Patrick J. Roache

Perspective: A Method for Uniform Reporting of Grid Refinement Studies

We propose the use of a Grid Convergence Index (GCI) for the uniform reporting of grid refinement studies in Computational Fluid Dynamics. The method provides an objective asymptotic approach to quantification of uncertainty of grid convergence. The basic idea is to approximately relate the results from any grid refinement test to the expected results from a grid doubling using a second-order method. The GCI is based upon a grid refinement error estimator derived from the theory of generalized Richardson Extrapolation. It is recommended for use whether or not Richardson Extrapolation is actually used to improve the accuracy, and in same cases even if the conditions for the theory do not strictly hold

Code Verification by the Method of Manufactured Solutions

Verification of Calculations involves error estimation, whereas Verification of Codes involves error evaluation, from known benchmark solutions. The best benchmarks are exact analytical solutions with sufficiently complex solution structure; they need not be realistic since Verification is a purely mathematical exercise. The Method of Manufactured Solutions (MMS) provides a straightforward and quite general procedure for generating such solutions. For complex codes, the method utilizes Symbolic Manipulation, but here it is illustrated with simple examples. When used with systematic grid refinement studies, which are remarkably sensitive, MMS produces strong Code Verifications with a theorem-like quality and a clearly defined completion point

Verification of Codes and Calculations

Background discussion, definitions, and descriptions are given for some terms related to confidence building in computational fluid dynamics. The two principal distinctions made are between verification vs validation and between verification of codes vs verification of individual calculations. Also discussed are numerical errors vs conceptual modeling errors; iterative convergence vs grid convergence (or residual accuracy vs discretization accuracy); confirmation, calibration, tuning, and certification; error taxonomies; and customer illusions vs customer care. Emphasis is given to rigorous code verification via systematic grid convergence using the method of manufactured solutions, and a simple method for uniform reporting of grid convergence studies using the Grid Convergence Index (GCI). Also discussed are surrogate single-grid error indicators.

Perspective: Validation—What Does It Mean?

Ambiguities, inconsistencies, and recommended interpretations of the commonly cited definition of validation for computational fluid dynamics codes/models are examined. It is shown that the definition-deduction approach is prone to misinterpretation, and that bottom-up descriptions rather than top-down legalistic definitions are to be preferred for science-based engineering and journal policies, though legalistic definitions are necessary for contracts.

A pseudo-spectral FFT technique for non-periodic problems

Abstract A technique is developed for the use of pseudo-spectral Fast Fourier Transform methods for non-periodic time-dependent problems in fluid dynamics. Called “reduction to periodicity,” it involves the evaluation of a polynomial function which approximates the departure from smooth periodicity of the dependent variable distribution at each time level. The FFT is then applied to the residual distribution. The accuracy is demonstrated in several one-dimensional problems. Stability and iterative convergence are demonstrated in one-dimensional problems with first order, second order, and fourth order time differencing, and in two-dimensional problems with first-order time differencing.

Mathematical aspects of variational grid generation II

Abstract Variational grid generation techniques are now used to produce grids suitable for solving numerical partial differential equations in irregular geometries. In this paper the existence and uniqueness of solutions of the volume and smoothness problems that are used in variational grid generation are studied. An analysis of the Euler-Lagrange (EL) equations near the identity shows that the volume problem is difficult. These variational problems use a reference grid to specify the properties of the desired grid. Replication of reference grid properties is analyzed. Examples are given that show the effectiveness of the reference grid concept.

Parameter estimation in variational grid generation

Two approaches to numerical variational grid generation use a linear combination of three variational problems to control the properties of the grid. Previously the parameters in the linear combination were determined by experimentation. Good estimates for these parameters can be obtained from simple model problems.

Using MACSYMA to write FORTRAN subroutines

This letter describes some ongoing work that uses the symbol manipulator MACSYMA to write FORTRAN subroutines which are incorporated into a large finite difference code that is used to solve boundary value problems for elliptic partial differential equations. The boundary value problems model the steady state of certain physical devices such as the electric field in a laser cavity or the flow around a blade in a turbine . The devices that interest us are those that occupy a fairly complicated region in three-dimensional space or, at best, have sufficient symmetry so that they can be modelled using a complicated two-dimensional region . The partial differential equations that are used to model the physics of the device are called the hosted equations. The hosted equations tend to be rather simple non-linear elliptic equations . However, our methods will apply to the most complicated hosted equations . The problem is complete when appropriate boundary conditions are specified: These problems are difficult to solve because of the complicated geometry of the region .

Variational curve and surface grid generation

Abstract Variational algorithms that control the lengths of grid lines, cell areas, and the orthogonality of grid lines can be used for generating boundary-conforming grids on surfaces. Additional geometric control is provided by using a reference grid, while solution adaptivity is achieved by using weights. In a typical application, the reference grid can be used to produce an exponential compression of the grid at a boundary, while the solution adaptive weights are used to make the grid spacing inversely proportional to the gradient (when the gradient is large) of some solution being computed on the grid. The grid is adapted on both the interior and boundary of the surface. The algorithm performs these tasks with exceptional precision, as demonstrated in the examples presented here.

Using VAXIMA to Write FORTRAN Code

This paper describes the symbol manipulation aspects of a project that produced a large FORTRAN program that is now used to model lasers and other physical devices. VAXIMA (MACSYMA) was used to write subroutines that were combined with standard software to produce the full program.