Verification of error estimators for the Euler equations

Abstract

In this paper, grid convergence studies are conducted to analyze four error estimators for their asymptotic behavior. The four error estimation techniques are based on residual, solution reconstruction, Richardson extrapolation and error equations respectively. Their accuracy, reliability and efficiency to control the true error have been verified on the quasi-one-dimensional Euler equations solved by a second-order accurate finite-volume method. Introduction The rapidly increasing power of computers allows very good approximation of solutions to partial differential equations. However, for complex problems highly accurate numerical solutions of partial differential equations can be achieved using locally adaptive grid. The generation of such adapted grid is usually guided by a proper error estimator or indicator. Analyses for error and error estimations abound for elliptic equations solved with finite-element methods. However, for hyperbolic problems, the theoretical foundation of a posteriori and a priori error analysis is far from satisfactory. Inspite of the lack of sound theoretical analyses, many practitioners simply choose one error indicator to guide mesh adaptation. Such indicators are usually based on the local gradient of a key variable,i12 such as fluid density, based on the idea that the error occurs when the variable of interest varies sharply. A more reasonable error indicator is based on the residual of the discretized equation3T4 as it is *Research Professional, Member AIAA tProfessor, Associate Fellow AIAA SAssociate ProfeSsor, Member AIAA §Professor, Member AIAA Copyright @ 2000 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. a more reliable indication of how accurately the differential equation has been solved. Another method is to use two different levels of approximation (solution reconstruction), 5, 6 which is suitable to estimate interpolation error. Another approach is to use solutions on two or three different levels of grid to perform Richardson extrapolation to estimate solution error.7 Finally, one can solve error equations with residuals as right-hand side to provide solution error*)” which account for the transport of errors. Most of these error estimatiors are extensions from finite element methods for elliptic equations. For hyperbolic equations, several of these techniques are questionable. One important issue for any a posteriori error estimator or indicator is its ability to control the discretization error of the numerical method. At least, one requires that the estimated error and the true error should have the same asymptotic convergence rate when the mesh is refined. For hyperbolic cases very little theoretical analyses can be found about convergence rate and asymptotic behaviors of the above mentioned error estimations because of the lack of mathematical foundations. As these properties are so important in the control and minimization of the discretization error, it is worthwhile to conduct numerical investigations or verifications for test problems. The motivation of this paper is to perform careful studies of various error estimators through numerical experiments. The main concern is to assess the asymptotic convergence rates and which norm is more appropriate to measure the true and estimated errors, and consequently to assess existing error estimators for their accuracy, reliability and efficiency. Error Estimation and its Control Global control of the solution error is generally termed a priori error estimate for a given numerical 1 American Institute of Aeronautics and Astronautics I (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization. method. It ensures the convergence of the method when the mesh size tends to zero, a basic requirement for any numerical methods. The global error estimation may be expressed as lb Uhllw,n = U(P), with h= rnrnh(h~) (1) where II * Ilw,n is a proper norm over the whole domain of concern fi in a Sobolev space W, P, the mesh partition established over a, hK represents the size of the cell K E Ph. The superscript Q: > 0 represents the convergence rate for the numerical method. For elliptic equations and finite element methods, this type of error estimation with W = H1(R) can be found in reference. lo For hyperbolic equations and finite volume methods, similar results with L2 norm were given in reference.lr It is evident that the order of convergence depends on both the problem and the method. It is also strongly dependent on the norm used to measure the error. The exact error is not always available for most problems of interest. To be able to control the true error globally, one relies on a specific error estimator and try to control the estimated error instead. However, the purpose of using adaptive technique to control the estimated error is to ultimately allow us to control the true error. This requires that the estimated error must have a behavior similar to the true one. The efficiency index defined by