth, and fifth order were derived for the general second order (vector) differential equation written as the second derivative of x = f(t, x, the first derivative of x). The formulas include a stepsize control procedure, based on a complete coverage of the leading term of the local truncation error in x, and they require no more evaluations per step than the earlier Runge-Kutta formulas for the first derivative of x = f(t, x). The developed formulas are expected to be time saving in comparison to the Runge-Kutta formulas for first-order differential equations, since it is not necessary to convert the second-order differential equations into twice as many first-order differential equations. The examples shown saved from 25 percent to 60 percent more computer time than the earlier formulas for first-order differential equations, and are comparable in accuracy.

Fehlberg, E.↗

Time elements

Time elements are introduced for use with Sundman time transformations of the type dt = r(alpha)ds for satellite equations of motion. Two time elements are given, one providing maximum accuracy for alpha = 1, the other for alpha = 2. Time elements and time transformations reduce local truncation error and Liapunov (in track) instability, and provide analytical step size control. Numerical results show accuracy improvements of more than one order of magnitude when time elements are employed with time transformations in the numerical integration of the satellite equations, compared with using time transformations alone.

Nacozy, P. E.↗

Time elements

Time elements are introduced for use with Sundman time transformations for satellite equations of motion. Two time elements are presented, one providing maximum accuracy when the exponent of the governing equation equals 1, the other when the exponent equals 2. Time elements and time transformations reduce local truncation error and Liapunov (in-track) instability, and provide analytical step size control. Numerical results show accuracy improvements of more than one order of magnitude when time elements are employed with time transformations in the numerical integration of the satellite equations, compared with using time transformations alone.

Nacozy, P. E.↗

Optimal nodal point distribution for improved accuracy in computational fluid dynamics

In applying finite-difference techniques to flow field problems, the accuracy attained for a fixed number of node points can be improved using unequally-spaced node points. The distribution of these node points is chosen here by minimizing a measure of local truncation error with respect to the parameters which define a transformation between the computational space of equally-spaced node points and the physical space of unequally-spaced node points. The problem then becomes a nonlinear programming problem. Numerical results are presented for two one-dimensional test problems: the Blasius boundary layer problem and the inviscid Burgers' equation.

Pierson, B. L.↗

An accurate method for two-point boundary value problems

A second-order method for solving two-point boundary value problems on a uniform mesh is presented where the local truncation error is obtained for use with the deferred correction process. In this simple finite difference method the tridiagonal nature of the classical method is preserved but the magnitude of each term in the truncation error is reduced by a factor of two. The method is applied to a number of linear and nonlinear problems and it is shown to produce more accurate results than either the classical method or the technique proposed by Keller (1969).

Walker, J. D. A.↗

Grid generation for time dependent problems: Criteria and methods

The problem of generating local mesh refinements when solving time dependent partial differential equations was examined. The problem of creating an appropriate grid, given a mesh function h defined over the spatial domain is discussed. A data structure which permits efficient use of the resulting grid is described. A good choice for h is an estimate of the local truncation error, and several ways to estimate it are discussed. The efficiency and implementation problems of these error estimates were compared.

Berger, M.↗

Influence of boundary approximations and conditions on finite difference solutions

Numerical representations of boundary approximations and conditions for three problems are investigated to determine the resulting global accuracy of the steady state solution. Numerical accuracy with various boundary approximations is determined for quasi-one-dimensional inviscid flow in a duct with the interior grid points evaluated using the MacCormack scheme. When an extrapolation approximation with first order local truncation error is used, the global second order accuracy of the difference scheme can be destroyed. For one dimensional flow in a porous medium, an implicit midpoint difference scheme which is consistent with the boundary conditions is developed without the need of boundary approximations. A dissipative model problem is solved with the boundary conditions discretized with first and second order accuracy. The overall second order accuracy of the difference scheme is destroyed if first order numerical representation of one of the boundary conditions is used. With a boundary approximation, the second order global accuracy of the model problem is retained if either second order extrapolation or first order representation of the governing equation is used.

Blottner, F. G.↗

Recent advances in methods for numerical solution of O.D.E. initial value problems

In the mathematical modeling of physical systems, it is often necessary to solve an initial value problem (IVP), consisting of a system of ordinary differential equations (ODE). A typical program produces approximate solutions at certain mesh points. Almost all existing codes try to control the local truncation error, while the user is really interested in controlling the true or global error. The present investigation provides a review of recent advances regarding the solution of the IVP, giving particular attention to stiff systems. Stiff phenomena are customarily defined in terms of the eigenvalues of the Jacobian. There are, however, some difficulties connected with this approach. It is pointed out that an estimate of the Lipschitz constant proves to be a very practical way to determine the stiffness of a problem.

Bui, T. D.↗

High accuracy solutions of incompressible Navier-Stokes equations

In recent years, high accuracy finite difference approximations were developed for partial differential equations of elliptic type, with particular emphasis on the convection-diffusion equation. These approximations are of compact type, have a local truncation error of fourth order, and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations. These high accuracy approximations are extended to the solution of Navier-Stokes equations. Solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors. It is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows.

Gupta, Murli M.↗

An efficient HZETRN (a galactic cosmic ray transport code)

An accurate and efficient engineering code for analyzing the shielding requirements against the high-energy galactic heavy ions is needed. The HZETRN is a deterministic code developed at Langley Research Center that is constantly under improvement both in physics and numerical computation and is targeted for such use. One problem area connected with the space-marching technique used in this code is the propagation of the local truncation error. By improving the numerical algorithms for interpolation, integration, and grid distribution formula, the efficiency of the code is increased by a factor of eight as the number of energy grid points is reduced. The numerical accuracy of better than 2 percent for a shield thickness of 150 g/cm(exp 2) is found when a 45 point energy grid is used. The propagating step size, which is related to the perturbation theory, is also reevaluated.

Shinn, Judy L.↗

Finite element/finite volume approaches with adaptive time stepping strategies for transient thermal problems

An adaptive time stepping strategy for transient thermal analysis of engineering systems is described which computes the time step based on the local truncation error with a good global error control and obtains optimal time steps to be used during the analysis. Combined mesh partitionings involving FEM/FVM meshes based on physical situations to obtain numerically improved physical representations are also proposed. Numerical test cases are described and comparative pros and cons are identified for practical situations.

Mohan, Ram V.↗

A new flux-conserving numerical scheme for the steady, incompressible Navier-Stokes equations

This paper is concerned with the continued development of a new numerical method, the space-time solution element (STS) method, for solving conservation laws. The present work focuses on the two-dimensional, steady, incompressible Navier-Stokes equations. Using first an integral approach, and then a differential approach, the discrete flux conservation equations presented in a recent paper are rederived. Here a simpler method for determining the flux expressions at cell interfaces is given; a systematic and rigorous derivation of the conditions used to simulate the differential form of the governing conservation law(s) is provided; necessary and sufficient conditions for a discrete approximation to satisfy a conservation law in E2 are derived; and an estimate of the local truncation error is given. A specific scheme is then constructed for the solution of the thin airfoil boundary layer problem. Numerical results are presented which demonstrate the ability of the scheme to accurately resolve the developing boundary layer and wake regions using grids which are much coarser than those employed by other numerical methods. It is shown that ten cells in the cross-stream direction are sufficient to accurately resolve the developing airfoil boundary layer.

Scott, James R.↗