On A Posteriori Estimation of the Approximation Error Norm for an Ensemble of Independent Solutions
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Posteriori Estimation of the Approximation Error Norm for an Ensemble of Independent Solutions A. K. Alekseev1* and A. E. Bondarev1** 1
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia Received September 6, 2018; in final form, May 7, 2019; accepted April 16, 2020
Abstract—An ensemble of independent numerical solutions makes it possible to construct a hypersphere around an approximate solution that contains the true solution. The analysis is based on some geometry considerations, such as the triangle inequality and the measure concentration in spaces of large dimensions. As a result, a nonintrusive postprocessor providing error estimation on an ensemble of solutions can be constructed. Some numerical tests for the two-dimensional compressible Euler equations are given to demonstrate the properties of such postprocessing. DOI: 10.1134/S1995423920030015 Keywords: discretization error, ensemble of numerical solutions, measure concentration, Euler equations.
INTRODUCTION Quantitative estimation of numerical calculation error for systems of partial differential equations is of great practical interest. However, in gas dynamics problems such estimation is often difficult to obtain. This fact has stimulated the present study. Let Au = f denote a system of partial differential equations, and let Ah uh = fh be its finite-dimensional approximation. The local approximation (truncation) error, δu, is obtained by expanding the discrete solution in a Taylor series [1]. Its asymptotic dependence on the grid spacing h is expressed as δu = O(hn ). The truncation error generates the approximation (calculation) error Δu = uh − u. For linear problems Δu = O(hn ), and both errors are of the same order n (approximation order) according to the Lax theorem [2] (if some important constraints are satisfied). Numerous papers based on various methods are devoted to estimation of calculation errors [3, 4]. They may be divided into two wide subclasses of a priori and a posteriori analysis of errors. The a priori analysis is of fundamental character and relates the calculation error to the local approximation (truncation) error according to the Lax theorem [2]. From a practical point of view, the a priori analysis justifies the current practice of mesh refinement. The norm of the calculation error in a priori analysis can be estimated as Δu < Chn , which contains an unknown constant. The presence of this unknown constant is the main shortcoming of the a priori approach, which does not allow obtaining any practical calculation error estimates. The a posteriori error estimation is more practical and can be presented in the form Δu ≤ ηh (uh ), where ηh (uh ) is a calculated error indicator that depends on the numerical solution uh . Currently, much progress has been achieved for elliptic and parabolic equations in the finite element analysis [5, 6] and using Richardson extrapolation [7]. This has become possible because the corresponding solutions have sufficient smoothness. For gas dynamic pro
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