Discretization Error Estimation Using the Error Transport Equations for Computational Fluid Dynamics Simulations

Computational Fluid Dynamics (CFD) has been widely used as a tool to analyze physical phenomena involving fluids. To perform a CFD simulation, the governing equations are discretized to formulate a set of nonlinear algebraic equations. Typical spatial discretization schemes include finite-difference methods, finite-volume methods, and finite-element methods. Error introduced in the discretization process is called discretization error and defined as the difference between the exact solution to the discrete equations and the exact solution to the partial differential or integral equations. For most CFD simulations, discretization error accounts for the largest portion of the numerical error in the simulation. Discretization error has a complicated nonlinear relationship with the computational grid and the discretization scheme, which makes it especially difficult to estimate. Thus, it is important to study the discretization error to characterize numerical errors in a CFD simulation.

Discretization error estimation is performed using the Error Transport Equations (ETE) in this work. The original nonlinear form of the ETE can be linearized to formulate the linearized ETE. Results of the two types of the ETE are compared. This work implements the ETE for finite-volume methods and Discontinuous Galerkin (DG) finite-element methods. For finite volume methods, discretization error estimates are obtained for both steady state problems and unsteady problems. The work on steady-state problems focuses on turbulent flow modelled by the Spalart-Allmaras (SA) model and Menter's $k-omega$ SST model. Higher-order discretization error estimates are obtained for both the mean variables and the turbulence working variables. The type of pseudo temporal discretization used for the steady-state problems does not have too much influence on the final converged solution. However, the temporal discretization scheme makes a significant difference for unsteady problems. Different temporal discretizations also impact the ETE implementation. This work discusses the implementation of the ETE for the 2-step Backward Difference Formula (BDF2) and the Singly Diagonally Implicit Runge-Kutta (SDIRK) methods. Most existing work on the ETE focuses on finite-volume methods. This work also extends ETE to work with the DG methods and tests the implementation with steady state inviscid test cases. The discretization error estimates for smooth test cases achieve the expected order of accuracy. When the test case is non-smooth, the truncation error estimation scheme fails to generate an accurate truncation error estimate. This causes a reduction of the discretization error estimate to first-order accuracy. Discussions are made on how accurate truncation error estimates can be found for non-smooth test cases.