Tyrone S. Phillips

Residual Methods for Discretization Error Estimation

The largest and most difficult numerical approximation error to estimate is discretization error. This study investigates the accuracy of two different residual-based discretization error estimators: discretization error transport equations and defect correction methods. Residual methods are a category of error estimators that use a discrete solution and information about the problem being solved (either the differential or the discrete equations) to calculate an error estimate using only one grid level. As few as two grid levels are needed if the reliability of the error estimate is to be assessed. This is advantageous because the reliability of all discretization error estimators require that the numerical solution, or solutions, be solved on sufficiently fine grids, which is often difficult to achieve for complex scientific computing applications. Two different linearization methods for the discretization error transport equation will also be studied. The estimated discretization error for these residual-based methods will be compared to Richardson extrapolation using exact solutions to 1D Burgers’ equation and the 2D Euler equations. The exact solutions to the Euler equations include two manufactured solutions, Ringlebs flow, and a supersonic vortex flow.

Finite Volume Solution Reconstruction Methods For Truncation Error Estimation

The numerical solution to differential equations results in a discrete solution space for the finite volume and finite difference discretization methods. For various reasons, it can be necessary to prolong the solution from a discrete space to a continuous space. The prolongation to a continuous space can be done using various curve-fitting methods which adds an additional level of approximation to the solution. The allowable error of a prolongation operation depends on the specific task required by the user. In this paper we investigate various prolongation methods and identify the minimal requirements specifically for the purpose of truncation error estimation (the difference between the discrete and integral governing equations) for finite volume methods. The reconstruction methods investigated will include k-exact and ENO methods. Truncation error estimation for 1D Burgers’ equation and the k-exact method suggest that the minimum polynomial order is dependent on the highest derivatives in the truncation error expression and, therefore, the discretization scheme. The effect of different reconstruction methods on truncation error estimation is investigated and the minimum polynomial order for accurate truncation error estimation is identified for the Euler equations and is found to be second-order for the weak formulation and third-order for the strong formulation.

SENSEI Computational Fluid Dynamics Code: A Case Study in Modern Fortran Software Development

interface subroutine calculate( a_in, b_in, c_out ) use set_precision, only : dr real(dr), intent(in) :: a_in, b_in real(dr), intent(out) :: c_out end subroutine calculate end interface procedure(calculate) :: calculation_routine public :: calculation_routine public :: subroutine_1

Evaluation of Extrapolation-Based Discretization Error and Uncertainty Estimators

The largest and most difficult numerical approximation error to estimate is discretization error. This study investigates the accuracy of various Richardson extrapolation-based discretization error and uncertainty estimators for problems in computational fluid dynamics. Richardson extrapolation uses two solutions on systematically refined grids to estimate the exact solution to the partial differential equations and is accurate only in the asymptotic range (i.e., when the grids are sufficiently fine). The uncertainty estimators that will be investigated are different implementations and the Grid Convergence Index and include a globally averaged observed order of accuracy, a least squares calculation of the observed order of accuracy, the Factor of Safety method, and the Correction Factor method. Two twodimensional, inviscid, supersonic flow fields with exact solutions and a twodimensional, turbulent flat plate are used to evaluate the extrapolation-based discretization error and uncertainty estimators. The conservativeness (percent of cases where the exact solution is bracketed by the estimated uncertainties) and the effectivity (how accurately the discretization error estimate approximates the exact error) are used to assess the relative merits of the different approaches. The overall trend suggests a trade-off between conservativeness and effectivity (e.g. the most conservative uncertainty estimator was the least accurate and vise-versa). The least squares method showed the best trade-off between conservativeness and effecitivity under specific conditions.

Adjoint and Truncation Error Based Adaptation for Finite Volume Schemes with Error Estimates

In addition to design, control, and optimization applications, adjoint methods can be used to provide discretization error estimation and solution adaptation for solution functionals. In this paper, adaptation based on estimates of truncation error is coupled with adjoint-based error estimation to provide improved estimates of discretization error in solution functionals. Comparisons between different types of adaptation indicators and error estimation techniques are made for the two-dimensional Euler equations.

A New Extrapolation-Based Uncertainty Estimator for Computational Fluid Dynamics

A new Richardson extrapolation-based uncertainty estimator is developed which utilizes a global order of accuracy. Various metrics are used to quantitatively evaluate the uncertainty estimators the most important of which is conservativeness which the percentage of uncertainty estimates which bracket the true solution. Another metric is the effectivity index which is a relative measure of accuracy (of either the error or the uncertainty estimate) compared to the true solution. Conservativeness and effectivity of the discretization uncertainty estimators are used to assess the proposed uncertainty estimator compared to other uncertainty estimators such as the GCI and the Factor of Safety method. Four two-dimensional, inviscid flow fields with exact solutions are used to calibrate the uncertainty estimator. Numerous additional solutions computed using different computational fluid dynamics codes with exact solutions and a zero pressure gradient, turbulent flat plate numerical benchmark are used to test the new uncertainty estimator. The proposed uncertainty estimator is developed with a focus on local variables and shows significant improvement in effectivity and conservativeness compared to existing extrapolation-based uncertainty estimates.

Richardson Extrapolation-Based Discretization Uncertainty Estimation for Computational Fluid Dynamics

This study investigates the accuracy of various Richardson extrapolation-based discretization error and uncertainty estimators for problems in computational fluid dynamics. Richardson extrapolation uses two solutions on systematically refined grids to estimate the exact solution to the partial differential equations and is accurate only in the asymptotic range (i.e., when the grids are sufficiently fine). The uncertainty estimators investigated are variations of the Grid Convergence Index and include a globally averaged observed order of accuracy, the Factor of Safety method, the Correction Factor method, and Least-Squares methods. Several 2D and 3D applications to the Euler, Navier-Stokes, and Reynolds-Averaged Navier-Stokes with exact solutions and a 2D turbulent flat plate with a numerical benchmark are used to evaluate the uncertainty estimators. Local solution quantities (e.g. density, velocity, and pressure) have much slower grid convergence on coarser meshes than global quantities resulting in non-asymptotic solutions and inaccurate Richardson extrapolation error estimates; however, an uncertainty estimate may still be required. The uncertainty estimators are applied to local solution quantities to evaluate accuracy for all possible types of convergence rates. Extensions were added where necessary for treatment of cases where the local convergence rate is oscillatory or divergent. The conservativeness and effectivity of the discretization uncertainty estimators are used to assess the relative merits of the different approaches.

Optimal Mesh Adaption for Burgers' Equation

Optimization methods are used for mesh adaption with the goal of reducing discretization error. Given that truncation error is the local source of discretization error, an objective function is created to minimize the truncation error therefore effectively minimizing the discretization error on a given mesh. As a proof of concept, the objective function and objective function gradients are calculated assuming that the exact solution and the exact truncation error are known. The mesh adaption procedure is explored using 1D and 2D Burgers’ equation for various Reynolds numbers. Other issues including adaption efficiency and global minimums are also addressed. For both 1D and 2D Burgers’ equation, results show several orders of magnitude reduction in discretization error as compared to that found on a uniform mesh.

Numerical Benchmark Solutions for Laminar and Turbulent Flows

Numerical benchmark solutions are numerical solutions that have been computed using a verified code and with a high degree of rigorously assessed numerical accuracy. They can bridge the gap between simple problems where the analytic solution to the differential equations is known and more complex problems where exact solutions are not known. In particular, benchmark numerical solutions can be used for code verification (i.e., algorithm and code correctness), assessing discretization error estimators, and evaluating solution adaptation strategies. The requirements for establishing a numerical benchmark solution are discussed. A numerical benchmark is created for a turbulent flat plate using the Spalart-Allmaras Reynolds-Averaged Navier-Stokes (RANS) turbulence model. Three computational fluid dynamics codes are employed to provide additional confidence in the final benchmark solution: Loci-CHEM, FUN3D, and CFL3D. A numerical benchmark is also created for a supersonic manufactured solution and Ringleb’s flow, both with known exact solutions, to ensure that the recommended guidelines for generating a numerical benchmark solution are sufficient.

Discretization error estimation and exact solution generation using the method of nearby problems.

The Method of Nearby Problems (MNP), a form of defect correction, is examined as a method for generating exact solutions to partial differential equations and as a discretization error estimator. For generating exact solutions, four-dimensional spline fitting procedures were developed and implemented into a MATLAB code for generating spline fits on structured domains with arbitrary levels of continuity between spline zones. For discretization error estimation, MNP/defect correction only requires a single additional numerical solution on the same grid (as compared to Richardson extrapolation which requires additional numerical solutions on systematically-refined grids). When used for error estimation, it was found that continuity between spline zones was not required. A number of cases were examined including 1D and 2D Burgers equation, the 2D compressible Euler equations, and the 2D incompressible Navier-Stokes equations. The discretization error estimation results compared favorably to Richardson extrapolation and had the advantage of only requiring a single grid to be generated.