Error Transport Equation Research Articles

Discretization Error Estimation Using the Unsteady Error Transport Equations

Abstract Computational Fluid Dynamics (CFD) has gained significant utility in the analysis of diverse fluid flow scenarios, thanks to advances in computational power. Accurate error estimation techniques are crucial to ensure the reliability of CFD simulations, as errors can lead to misleading conclusions. This study focuses on the estimation of discretization errors in time-dependent simulations, building upon prior work addressing steady-state problems Wang et al. (2020, “Error Transport Equation Implementation in the Sensei CFD Code,” AIAA Paper No. 2020-1047). In this research, we employ unsteady error transport equations (ETE) to generate localized discretization error estimates within the framework of the finite volume CFD code SENSEI. For steady-state problems, the ETE only need to be solved once after the solution has converged, whereas the unsteady ETE need to be co-advanced with the primal solve. To enhance efficiency, we adopt a one-sided temporal stencil and develop a modified iterative correction process tailored to the unsteady ETE. The time-marching schemes utilized encompass second-order accurate singly diagonally implicit Runge–Kutta (SDIRK) and second-order backward differentiation formula (BDF2), both being implicit. To rigorously assess the accuracy of our error estimates, all test cases feature known analytical solutions, facilitating order-of-accuracy evaluations. Two test cases are considered: the 2D convected vortex for inviscid flow and a cross-term sinusoidal (CTS) manufactured solution for viscous flow. Results indicate higher-order convergence rates for the 2D convected vortex test case even when iterative correction is not applied, with similar observations in the CTS case, albeit not at the finest grid levels. Although the current implementation of iterative correction exhibits lower stability compared to the primal solve, it generally enhances the discretization error estimate. Notably, after iterative correction, the discretization error estimate for the unsteady ETE achieves higher-order accuracy across all grid levels in the 2D CTS manufactured solution.

Error transport equations implementation for discontinuous Galerkin methods

The procedure for performing higher-order discretization error estimation for Discontinuous Galerkin (DG) methods using Error Transport Equations (ETE) is presented in this work. The results obtained using the linearized ETE and the nonlinear ETE are compared for several steady state inviscid test cases containing completely smooth cases and cases with singularities or discontinuities. Prior work with the linearized version of ETE achieved improved results by using a relinearization process. This process is generalized to the nonlinear version of ETE and is more appropriately termed iterative correction when applied to both the linear and nonlinear versions. For smooth test cases, the discretization error estimates show expected order of accuracy which is higher than the order of accuracy of the primal solution. This holds for both the linearized and the nonlinear ETE. Discontinuities and singularities in the solution cause trouble in the discretization error estimation process. Discussion is made on the behavior of the ETE error estimates for non-smooth test cases to figure out potential solutions to the existing problems.

Iterated discretization error transport equations for laminar and turbulent flows

AbstractThe focus of this work is on discretization error estimation for finite‐volume schemes using iterated forms of the linear and nonlinear error transport equations. The accuracy of the discretization error estimates is studied by applying the error estimates as corrections to the primal problem and calculating the observed order of accuracy of the corrected solution. The test cases used in this work include two‐dimensional Cross Term Sinusoidal manufactured solutions for laminar flow and turbulent flow modeled by the Spalart–Allmaras and Menter's Shear Stress Transport model and a boundary layer manufactured solution for the Spalart–Allmaras model. The discretization error estimates of the linearized error transport equations and the nonlinear error transport equations are compared. Higher‐order accurate corrected solutions have been obtained which shows that the discretization error estimates are sufficiently accurate.

Adaptive Refinement Method for Solving Ion-Flow Field of HVDC Transmission Line

To improve the accuracy for solving the ion-flow field problem, an adaptive refinement method based on the streamline upwind Petrov–Galerkin (SUPG) formula is proposed. The error transport equation has been decomposed to a series of local boundary value problems and solved by the method of local characteristic. An equidistributional strategy is applied to steer the refinement. Numerical examples show that the results using the grid generated by the proposed method are more accurate and efficient than the ones with the regular refinement strategy. An example of application shows that this method is suitable for simulating the ion-flow problem with a complex geometry.

Relinearization of the error transport equations for arbitrarily high-order error estimates

A higher-order accurate discretization error estimation procedure for finite-volume schemes is presented. Discretization error estimates are computed using the linearized error transport equations (ETE). ETE error estimates are applied as a correction to the primal solution. The ETE are then relinearized about the corrected primal solution, and discretization error estimates are recomputed. This process, referred to as ETE relinearization, is performed in an iterative manner to successively increase the accuracy of discretization error estimates. Under certain conditions, ETE relinearization is shown to correct error estimates, or equivalently the entire primal solution, to higher-order accuracy. In terms of computational cost, ETE relinearization has a competitive advantage over conventional higher-order discretizations when used as a form of defect correction for the primal solution. Furthermore, ETE relinearization is shown to be particularly useful for problems where the error incurred by the linearization of the ETE cannot be neglected. Results are presented for several steady-state inviscid and viscous flow problems using both structured and unstructured meshes.

A higher-order error estimation framework for finite-volume CFD

Computational fluid dynamics is an invaluable tool for both the design and analysis of aerospace vehicles. Reliable error estimation techniques are needed to ensure that simulation results are accurate enough to be used in engineering decision-making processes. In this work, a framework for estimating error and improving solution accuracy is presented. A linearized error transport equation (ETE) is used to estimate local discretization errors. A truncation error estimation technique is proposed which combines aspects of higher-order residual methods and continuous residual methods. The equivalence between adjoint and ETE methods for functional error estimation is demonstrated. Using adjoint/ETE equivalence, the higher-order properties of adjoint methods are extended to ETE methods. Consequently, ETE error estimates are shown to converge to the true discretization error at a higher-order rate. ETE error estimates are then used to correct the entire primal solution, and by extension, all output functionals, to higher order. The computational advantages of this ETE approach are discussed. Results are presented for 1D and 2D inviscid and viscous flow problems on grids with smoothly varying and non-smoothly varying grid metrics.

Application of Exactly Linearized Error Transport Equations to Sonic Boom Prediction Workshop

The computational fluid dynamics (CFD) prediction workshops sponsored by the AIAA have created invaluable opportunities to discuss the predictive capabilities of CFD in areas in which it has struggled, for example, sonic boom prediction. While there are many factors that contribute to disagreement between simulated and experimental results, such as modeling or discretization errors, quantifying the errors contained in a simulation is important for those who make decisions based on the computational results. The linearized error transport equations (ETE) combined with a truncation error estimate is a method to quantify one source of error. The ETE are implemented with a complex-step method to provide an exact linearization with minimal source code modifications to CFD and multidisciplinary analysis methods including atmospheric propagation of sonic boom signatures. Uniformly refined grids from the 2nd AIAA Sonic Boom Prediction Workshop demonstrate the utility of the linearized ETE for multidisciplinary analysis, capturing the general trends of the discretization error when flow features are resolved while also demonstrating issues related to error prediction near strong shocks or when under-resolved flow features are present.

Applications of the Unsteady Error Transport Equation on Unstructured Meshes

A numerical study of using the error transport equation, an auxiliary problem to a set of model equations, is performed to obtain higher-order accurate error estimates and corrections for applications in unsteady compressible flow. In many such applications, a functional of the solution is often examined as a proxy for the accuracy of the solution itself. Several measures of time-dependent functionals are used for test cases that have a periodic steady-state solution. The approach is verified by examining unsteady functionals for diffusion and advection model problems, and then the error transport method is applied to the von Kármán vortex shedding test case, a difficult problem to establish accuracy properties on its own. It was found that scalar measures of the unsteady functionals not only give the order of accuracy that would be expected from a primal discretization only but also the expected higher-order accuracy if the functionals were computed using the solutions corrected by this accurate error estimate, all without the otherwise necessary requirement of discretizing to higher order in both time and space for the primal problem. The results are consistent with previous studies on solution error using simple test cases with manufactured solutions.

On Efficiently Obtaining Higher Order Accurate Discretization Error Estimates for Unstructured Finite Volume Methods Using the Error Transport Equation

A numerical estimation of discretization error for steady compressible flow solutions is performed using the error transport equation (ETE). There is a deficiency in the literature for obtaining efficient, higher order accurate error estimates for finite volume discretizations using nonsmooth unstructured meshes. We demonstrate that to guarantee sharp, higher order accurate error estimates, one must discretize the ETE to a higher order than the primal problem, a requirement not necessary for uniform meshes. Linearizing the ETE can limit the added cost, rendering the overall computational time competitive, while retaining accuracy in the error estimate. For the Navier–Stokes equations, when the primal solution is corrected using this error estimate, for the same level of solution accuracy the overall computational time is more than two times faster compared to solving the higher order primal problem. In addition, our scheme has robustness advantages, because we solve the primal problem only to lower order.

Towards higher order discretization error estimation by error transport using unstructured finite-volume methods for unsteady problems

Abstract A numerical estimation of discretization error for solutions to unsteady laminar compressible flow equations is performed using the error transport equation (ETE) on unstructured meshes. This method is an extension to our previous work on steady problems, where it was found that solving the ETE can be more efficient and robust than solving the higher order primal problem. Computing the time-dependent ETE source term accurately is critical to the accuracy of the discretization error estimate, and several methods of doing so are considered. It was found that computing the ETE source term directly by a finite-difference approximation in time gives accurate error estimates, which we show is equivalent to an accurate corrected solution. A truncation error analysis was performed for the ETE to determine the expected accuracy of the error estimate, where a term that mixes the space and time discretization was observed. Although more stringent requirements for error estimation are needed when using unstructured meshes, constant time steps can be used and the best schemes we found were still able to attain an estimate of the discretization error that is higher order accurate in space and time, without discretizing both to higher order. Furthermore, unlike unsteady adjoints, the ETE requires only one other auxiliary equation to be solved, agnostic to the choice and number of output functionals, and co-advancing with the primal problem requires the storage of only local solutions in time, reducing memory requirements.

Prediction of discretization error using the error transport equation

This study focuses on an approach to quantify the discretization error associated with numerical solutions of partial differential equations by solving an error transport equation (ETE). The goal is to develop a method that can be used to adequately predict the discretization error using the numerical solution on only one grid/mesh. The primary problem associated with solving the ETE is the formulation of the error source term which is required for accurately predicting the transport of the error. In this study, a novel approach is considered which involves fitting the numerical solution with a series of locally smooth curves and then blending them together with a weighted spline approach. The result is a continuously differentiable analytic expression that can be used to determine the error source term. Once the source term has been developed, the ETE can easily be solved using the same solver that is used to obtain the original numerical solution. The new methodology is applied to the two-dimensional Navier–Stokes equations in the laminar flow regime. A simple unsteady flow case is also considered. The discretization error predictions based on the methodology presented in this study are in good agreement with the ‘true error’. While in most cases the error predictions are not quite as accurate as those from Richardson extrapolation, the results are reasonable and only require one numerical grid. The current results indicate that there is much promise going forward with the newly developed error source term evaluation technique and the ETE.

USM3D Simulations for the F-16XL Aircraft Configuration

Numerical simulations of flow around the F-16XL are presented as a contribution to the Cranked Arrow Wing Aerodynamics Project International II. The NASA Tetrahedral Unstructured Software System with its USM3D solver is used to perform numerical simulations. This computational fluid dynamics suite is developed and maintained by the NASA Langley Research Center. The CRISP CFD package is used to provide error estimates and grid adaptation for validation of USM3D results. A subsonic high-angle-of-attack case flight condition 25 is computed and analyzed. Three turbulence models are used in the calculations: the one-equation Spalart–Allmaras, the two-equation shear-stress transport, and the turbulence models. Computational results and surface-static-pressure profiles are presented and compared with flight data. Solution validation is performed using formal grid-refinement studies, the solution of error-transport equations, and adaptive mesh refinement. The current study shows that the USM3D solver coupled with CRISP CFD can be used to provide valuable data in predicting vortex-flow physics on a complex configuration.

Error transport equation boundary conditions for the Euler and Navier–Stokes equations

Discretization error is usually the largest and most difficult numerical error source to estimate for computational fluid dynamics, and boundary conditions often contribute a significant source of error. Boundary conditions are described with a governing equation to prescribe particular behavior at the boundary of a computational domain. Boundary condition implementations are considered sufficient when discretized with the same order of accuracy as the primary governing equations; however, careless implementations of boundary conditions can result in significantly larger numerical error. Investigations into different numerical implementations of Dirichlet and Neumann boundary conditions for Burgers' equation show a significant impact on the accuracy of Richardson extrapolation and error transport equation discretization error estimates. The development of boundary conditions for Burgers' equation shows significant improvements in discretization error estimates in general and a significant improvement in truncation error estimation. The latter of which is key to accurate residual-based discretization error estimation. This research investigates scheme consistent and scheme inconsistent implementations of inflow and outflow boundary conditions up to fourth order accurate and a formulation for a slip wall boundary condition for truncation error estimation are developed for the Navier–Stokes and Euler equations. The scheme consistent implementation resulted in much smoother truncation error near the boundaries and more accurate discretization error estimates.

Comparisons of p-adaptation strategies based on truncation- and discretisation-errors for high order discontinuous Galerkin methods

Abstract In this work, local p-adaptation strategies for high order discontinuous Galerkin spectral element methods are compared. The principal aim of the paper is to determine the advantages and drawbacks of various sensors, based on truncation or discretisation errors, to detect the regions that require adaption. A well-established discretisation error based approach, estimated through the decay of the energy associated to the approximated solution modes, is compared to recently proposed truncation error adaptation methodologies (i.e., isotropic and anisotropic versions). The truncation error technique detects the regions that require refinement by estimating the truncation error in meshes with varying degrees of freedom. This comparison is particularly interesting since the truncation error is related to the discretisation error through a discretisation error transport equation, where the truncation error appears as a source term for the discretisation error. The comparisons included quantify the accuracy of the flow solutions resulting from meshes adapted with these sensors and hence provide guidelines into which sensors are better suited to adapt meshes for inviscid and viscous flows. All the adaption strategies are tailored to high order methods and particularly implemented and tested in a discontinuous Galerkin solver. Results include an inviscid NACA0012 and a viscous flat plate boundary layer. Output functionals (e.g., lift, drag) resulting from adapted meshes are compared in terms of the number of degrees of freedom, providing a critical assessment of the performance of each sensor. Namely, it is shown that both truncation error adaptation methodologies provide meshes with polynomial order distributions that lead to fewer degrees of freedom than when using the discretisation error based adaptation. In addition, the examples illustrate the outperforming advantage of anisotropic over isotropic adaptation.

On asymptotic global error estimation and control of finite difference solutions for semilinear parabolic equations

The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete $L_2$-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies.

Error Quantification for Computational Aerodynamics Using an Error Transport Equation

Error quantification in computational fluid dynamics is a subject of increasing interest and research. Although solution verification is conventionally performed using systematic grid refinement, Richardson extrapolation can be restrictive in its applicability, possibly requiring more than three grids to establish monotonic behavior. Ultimately, a single grid error estimation method may prove of greater utility for practical application in computational aerodynamics. In this work, an error transport equation is implemented in a three-dimensional upwind unstructured Navier-Stokes solver. An approach for deriving errors in related quantities of interest from the solution of the error transport equation is presented. Error quantification is demonstrated in two and three dimensions for aerodynamic flows where experimental data are available for comparison. Predicted error bars are found to contain fine grid solutions, test data, and the results of Richardson extrapolation. Furthermore, the error transport equation provides meaningful error quantification for aerodynamic coefficients in cases where Richardson extrapolation cannot be applied.

Single Grid Error Estimation Using Error Transport Equation

This paper presents an approach to quantify the discretization error as well as other errors related to mesh size using the error transport equation (ETE) technique on a single grid computation. The goal is to develop a generalized algorithm that can be used in conjunction with computational fluid dynamics (CFD) codes to quantify the discretization error in a selected process variable. The focus is on applications where the conservation equations are solved for primitive variables, such as velocity, temperature and concentration, using finite difference and/or finite volume methods. An error transport equation (ETE) is formulated. A generalized source term for the ETE is proposed based on the Taylor series expansion and accessible influence coefficients in the discretized equation. Representative examples, i.e., one-dimensional convection diffusion equation, two-dimensional Poisson equation, two-dimensional convection diffusion equation, and non-linear one-dimensional Burger’s equation are presented to verify this method and elucidate its properties. Discussions are provided to address the significance and possible potential applications of this method to Navier-Stokes solvers.