Václav Kučera

On a robust discontinuous Galerkin technique for the solution of compressible flow

In the paper, we describe a numerical technique allowing the solution of compressible inviscid flow with a wide range of Mach numbers. The method is based on the application of the discontinuous Galerkin finite element method for the space discretization of the Euler equations written in the conservative form, combined with a semi-implicit time discretization. Special attention is paid to the treatment of boundary conditions and to the stabilization of the method in the vicinity of discontinuities avoiding the Gibbs phenomenon. As a result we obtain a technique allowing the numerical solution of flows with practically all Mach numbers without any modification of the Euler equations. This means that the proposed method can be used for the solution of high speed flows as well as low Mach number flows. Presented numerical tests prove the accuracy of the method and its robustness with respect to the Mach number.

Simulation of compressible viscous flow in time-dependent domains

The paper is concerned with the simulation of viscous compressible flow in time dependent domains. The dependence on time of the domain occupied by the fluid is taken into account with the aid of the Arbitrary Lagrangian-Eulerian (ALE) formulation of the compressible Navier-Stokes equations. They are discretized by the discontinuous Galerkin finite element method using piecewise polynomial discontinuous approximations. The time discretization is based on a semi-implicit linearized scheme, which leads to the solution of a linear algebraic system on each time level. A suitable treatment of boundary conditions and shock capturing are used, allowing the solution of flow with a wide range of Mach numbers. The applicability of the developed method is demonstrated by computational results obtained for compressible viscous flow in a channel with moving walls and flow induced airfoil vibrations.

Discontinuous Galerkin solution of compressible flow in time-dependent domains

This work is concerned with the simulation of inviscid compressible flow in time-dependent domains. We present an arbitrary Lagrangian-Eulerian (ALE) formulation of the Euler equations describing compressible flow, discretize them in space by the discontinous Galerkin method and introduce a semi-implicit linearized time stepping for the numerical solution of the complete problem. Special attention is paid to the treatment of boundary conditions and the limiting procedure avoiding the Gibbs phenomenon in the vicinity of discontinuities. The presented computational results show the applicability of the developed method.

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L2(L2)”- and “DG”-norm formed by the “L2(H1)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.

DGFEM for dynamical systems describing interaction of compressible fluid and structures

Abstract The paper is concerned with the numerical solution of flow-induced vibrations of elastic structures. The dependence on time of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian–Eulerian) formulation of the compressible Navier–Stokes equations. The deformation of the elastic body, caused by aeroelastic forces, is described by the linear dynamical elasticity equations. These two systems are coupled by transmission conditions. The flow problem is discretized by the discontinuous Galerkin finite element method (DGFEM) in space and by the backward difference formula (BDF) in time. The structural problem is discretized by conforming finite elements and the Newmark method. The fluid–structure interaction is realized via weak or strong coupling algorithms. The developed technique is tested by numerical experiments and applied to the simulation of vibrations of vocal folds during phonation onset.

The ALE Discontinuous Galerkin Method for the Simulatio of Air Flow Through Pulsating Human Vocal Folds

The aim of this work is the simulation of viscous compressible flows in human vocal folds during phonation. The computational domain is a bounded subset of IR2, whose geometry mimics the shape of the human larynx. During phonation, parts of the solid impermeable walls are moving in a prescribed manner, thus simulating the opening and closing of the vocal chords. As the governing equations we take the compressible Navier‐Stokes equations in ALE form. Space semidiscretization is carried out by the discontinuous Galerkin method combined with a linearized semi‐implicit approach. Numerical experiments are performed with the resulting scheme.

L ∞ (L 2)-error estimates for the DGFEM applied to convection–diffusion problems on nonconforming meshes

Abstract This paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection–diffusion Dirichlet problem. General nonconforming simplicial meshes are considered and the SIPG scheme is used. Under the assumption that the exact solution is sufficiently regular an L ∞ (L 2)-optimal error estimate is derived. The theoretical results are illustrated by numerical experiments.

Optimal L infinito (L 2)-Error Estimates for the DG Method Applied to Nonlinear ConvectionDiffusion Problems with Nonlinear Diffusion

This article is concerned with the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonstationary convection–diffusion problem with nonlinear convection and nonlinear diffusion. Optimal estimates in the L ∞(L 2)-norm are derived for the symmetric interior penalty (SIPG) scheme in two dimensions. The error analysis is carried out for nonconforming triangular meshes under the assumption that the exact solution of the problem and the solution of a linearized elliptic dual problem are sufficiently regular.

Two Dimensional Compressible Fluid-Structure Interaction Model Using DGFEM

The subject of this paper is the numerical solution of the interaction of compressible flow and an elastic body with a special emphasis on the simulation of vibrations of vocal folds during phonation onset. The time-dependence of the domain occupied by the fluid is treated by the ALE (Arbitrary Lagrangian-Eulerian) method and the compressible Navier-Stokes equations are written in the ALE form. The deformation of the elastic body, caused by the aeroelastic forces, is described by the linear dynamical elasticity equations. Both these systems are coupled by transmission conditions. For the space-discretization of the flow problem the discontinuous Galerkin finite element method (DGFEM) is used. The time-discretization is realized by the backward difference formula (BDF). The structural problem is discretized by the conforming finite element method and the Newmark method. The results of the use of two different couplings and their comparison are presented.

Discontinuous Galerkin Method – A Robust Solver for Compressible Flow

The subject of the paper is the numerical simulation of inviscid and viscous compressible flow in time dependent domains. The motion of the boundary of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian-Eulerian) formulation of the Euler and Navier-Stokes equations describing compressible flow. They are discretized in space by the discontinuous Galerkin (DG) finite element method using piecewise polynomial discontinuous approximations. For the time discretization the BDF method or DG in time is used. Moreover, we use a special treatment of boundary conditions and shock capturing, allowing the solution of flow with a wide range of Mach numbers. As a result we get an efficient and robust numerical process. We show that the method allows to solve numerically the flow with a wide range of Mach numbers and it is applicable to the solution of practically relevant problems of flow induced airfoil vibrations.