A higher-order error estimation framework for finite-volume CFD
Introduction
In recent years, computational fluid dynamics (CFD) has been extensively employed in the design of aerospace vehicles. This increased reliance upon CFD necessitates the development of accurate and robust error estimation techniques to avoid making decisions based on inaccurate information. The dominant errors in any given CFD simulation are typically modeling errors and numerical errors, with the latter being the focus of this paper. Numerical errors are generated through a variety of mechanisms including finite-precision arithmetic, incomplete convergence of iterative procedures, and discretization of the governing equations. With advances in numerical algorithms and modern computing architectures, the errors associated with discretization, or discretization errors, are often the primary source of numerical errors.
Discretization error can be difficult to quantify as the generation, transport, and diffusion of these errors is a highly nonlinear function of the mesh and flow solution. Historically, discretization error has been estimated using Richardson extrapolation [1] which establishes an estimate using discrete solutions from multiple systematically-refined grid levels. Roache's grid convergence index (GCI) [2] can then be used in conjunction with Richardson extrapolation to establish an uncertainty bound about a given discrete solution. While Richardson extrapolation and GCI are non-intrusive to existing flow solvers and can easily be performed as post-processing steps, they are often not practical for large problems as meshing and solving on a family of grids can be computationally expensive. Also, asymptotic grid convergence for all grid levels is necessary to obtain an accurate error estimate. This requirement can be challenging to meet especially for complex flow fields [3], [4]. Another method known as defect correction [5], [6], [7], [8], [9] can provide an error estimate by re-solving the governing equations with a truncation error estimate added as a source term. In this way, the discrete solution is driven toward the exact solution to the PDEs. The difference between discrete solutions can then be used as an error estimate. Although defect correction is generally less expensive than Richardson extrapolation, a significant computational cost is incurred from the need to re-solve the original nonlinear problem, albeit with a better initial guess. More recently, advancements have been made with the introduction of adjoint methods [10], [11], [12], [13], [14] and error transport equation (ETE) methods [15], [16], [17]. Adjoint methods provide an error estimate and adaptation indicator for any output functional of interest. However, adjoint methods can be prohibitively expensive when multiple functionals are required, since an adjoint solution must be obtained for each functional of interest. Adjoint methods also do not provide local discretization error estimates. ETE methods, on the other hand, can simultaneously provide local discretization error estimates and error estimates for multiple functionals due to an equivalency that exists between adjoint methods and ETE methods [18], [19].
The vast majority of error estimation techniques currently available require an accurate estimate of truncation error. Truncation error, which is defined as the difference between the continuous and discrete governing equations, acts as the source term for the generation, transport, and diffusion of discretization errors. Common truncation error estimation methods include traditional analysis via operator expansion [15] (i.e., the modified equation approach), embedded grid methods [20], higher-order discrete residual methods [21], and continuous residual methods [22]. Once an accurate truncation error estimate has been computed, functional errors and discretization errors may be estimated. Under certain conditions, error estimates can be obtained which converge to the true error at a faster rate than the primal order of accuracy. This has been exploited within the adjoint community for functional error estimation [12], [23]. Within the ETE community, Hay and Visonneau [24] as well as Banks et al. [25] observed higher-order (i.e., greater than second-order accurate) error estimates for the continuous implementation of the ETE. Similarly, Yan and Ollivier-Gooch [26], [27] obtained higher-order error estimates using the discrete ETE on unstructured grids, albeit with a higher-order discretization of the ETE.
In this work, a comprehensive ETE framework is presented for estimating discretization error and increasing solution accuracy. The goal of this paper is to improve each component of this error estimation framework including solution reconstruction, truncation error estimation, and discretization error estimation. Furthermore, we aim to demonstrate and explain continued deficiencies of error estimators for finite-volume schemes. For truncation error estimation, a technique is presented which combines aspects of both higher-order residual methods and continuous residual methods. In particular, we examine how enforcing boundary conditions on the reconstruction affects the accuracy of discretization error estimates. For discretization error estimation, we extend the higher-order properties of discrete adjoint methods to discrete ETE methods using adjoint/ETE equivalency [19]. In contrast to other error estimators which require higher-order discretizations [26], the ETE are discretized at the formal order of the primal problem to improve computational efficiency and reduce the adoption cost of the ETE as an error estimator. On “structured grids” (i.e., grids with smoothly varying grid metrics), ETE error estimates are shown to converge to the true discretization error at a higher-order rate. On “unstructured grids” (i.e., grids without smoothly varying grid metrics), the observed order of accuracy of ETE error estimates degrades to that of the original primal problem. This degradation has been demonstrated by previous authors [26], [28] when the ETE are discretized at the same order of the primal problem. However, we further demonstrate that this accuracy loss is a direct result of an inaccurate truncation error estimate, not because the ETE are discretized at the formal order of the primal problem. Suggestions for recovering the higher-order properties of the error estimate are also provided. ETE error estimates are applied as corrections to the primal solution in a similar manner to adjoint methods for output functionals in order to increase the observed order of accuracy of the primal solution. Efficiency and accuracy comparisons are made between these ETE-based corrections and a conventional higher-order method. The proposed ETE framework is applied to 1D and 2D inviscid and viscous flow problems.
Section snippets
Finite-volume discretization
Consider a general conservation law of the form defined over the domain where is the vector of conserved quantities, is a flux tensor, and is a source term. In general, the flux tensor and source term in Eq. (1) may also be functions of , but this dependency has been omitted here for the sake of brevity. The solution domain, Ω, is decomposed into a set of non-overlapping control volumes, , such that where is the
Pseudo-CR truncation error estimation
We propose a truncation error estimation technique for finite-volume methods which combines aspects of both higher-order residual methods and continuous residual (CR) methods. The goal of this approach is to maintain the accuracy of CR methods while leveraging the implementation advantages of higher-order residual methods. With CR truncation error estimation, the discrete solution is first projected onto an rth order polynomial space. It is then assumed that the reconstructed solution, ,
Quasi-1D Euler equations
The quasi-1D Euler equations are a statement of conservation of mass, momentum, and energy for a fluid where the computational domain is one dimensional with a given area distribution, . The quasi-1D Euler equations in strong, conservation form are where u is the vector of conserved variables, is the vector of inviscid fluxes, and is a source term. The conserved variables, inviscid fluxes, and source term are given by
Conclusions
In this work, a comprehensive ETE framework for estimating discretization error and improving solution accuracy was presented. A truncation error estimation technique was proposed which combines aspects of higher-order residual methods and continuous residual methods. In particular, the continuous boundary conditions for the primal problem were enforced on the reconstruction of the discrete solution to improve the accuracy of truncation error estimates. Additionally, the equivalence between
Acknowledgements
This work was supported by the Collaborative Center for Aeronautical Sciences with Dr. John Benek of the Air Force Research Laboratory in Dayton, Ohio serving as the program manager. Additional support was provided by the NASA Graduate Aeronautics Scholarship (grant number NNX16AT28H) as part of the Aeronautics Scholarship and Advanced STEM Training and Research Fellowship (AS&ASTAR) Program. We wish to thank Dr. Michael Park and Dr. Joseph Derlaga of NASA Langley Research Center for serving as
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