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AMA V. Dolejsi: Anisotropic mesh adaptation method for compressible flow

AMA1 V. Dolejsi: Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes

CONV P. Angot, V. Dolejsi, M.Feistauer, J. Felcman: Analysis of a combined barycentric finite volume - nonconforming finite element method for nonlinear convection - diffusion problems

HORS J. Felcman, V. Dolejsi: Investigation of order of accuracy for higher order schemes for finite volumes method on triangular meshes

THES V. Dolejsi: Combined finite volume - finite element methods for compressible flow on unstructured meshes (partly written in French)

HABIL V. Dolejsi: Adaptive higher order methods for compressible flow, Prague, 2003

TURB V. Dolejsi, M.Feistauer, J. Felcman: Algebraic turbulence models for unstructured meshes

PROF M.Feistauer, J. Felcman, V. Dolejsi: Numerical Simulation of Compressible Viscous Flow through Cascades of Profiles

VISC M.Feistauer, J. Felcman: Theory and applications of numerical schemes for nonlinear convection-diffusion problems and compressible viscous flow

AMA2 V. Dolejsi: Anisotropic Mesh Adaptation Technique for Viscous Flow Simulation

AMA3 V. Dolejsi: Anisotropic mesh adaptation and its application for scalar diffusion equations

DGM1 V. Dolejsi, M. Feistauer, C. Schwab: A Finite Volume Discontinuous Galerkin Scheme for Nonlinear Convection-Diffusion Problems

DGM2 V. Dolejsi, M. Feistauer, C. Schwab: On some aspects of the Discontinuous Galerkin finite element method for conservation laws

DGM3 V. Dolejsi, M. Feistauer and C. Schwab: On Discontinuous Galerkin Methods for Nonlinear Convection-Diffusion Problems and Compressible Flow

DGM5 V. Dolejsi, M. Feistauer: On the Discontinuous Galerkin Method for the Numerical Solution of Compressible High-Speed Flow

DGM6 V. Dolejsi: On the Discontinuous Galerkin Method for the Numerical Solution of the Euler and the Navier-Stokes Equations

DGM7 V. Dolejsi, M. Feistauer: A Semi-Implicit Discontinuous Galerkin Finite Element Method for the Numerical Solution of Inviscid Compressible Flow

DGM8 V. Dolejsi, M. Feistauer, V. Sobotikova: Analysis of the Discontinuous Galerkin Method for Nonlinear Convection--Diffusion Problems

DGM9 V. Dolejsi, M. Feistauer: Error estimates of the discontinuous Galerkin method for nonlinear nonstationary convection-diffusion problems

DGM10 V. Dolejsi, M. Feistauer: Analysis of Semi-Implicit DGFEM for Nonlinear Convection-Diffusion Problems



V. Dolejsi: Anisotropic mesh adaptation method for compressible flow

part of the habilitation thesis, 2003.

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V. Dolejsi: Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes

Computing and Visualisation in Science, 1:165-178, 1998.

Abstract The present paper deals with an anisotropic mesh adaptation (AMA) of triangulation which can be employed for the numerical solution various problems of physics. AMA tries to construct an optimal triangulation of the domain of computation in the sense that an ``error'' of the solution of the problem considered is uniformly distributed over the whole triangulation. First, we describe the main idea of AMA. We define an optimal triangle and an optimal triangulation. Then we describe the process of optimization of the triangulation and the complete multilevel computational process. We apply AMA to a problem of CFD, namely to inviscid compressible flow. The computational results for a channel flow are presented.

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P. Angot, V. Dolejsi, M.Feistauer, J. Felcman: Analysis of a Combined Barycentric Finite Volume - Nonconforming Finite Element Method for Nonlinear Convection - Diffusion Problems

Aplications of Mathematics, 43(4):263-310, 1998.

Abstract We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided the mesh size tends to zero.

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J. Felcman and V. Dolejsi: Investigation of order of accuracy for higher order schemes for finite volumes method on triangular meshes

Journal of Engineering Mechanics, 5: 327-337, 1998.

Abstract The finite volumes method (FVM) is very used in computational flow problems. The basic scheme of FVM which uses for approximation of integral over an edge of a finite volume only the values from adjacent elements is only of first order of accuracy. Many authors tried to increase the order of accuracy using a higher order reconstruction for approximation of integral over an edge. We have tested several methods of higher order for problems of compressible inviscid flows and determined the order of accuracy for Sod's channel where the analytical solution is known. The computation is performed for three different triangulation with seizes of mesh $h,\ h/2$ and $h/4$ and the determination of the order of accuracy is done using the method of the least squares.

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V. Dolejsi: Combined finite volume - finite element methods for compressible flow on unstructured meshes

Doctoral Thesis, Charles University Prague, 1998.

Abstract We deal with the numerical simulation of compressible flow in two dimensions. Taking into account the mathematical formulation of conservation laws, we obtain a system of differential equations describing the behavior of fluids. To close the system, these equation must be accompanied by constitutive relations. We determined the final system of the Navier-Stokes equations. Since the viscosity and heat conduction coefficients of gases are small, the viscous dissipative terms are often considered as perturbations of the inviscid Euler system. Very often we are interested in steady flow. In this case, the operator splitting method can be applied. We divide the system of the Navier-Stokes equations into the inviscid system (the Euler equations) and the purely viscous system and discretize these separately. The inviscid system is a hyperbolic system. The nonlinear character of the hyperbolic system lead us to apply the finite volume method (FVM) for its discretization. The purely viscous system is parabolic and its structure offers us the application of the finite element method (FEM). Furthermore, to solve the complete system of the Navier-Stokes equations, the combined finite volume - finite element method is applied. We have developed two numerical schemes: the former uses the triangular finite volumes and conforming finite elements and the latter the barycentric finite volumes and nonconforming finite elements. To improve the order of accuracy of our schemes, a higher order reconstruction of piecewise constant functions for FVM is implemented. The first part of this work is devoted to the numerical schemes developed for the solution of the compressible Navier-Stokes equations. We present a number of numerical examples of inviscid as well as viscous flow. Both the basic test problems and industrial applications are presented. The comparison with experimental measurement is performed. In the second part we develop the strategies of a mesh adaptation. We have developed two strategies: multilevel mesh refinement (MMR) and anisotropic mesh adaption (AMA). The former is based on the shock indicator, which seems to be very suitable namely for a precise resolution of shock waves. The latter strategy is AMA, which tries to construct an optimal triangulation of the computational domain in the sense that an ``interpolation error'' of the solution of the considered problem is uniformly distributed over the whole triangulation. Both of these adaptive strategies are compared. The last part is devoted to the investigation of the convergence of our numerical method for viscous flow simulations. We confine our convergence analysis to a model scalar convection-diffusion conservation law equation.

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V. Dolejsi: Adaptive higher order methods for compressible flow

Habilitation Thesis, Charles University Prague, 2003.

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V. Dolejsi, M.Feistauer, J. Felcman: Algebraic turbulence models for unstructured meshes

(unpublished)

Abstract We deal with the numerical simulation of compressible turbulent flow. To discretize the system of the Navier-Stokes equations the operator splitting method is applied. To simulate the turbulence effect two algebraic turbulence models are implemented in our codes. To determine the turbulence viscosity the local structured mesh is constructed. At each time step, the values of physical quantities are interpolated from the global unstructured mesh to the local structured one, the turbulence model is executed on this mesh, and the resulting turbulence viscosity values are interpolated back to the global triangulation. Both method, Baldwin-Lomax model and Cebeci-Smidt model, were tested for the flow past cascade of profiles and the results are compared.

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M.Feistauer, J. Felcman, V. Dolejsi: Numerical Simulation of Compressible Viscous Flow through Cascades of Profiles

ZAMM, 76:301-304, 1996.

Abstract The paper presents the use of an adaptive mesh refinement method devised for flow problems with shocks. The goal is to detect automatically the regions where the local structures such as shocks and contact discontinuities occur. During the computational algorithm these regions are refined with the aim to improve the quality of the previously computed approximation in a further computation. This can be accomplished with the aid of a suitable error indicator or by a shock indicator. Here the numerical study of several shock indicators is presented.

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M.Feistauer, J. Felcman: Theory and applications of numerical schemes for nonlinear convection-diffusion problems and compressible viscous flow

In Proc. of the MAFELAP96 Conf., Willey 1996.

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V. Dolejsi: Anisotropic Mesh Adaptation Technique for Viscous Flow Simulation

East-West Journal of Numerical Mathematics, 9(1):1-24,2001.

Abstract We present an efficient adaptive method for unstructured triangular meshes especially developed for the numerical simulation of viscous compressible flow. This method uses the anisotropic mesh adaptation (AMA) technique introduced in \cite{AMA-98}, \cite{AMA-99} with some additional improvement reflecting physical properties of viscous high speed flow. We summarize the AMA method with its geometrical interpretation and we describe the numerical solver, which uses the combination of finite volume and finite element methods. We show that the direct application of AMA technique is not suitable for the viscous flow simulation and we propose some modifications, namely for capturing boundary layers and a wake. Moreover an additional smoothing technique increasing the regularity of the triangulation is presented.

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V. Dolejsi:

Anisotropic mesh adaptation and its application for scalar diffusion equations

Numerical Methods for Partial Differential Equations (submitted)

Abstract We present an efficient mesh adaptation algorithm, which can be successfully applied to numerical solutions of a wide range of 2D problems of physics and engineering described by partial differential equations. We are interested in the numerical solution of a general boundary value problem discretized on triangular grids. We formulate the necessary condition for the properties of the triangulation on which the discretization error is below the prescribed tolerance and control this necessary condition by the interpolation error. For the sufficiently smooth function, we recall the strategy how to construct the mesh on which the interpolation error is below the prescribed tolerance. Solving the boundary value problem we apply this strategy to the smoothed approximate solution. The novelty of the method lies in the smoothing procedure that, followed by the anisotropic mesh adaptation (AMA) algorithm, leads to the significant improvement of the numerical results. We apply AMA to the numerical solution of an elliptic equation, where the exact solution is known and demonstrate the practical aspects of the adaptation procedure: how to control the ratio between the longest and the shortest edge of the triangulation and how to control the transition of the coarsest part of the mesh to the finest one if the two length scales of all the triangles are clearly different. An example of the use of AMA for the physically relevant numerical simulation of a geometrically challenging industrial problem (inviscid transonic flow around NACA0012 profile) is presented.

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V. Dolejsi, M. Feistauer, C. Schwab: A Finite Volume Discontinuous Galerkin Scheme for Nonlinear Convection-Diffusion Problems

Calcolo (submitted), also Preprint ETH Zurich, January 2001.

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V. Dolejsi, M. Feistauer, C. Schwab: On some aspects of the Discontinuous Galerkin finite element method for conservation laws

Mathematics and Computers in Simulation (submitted), also The Preprint Series of the School of Mathematics, Charles University Prague, No. MATH-KNM-2001/5, 2001.

Abstract The paper is concerned with the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems. We discuss two versions of this method: a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume - finite element method. Its advantage is the use of only one mesh (in contrast to the combined finite volume - finite element schemes). However, it is of the first order only. b) Further, the pure discontinuous Galerkin finite element method of higher order is considered. In this case a new limiting is developed to avoid spurious oscillations in the vicinity of shocks.

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V. Dolejsi, M. Feistauer, C. Schwab: On Discontinuous Galerkin Methods for Nonlinear Convection-Diffusion Problems and Compressible Flow

Mathematica Bohemica, (to appear) also The Preprint Series of the School of Mathematics, Charles University Prague, No. MATH-KNM-2001/9, 2001.

Abstract The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flow. We discuss two versions of this method: a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume - finite element method. Its advantage is the use of only one mesh (in contrast to the combined finite volume - finite element schemes). However, it is of the first order only. b) Further, the pure discontinuous Galerkin finite element method of higher order is considered combined with a technique avoiding spurious oscillations in the vicinity of shock waves.

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V. Dolejsi, M. Feistauer: On the Discontinuous Galerkin Method for the Numerical Solution of Compressible High-Speed Flow

Proceeding of ENUMATH'01 conference, (submitted) also The Preprint Series of the School of Mathematics, Charles University Prague, No. MATH-KNM-2002/1, 2001.

Abstract The paper deals with an application of the discontinuous Galerkin finite element (DG FE) method to the numerical solution of the system of hyperbolic equations. We extend our results from DGM1, DGM2, where two versions of the DG FE method were applied to the scalar convection-diffusion equation. In order to avoid spurious oscillations near discontinuities we develop a new limiting which is based on the control of interelement jumps and switches from piecewise linear to piecewise constant approximations. Isoparametric finite elements are used near a curved boundary of nonpolygonal computational domain in order to achieve a physically admissible and sufficiently accurate numerical solution. Numerical examples of transonic flow through the GAMM channel and around the NACA0012 airfoil are presented. Finally, we mention some theoretical results obtained for a modified DG FE method applied to a nonlinear convection-diffusion problem.

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V. Dolejsi: On the Discontinuous Galerkin Method for the Numerical Solution of the Euler and the Navier-Stokes Equations

Int. J. Numer. Methods Fluids (submitted). Also The Preprint Series of the School of Mathematics, Charles University Prague, No. MATH-KNM-2002/2, 2002

Abstract The paper deals with the use of the discontinuous Galerkin finite element method (DG FE) for the numerical solution of compressible flows. We start with a scalar convection-diffusion equation and present a discretization with the aid of the antisymmetric variant of DG FE. We also mention some theoretical results. Then we extend the scheme to the system of the Euler and Navier-Stokes equations. The use of order limiting in vicinity of discontinuities and steep gradients and the use of superparametric finite elements are discussed. Several numerical examples are presented.

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V. Dolejsi, M. Feistauer: A Semi-Implicit Discontinuous Galerkin Finite Element Method for the Numerical Solution of Inviscid Compressible Flow

J. Comp. Phys. (submitted). Also The Preprint Series of the School of Mathematics, Charles University Prague, No. MATH-KNM-2003/2, 2003

Abstract We deal with the numerical solution of an inviscid compressible flow with the aid of the discontinuous Galerkin finite element method. Since the explicit discretization of the time derivative requires a high restriction of the time step, we propose a semi-implicit time discretization approach. The homogeneity of inviscid fluxes allows us a simple linearization of the Euler equations which leads to a linear algebraic system. Numerical experiments performed for the Ringleb flow problem verify a higher order of accuracy of the presented method and demonstrate lower CPU-time costs in comparison with an explicit method.

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V. Dolejsi, M. Feistauer, V. Sobotikova: Analysis of the Discontinuous Galerkin Method for Nonlinear Convection-Diffusion Problems

Comput. Methods Appl. Mech. Eng. (submitted). Also The Preprint Series of the School of Mathematics, Charles University Prague, No. MATH-KNM-2004/4, 2003

Abstract The subject-matter is the analysis of the discontinuous Galerkin finite element method applied to a nonlinear convection-diffusion problem. In the contrary to the standard FEM the requirement of the conforming properties is omitted. This allows us to consider general polyhedral elements with mutually disjoint interiors. We do not require their convexity, but assume only that they are uniformly star-shaped. We present an error analysis for the case of a nonsymmetric discretization of diffusion terms. Theoretical results are accompanied by numerical experiments.

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V. Dolejsi, M. Feistauer: Error estimates of the discontinuous Galerkin method for nonlinear nonstationary convection-diffusion problems

Numerical Functional Analysis and Optimization, (submitted). Also The Preprint Series of the School of Mathematics, Charles University Prague, No. MATH-KNM-2004/5, 2004

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V. Dolejsi, M. Feistauer: Analysis of Semi-Implicit DGFEM for Nonlinear Convection-Diffusion Problems

Numerische Mathematik, (submitted). Also The Preprint Series of the School of Mathematics, Charles University Prague, No. MATH-KNM-2004/1, 2004

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