A posteriori error estimation for finite-volume solutions of hyperbolic conservation laws
Introduction
The use of locally adaptive computational grids is one of the most attractive strategies to achieve highly accurate solutions of partial differential equations. The generation of such adapted grids is usually guided by a proper a posteriori error estimation technique. Such techniques abound for finite-element methods, but for finite-volume methods, the theoretical foundation of a posteriori and a priori error analysis is far from satisfactory. This is especially true for hyperbolic problems [1].
Most error estimation techniques used for grid adaptation in finite-volume methods are based on the local flow structure, which means that the predicted errors are not influenced by distant information (i.e. convection of error is not accounted for). Many widely used error estimators and indicators are based on the local gradient of density or Mach number [2], [3]. In these cases, the predicted error may become unbounded with the grid refinement near discontinuities. Alternatives have been proposed based on cell-length adjusted velocity derivatives [4] or on cell-length weighted density gradients [5]. Another strategy using residual as a mesh refinement indicator has been used in Ref. [6] for compressible Euler equations. However, the residual represents the error source, not necessarily the solution error as will be demonstrated in this paper.
In fact, for hyperbolic systems, the errors arising from the numerical discretization act as erroneous waves which are propagated by the equations in the same manner as physical waves. Thus the source of an error may manifest itself far away from its active point. Error indicators based on a local analysis of the solution may produce a misleading error distribution and may not be able to identify the error source. Solving an appropriate equation for the error may prove to be a more effective and reliable approach to error estimation.
In Ref. [7], an error equation was derived for scalar linear advection–diffusion problems and a discrete residual estimate is used as an error source for the error equation by assuming that the approximated solution is smooth across the interfaces of each control volume. For non-smooth or discontinuous approximation often arising in finite-volume methods, the error source may be modified by accounting for the jump in the solution across the element boundaries, as discussed in Refs. [8], [9] in a finite-element context. To the authors' knowledge, very little work has been done for non-linear hyperbolic systems of conservation laws solved using a finite-volume method.
In the present paper, a dynamic procedure for a posteriori error estimation for hyperbolic conservation laws is proposed and tested. This method is based on solving linearized hyperbolic equations for the errors with source terms obtained using the modified equation analysis [10], [11]. The dominant term in the modified equation is used as the error source. This provide an alternative way of accurately estimating the solution error for hyperbolic equations. This technique accounts for the wave structure of the solution. In particular it will detect convection of errors which is a non-local phenomena. Also, it will be shown that efficient grid adaptation can be achieved by using the estimated error source instead of the solution error itself. The estimated a posteriori error distribution can be used as a verification of the grid adaptation algorithm. In 2 Equations and the finite-volume scheme, 3 Error equation, the hyperbolic system of conservation laws and the corresponding error equations are presented. Section 4 details the method to estimate the error source term. Numerical experiments for one-dimensional test cases are presented in Section 5.
Section snippets
Equations and the finite-volume scheme
In this paper, we consider the following one-dimensional hyperbolic system of conservation lawswhere u=u(x,t) is an unknown vector, f(u) the vector-valued flux function of u. For hyperbolic conservation laws, one assumes that the Jacobian matrixhas only real eigenvalues and can be diagonalized aswhere D=diag(λ1,λ2,…) is the diagonal matrix consisting the eigenvalues of A, L and R(=L−1) are formed from the left and right eigenvectors of A, respectively.
The
Derivation
Consider the hyperbolic system of conservation laws given in Eq. (1) and its finite volume approximation uh, the following residual can be definedDefining the error vector as ϵ=u−uh, subtracting Eq. (7) from Eq. (1) and using relations given in Eq. (6), it is clear thatorwhich is a non-linear equation for the error vector ϵ. In practice, it is reasonable to replace u by uh in the Jacobian matrix and the above equation becomes
Error source approximation
The modified equation approach is used for approximating the error source −r(uh) for Eq. (9). The idea of using the modified equation analysis to identify and remove the dominant errors for hyperbolic equations can be found in Refs. [10], [11]. The modified equation analysis is basically designed for the finite difference scheme using Taylor series expansion based on a set of discrete data. An intrinsic assumption of smooth interpolation between those discrete data is made. To apply this
Results and discussion
We now present tests of the previously described procedure for error estimation. Four cases possessing closed form solutions are used:
(i) a linear advection equation,
(ii) a non-linear Burgers' equation,
(iii) the unsteady Euler equations,
(iv) the steady Euler equations with variable cross-section.
It is well known that the first-order upwind scheme is very diffusive. If the diffused solution is fed into the error Eq. (9) and a first-order scheme is used to solve this equation, then the estimated
Concluding remarks
An a posteriori error estimation technique has been proposed for hyperbolic conservation laws. The method is based on the solution of a linearized hyperbolic equation with source term for the error. The error source is approximated using the modified equation approach.
Many useful features of this technique have been demonstrated for one-dimensional unsteady and steady test cases. First, in all the test problems, the estimated errors are a good approximation of the true errors in terms of both
Acknowledgements
The authors would like to express their appreciation to Dominique Pelletier of École Polytechnique de Montréal and Paul Labbé of CERCA for several helpful discussions and suggestions. Thanks also go to the reviewers for their useful comments and suggestions to improve the paper. The financial support provided by the Natural Science and Engineering Research Council (NSERC) of Canada and the Centre de Recherche en Calcul Appliqué (CERCA) is gratefully acknowledged.
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