Discretization Error Estimation by the Error Transport Equation on Unstructured Meshes - Applications to Viscous Flows

2016 ◽  
Author(s):  
Gary Yan ◽  
Carl F. Ollivier Gooch
Keyword(s):  
Transport Equation ◽  
Error Estimation ◽  
Viscous Flows ◽  
Unstructured Meshes ◽  
Discretization Error ◽  
Error Transport Equation

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

2017 ◽  
Vol 2 (4) ◽  
Author(s):  
Gary K. Yan ◽  
Carl Ollivier-Gooch
Keyword(s):  
Error Estimate ◽  
Transport Equation ◽  
Error Estimates ◽  
Finite Volume ◽  
Stokes Equations ◽  
Higher Order ◽  
Discretization Error ◽  
Computational Time ◽  
Primal Problem ◽  
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.

Applications of the Unsteady Error Transport Equation on Unstructured Meshes

AIAA Journal ◽  
2018 ◽  
Vol 56 (11) ◽  
pp. 4463-4473 ◽  
Author(s):  
Gary Yan ◽  
Carl Ollivier-Gooch
Keyword(s):  
Transport Equation ◽  
Unstructured Meshes ◽  
Error Transport Equation

Discretization Error Estimation Using Error Transport Equation

2002 ◽  
Author(s):  
Ismail Celik ◽  
Gusheng Hu
Keyword(s):  
Fluid Dynamics ◽  
Transport Equation ◽  
Cfd Simulation ◽  
Control Volume ◽  
Process Variable ◽  
Discretization Error ◽  
Computational Fluid Dynamics Cfd ◽  
Finite Volume Approach ◽  
Error Transport Equation ◽  
Modified Equation

The goal of this paper is to develop a dynamic algorithm that can be used in conjunction with computational fluid dynamics (CFD) simulation codes to quantify the discretization error in a selected process variable. The focus is on fluid dynamics applications where conservation equations are solved for primitive variables using finite difference and/or control volume approach. A transport equation for the error is formulated and solved along with a localized residual estimation based on modified equation concept. Spatiotemporal evolution of the error distribution is mapped and compared to exact error for various cases. A new method is suggested for deriving the modified equation specifically aimed at using it with commercial CFD codes which use finite volume approach.

Single Grid Error Estimation Using Error Transport Equation

2004 ◽  
Vol 126 (5) ◽  
pp. 778-790 ◽  
Author(s):  
Ismail Celik ◽  
Gusheng Hu
Keyword(s):  
Diffusion Equation ◽  
Transport Equation ◽  
Mesh Size ◽  
Taylor Series Expansion ◽  
Discretization Error ◽  
Two Dimensional ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
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.

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