For two-dimensional Euler equations, flow control using spatially distributed stationary heat sources is considered. The initial flow, characterized by the interaction of Edney - type shock waves, was changed with the help of heat sources in such a way as to reduce the maximum pressure on the surface of the body. The problem was solved in an optimization formulation using conjugate equations to calculate the gradient of the target functional. In addition to visualizing flow parameters in this problem, it is of interest to visualize the distribution of adjoint parameters and heat sources.

The numerical solution in sense of Prager Synge is defined as a hypersphere containing a true solution of a system of partial differentiation equations (PDE). In the original variant Prager Synge method is based on special orthogonal properties of PDE and may be applied only to several equations. Herein, the Prager Synge solution (center and radius of the hypersphere) is estimated using the ensemble of numerical solutions obtained by independent algorithms. This approach is not problem dependent and may be applied to arbitrary system of PDE. Several options for computation of the Prager Synge solution are considered. The first one is based on the search for the orthogonal truncation errors and their transformation. The second is based on the orthogonalization of approximation errors obtained using the defect correction method. It applies the superposition of numerical solutions. The third option uses the width of the ensemble of numerical solutions. The numerical tests for the two dimensional inviscid flows are presented that demonstrate the acceptable effectivity of the approximation error estimates based on the solution in the Prager Synge sense.

The approximation of the tensor appearing at a discretization of the multidimensional function is considered from the viewpoint of storing and treating of the results of parametric computations obtained in computational aerogasdynamics. The new algorithm for the computation of the canonical decomposition using gradient descent and approximately decomposable goal functional is described. This algorithm applies the random set of points on the hyperplane orthogonal to the computed core of the canonical decomposition (“umbrella”) that ensures its flexible application for an approximation of the tensors with a priori unknown rank and may be naturally transferred on such tensor decomposition as the tensor train. The results of the numerical tests are presented for the model six-dimensional functions and for an ensemble of the numerical solutions for the two-dimensional Euler equations. These equations describe the flow of the compressible gas with two crossing shock waves. The Mach number and angles of the flow deflection serve as the flow parameters. The results are provided for the dimensionality 3 (simple numerical solution) and 4 (the ensemble of the numerical solutions in dependence on the Mach number).

The epistemic uncertainty quantification concerning the estimation of the approximation error using the differences between numerical solutions treated in the Inverse Problem statement is addressed and compared with the Richardson extrapolation. The Inverse Problem is posed in the variational statement with the zero order Tikhonov regularization. The ensemble of numerical results, obtained by the OpenFOAM solvers for the inviscid compressible flow with a shock wave is analyzed. The approximation errors, obtained by the Richardson extrapolation and the Inverse Problem are compared with the exact error, computed as the difference of numerical solutions and the analytical solution. The Inverse problem based approach is demonstrated to be an inexpensive alternative to the Richardson extrapolation.

This paper deals with estimation of local (pointwise) approximation error on an ensemble of numerical solutions obtained by using independent algorithms. A variational inverse problem is posed for approximation error estimation. This problem is ill-posed due to translation-invariance of the governing equations. Zero order Tikhonov regularization is applied to obtain stable solutions. Numerical tests for two-dimensional equations describing inviscid compressible flow are performed to verify the efficiency of the algorithm. The approximation error estimates obtained by using the inverse problem are in satisfactory agreement with those obtained by Richardson extrapolation, but with significantly less computational costs.

The estimation of the approximation errors using the ensemble of numerical solutions is considered in the Inverse Problem statement. The Inverse Problem is set in the nonlinear variational formulation that provides additional opportunities. The ensemble of numerical results is analyzed, which is obtained by the OpenFOAM solvers for the inviscid compressible flow containing an oblique shock wave. The numerical tests demonstrated feasibility to compute the approximation errors without any regularization. The refined solution, corresponding the mean of numerical solutions with the approximation error correction, is also computed and compared with the analytic one.

The geometric properties of the ensemble of numerical solutions obtained by the algorithms of different inner structure are addressed from the prospects for a posteriori error estimation. The numerical results are presented for the two-dimensional inviscid supersonic flows, containing shock waves. The truncation errors are computed using a postprocessor, the approximation errors are calculated by the subtraction of the numerical and the analytic solutions. The angles between the approximation errors are found to be far from zero that enables a posteriori estimation of the error norm. The correlation of the angles between the approximation errors and the corresponding angles between the computable truncation errors is observed in numerical tests and discussed from the viewpoint of the measure concentration effect and the algorithmic randomness. The analysis of the truncation errors’ geometry and the distances between solutions enables the estimation of the approximation error norm on the ensemble of numerical solutions obtained by the independent algorithms.

The Yakutsk Extensive Air Shower Array has been continuously operating for more than 50 years (since 1970) and up until recently it has been one of world's largest ground-based instruments aimed at studying the properties of cosmic rays in the ultra-high energy domain. In this report we discuss results recently obtained at the array - on cosmic rays energy spectrum, mass composition and directional anisotropy - and how they fit into the world data. Special attention is paid to the measurements of muonic component of extensive air showers. Theoretical results of particle acceleration at shocks are also briefly reviewed. Future scientific and engineering plans on the array modernization are discussed.

In this paper, we consider the inverse problem for the estimation of a point-wise approximation error occurring at the discretization of the system of partial differential equations. We analyse the set of the solutions, obtained by the numerical algorithms of the dissimilar structures on the same grid. The differences between the numerical solutions are used as the input data for the inverse problem, which is posed in the variational statement with the zero-order Tikhonov regularization. The numerical tests, performed for the two-dimensional inviscid compressible flows corresponding to Edney-I and Edney-VI shock wave interference modes, are provided. The comparison of the estimated error and the exact error, obtained by subtraction of numerical and analytic solutions, is presented.

— The impact of the choice of the proximity measure for the numerical and reference solutions is discussed in terms of the verification of the calculations and software. If no reference solution is available, the deterministic and stochastic options for estimating computational errors are considered using an ensemble of solutions obtained by different numerical algorithms. The relation between the norm of the solution error and the error of valuable functionals is studied via the Cauchy–Bunyakovsky–Schwarz inequality. The results of numerical tests for the two-dimensional Euler equations, which demonstrate how the choice of the proximity measure affects the estimation of the approximation error on the ensemble of solutions and show the efficiency of the considered algorithms, are presented. The comparison of different proximity measures (norms and metrics) both for estimating the computational error and for comparing the flow fields that correspond to both small variations in the flow structure and qualitatively different flow patterns is a new element of the paper. The application of the errors of valuable functionals for the evaluation of the approximation errors in practical terms is also novel. The feasibility for computationally cheap (single-grid, in contrast to the Richardson extrapolation method) quantitative verification of solutions considered and analyzed in the paper seems useful for the implementation of the Russian standards for numerical solution verification and CFD code validation.

This paper considers the construction of a generalized computational experiment for solving verification problems. The problem of comparative accuracy assessment of numerical methods is currently acquiring special relevance due to the introduction of published standards and widespread use of software packages that include a large number of different solvers. A generalized computational experiment makes it possible to obtain a numerical solution for a class of problems determined by variation ranges of their governing parameters. Analysis of results represented as multidimensional arrays, where the number of measurements depends on the dimension of the space of governing parameters, requires the use of scientific visualization and visual analytics tools. Some approaches to the application of generalized computational experiments with and without a reference solution are discussed. An example of constructing error surfaces when comparing some solvers from the OpenFOAM software package is considered. The classical problem of an inviscid oblique shock wave is used as a basic problem. Certain variations of its main parameters-Mach number and angle of attack-are analyzed. In addition, we consider an example of the cone flow problem with variable Mach number, cone angle, and angle of attack. The concept of an error index is introduced as an integral characteristic of deviations from the exact solution for each solver in the class of problems under consideration.

The instance of the epistemic uncertainty quantification concerning the estimation of the approximation error norm is analyzed using the ensemble of numerical solutions obtained via independent numerical algorithms. The analysis is based on the geometry considerations: the triangle inequality and the diameter of the ensemble related with the measure concentration phenomenon in spaces of great dimension. In result, nonintrusive postprocessing may be performed that provides the approximation error norm estimation on the ensemble of the solutions. The ensemble of numerical results obtained by five OpenFOAM solvers (based on independent algorithms) is analyzed from this viewpoint. The numerical tests are made for the inviscid compressible flow around a cone at zero angle of attack. The norm of the approximation error and the error of the valuable functional (drag coefficient) are successfully estimated via ensemble based approach that is confirmed by the comparison with the etalon precise solution. The considered approach provides the error estimation with the acceptable value of the efficiency index. (C) 2020 Elsevier B.V. All rights reserved.

The paper is devoted to comparison of a posteriori methods (based on the precomputed solutions) for approximation error estimation. Rigorous a posteriori error estimation for computational Fluid Dynamics at present is practically impossible due to nonlinearity and the discontinuities that may occur and migrate along the flow field. In this situation, several nonstrict (weak) forms of a posteriori estimation of the approximation error may be considered. They either do not provide the error norm estimation in the form of inequalities or provide values of the effectivity index to be less than unit. The best quality of estimates are provided by the Richardson extrapolation, unfortunately for the cost of extremely high computational burden. We pay the special attention to the nonstrict methods that either cannot be presented in a form of inequalities, or demonstrate the effectivity index of an estimator to be below unit. Several new, computationally inexpensive methods for both the point-wise error and the error norm estimation are considered. They are nonintrusive, realized by postprocessing and provide a successful compromise of the reliability and computational efforts. Methods based on the use of an ensemble of independent solutions can be implemented by constructing a generalized computational experiment, which sharply increases the speed and efficiency of the assessment.

This work is devoted to the application of a generalized computational experiment for a comparative assessment of numerical methods accuracy. A generalized computational experiment allows one to obtain a numerical solution for a class of problems determined by the ranges of defining parameters variation. The approaches to the application of a generalized computational experiment in the presence of a reference solution and in its absence are dis-cussed. An example of constructing error surfaces is given when the solvers of the OpenFOAM software package are compared. The classic inviscid problem of oblique shock wave is used as a basic task. Variations of the key parameters of the problem — the Mach number and angle of attack — are considered. An example of the problem of flow around a cone at an angle of attack with varying Mach number, cone angle and angle of attack is also considered. The concept of an error index is introduced as an integral characteristic of deviations from the exact solution for each solver in the class of problems under consideration.

In this work we address the problem of the estimation of the approximation error that arise at a discretization of the partial differential equations. For this we take advantage of the ensemble of numerical solutions obtained by independent numerical algorithms. To obtain the approximation error, the differences between numerical solutions are treated in the frame of the Inverse Problem that is posed in the variational statement with the zero order regularization. In this work we analyse the ensemble of numerical results that is obtained by five OpenFOAM solvers for the inviscid compressible flow around a cone at zero angle of attack. We present the comparison of approximation errors that are obtained by the Inverse Problem, and the exact error that is computed as the difference of numerical solutions and a high precision solution.

The truncation and approximation errors for the set of numerical solutions computed by methods based on the algorithms of different structure are calculated and analyzed for the case of the two-dimensional steady inviscid compressible flow. The truncation errors are calculated using a special postprocessor, while the approximation errors are obtained by the comparison of the numerical solution and the analytic one. The extent of the independence of errors for the numerical solutions may be estimated via the Pearson correlation coefficient that may be geometrically expressed by the angle between errors. Due to this reason, the angles between the approximation errors are computed and related with the corresponding angles between the truncation errors. The angles between the approximation errors are found to be far from zero that enables a posteriori estimation of the error norm. The analysis of the distances between these solutions provides another approach to the estimation of the error. The comparison of the error norms, obtained by these two procedures, is provided that demonstrates the acceptable values of their effectivity indices. The results of the approximation error norm estimation for the supersonic flows, containing shock waves, are presented. The measure concentration phenomenon and the algorithmic randomness give some insights into these results.

An ensemble of independent numerical solutions makes it possible toconstruct a hypersphere around an approximate solution that contains thetrue solution. The analysis is based on some geometry considerations,such as the triangle inequality and the measure concentration in spacesof large dimensions. As a result, a nonintrusive postprocessor providingerror estimation on an ensemble of solutions can be constructed. Somenumerical tests for the two-dimensional compressible Euler equations aregiven to demonstrate the properties of such postprocessing.

Formation of polymer films under the action of mass forces has been modeled. The instability of a non-Newtonian liquid front edge at the initial stage of flow over disc is examined. The factors that determine the final shape of a surface of polymer coating are discussed.

The element of the epistemic uncertainty quantification concerning the estimation of the approximation error is analyzed from the viewpoint of the ensemble of numerical solutions obtained via independent numerical algorithms. The analysis is based on the geometry considerations: the triangle inequality and measure concentration in spaces of great dimension. In result, the feasibility for nonintrusive postprocessing appears that provides the approximation error estimation on the ensemble of the solutions. The ensemble of numerical results obtained by five OpenFOAM solvers is analyzed. The numerical tests were made for the inviscid compressible flow around a cone at zero angle of attack and demonstrated the successful estimation of the approximation error.