main-content

## Über dieses Buch

This first volume of the proceedings of the 8th conference on "Finite Volumes for Complex Applications" (Lille, June 2017) covers various topics including convergence and stability analysis, as well as investigations of these methods from the point of view of compatibility with physical principles. It collects together the focused invited papers comparing advanced numerical methods for Stokes and Navier–Stokes equations on a benchmark, as well as reviewed contributions from internationally leading researchers in the field of analysis of finite volume and related methods, offering a comprehensive overview of the state of the art in the field.

The finite volume method in its various forms is a space discretization technique for partial differential equations based on the fundamental physical principle of conservation, and recent decades have brought significant advances in the theoretical understanding of the method. Many finite volume methods preserve further qualitative or asymptotic properties, including maximum principles, dissipativity, monotone decay of free energy, and asymptotic stability. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete level. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications.

The book is a valuable resource for researchers, PhD and master’s level students in numerical analysis, scientific computing and related fields such as partial differential equations, as well as engineers working in numerical modeling and simulations.

## Inhaltsverzeichnis

### Bound-Preserving High Order Finite Volume Schemes for Conservation Laws and Convection-Diffusion Equations

Finite volumeShu, Chi-Wang schemes evolve cell averages based on high order reconstructions to solve hyperbolic conservation laws and convection-diffusion equations. The design of the reconstruction procedure is crucial for the stability of the finite volume schemes. Various reconstruction procedures, such as total variation diminishing (TVD), total variation bounded (TVB), essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstructions have been developed in the literature to obtain non-oscillatory and high order finite volume schemes. However, it is a challenge to design strictly bound-preserving finite volume schemes which are genuinely high order accurate, including at smooth extrema. These include maximum-principle-preserving schemes for scalar conservation laws and convection-diffusion equations, and positivity-preserving (for relevant physical quantities such as density, pressure or water height) for systems. In this presentation we survey strategies in the recent literature to design high order bound-preserving finite volume schemes, including a general framework in constructing high order bound-preserving finite volume schemes for scalar and systems of hyperbolic conservation laws through a simple scaling limiter and a convex combination argument based on first order bound-preserving building blocks, and a non-standard finite volume scheme which evolves the so-called “double cell averages” for solving convection-diffusion equations which can maintain the bound-preserving property and high order accuracy simultaneously.

Chi-Wang Shu

### Some Geophysical Applications with Finite Volume Solvers of Two-Layer and Two-Phase Systems

There existsFernandez-Nieto, Enrique in the literature a huge range of geophysical applications that have been modeled trough two-layer or two-phase models. In this work first some averaged two-layer and two-phase models are presented. We focus on applications to submarine avalanches, debris flows and sediment transport in rivers. Secondly, their numerical approximation by a finite volume method is discussed and a numerical test is presented.

E. D. Fernández-Nieto

### Some Discrete Functional Analysis Tools

The objectiveGallouet, Thierry of this short paper is to present discrete functional analysis tools for proving the convergence of numerical schemes, mainly for elliptic and parabolic equations (Stefan problem and incompressible and compressible Navier–Stokes equations, for instance). The main part of these results are given in some papers coauthored with several coworkers.

Thierry Gallouët

### A Simple Finite-Volume Method on a Cartesian Mesh for Pedestrian Flows with Obstacles

We considerCheng, Yuanzhen a two-dimensionalChertock, AlinapedestrianKurganov, Alexander flow model with obstacles governed by scalar hyperbolic conservation laws, in which the flux is implicitly dependent on the density through the Eikonal equation. We propose a simple second-order finite-volume method, which is applicable to the case of obstacles of arbitrary shapes. Though the method is only first-order accurate near the obstacles, it is robust and provides sharp resolution of discontinuities as illustrated in a number of numerical experiments.

Yuanzhen Cheng, Alina Chertock, Alexander Kurganov

### Benchmark Proposal for the FVCA8 Conference: Finite Volume Methods for the Stokes and Navier–Stokes Equations

This benchmarkBoyer, FranckproposesOmnes, Pascal test-cases to assess innovative finite volume type methods developped to solve the equations of incompressible fluid mechanics. Emphasis is set on the ability to handle very general meshes, on accuracy, robustness and computational complexity. Two-dimensional as well as three-dimensional tests with known analytical solutions are proposed for the steady Stokes and both steady and unsteady Navier–Stokes equations, as well as classical lid-driven cavity tests.

Franck Boyer, Pascal Omnes

### A High-Order Finite Volume Solver on Locally Refined Cartesian Meshes—Benchmark Session

This paper providesLe Touze, DavidnumericalDe Leffe, MatthieuresultsLi, Zhe of a finite volumeVittoz, Louis solver basedOger, Guillaume on high-order schemes for Cartesian Meshes. This solver is dedicated to the computation of complex flows in marine and ocean engineering. It aims at solving complex hydrodynamic flows that cannot be yet propelery solved such as breaking waves, fluid-structure interactions, turbulent flows, etc. Since high-order schemes are highly recommended for under-resolved simulations, a WENO5 reconstruction is combined with a 4th order Runge–Kutta scheme for time integration.

Louis Vittoz, Guillaume Oger, Zhe Li, Matthieu de Leffe, David Le Touzé

### Benchmark Session: The 2D Hybrid High-Order Method

We consider hereDi Pietro, Daniele the two-dimensional version of the Hybrid High-Order (HHO) methodKrell, Stella for the steady incompressible Navier–Stokes equations originally introduced in [Di Pietro, Krell, A Hybrid High-Order method for the steady incompressible Navier–Stokes problem, preprint arXiv:1607.08159 math.NA]. This method displays several advantageous features: it is inf-sup stable on general meshes including polyhedral elements and nonmatching interfaces, it supports arbitrary approximation order, and has a reduced computational cost thanks to the possibility of statically condensing a subset of both velocity and pressure degrees of freedom (DOFs) at each nonlinear iteration.

Daniele A. Di Pietro, Stella Krell

### Benchmark: Two Hybrid Mimetic Mixed Schemes for the Lid-Driven Cavity

We brieflyDroniou, JeromepresentEymard, Robert the Hybrid Mimetic Mixed scheme for the steady incompressible Navier–Stokes equations. Two centred approximations of the nonlinear convection term are proposed and compared, between themselves as well as with reference results from the literature, on the lid driven cavity test case applied to various grid types.

Jérôme Droniou, Robert Eymard

### Results with a Locally Refined MAC-Like Scheme—Benchmark Session

We recall theChenier, EricextensionEymard, Robert of the Marker and Cell (MAC) scheme for locally refinedHerbin, Raphaèle grids which was introduced in Chénier et al. (Calcolo 52(1), 69–107 (2015), [3]) and present the results obtained on the lid driven cavity test.

Eric Chénier, Robert Eymard, Raphaèle Herbin

### Numerical Results for a Discrete Duality Finite Volume Discretization Applied to the Navier–Stokes Equations

We presentOmnes, Pascal an applicationDelcourte, Sarah of the discrete duality finite volume method to the numerical approximation of the 2D Stokes or (unsteady) Navier–Stokes equations associated to Dirichlet boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical MAC scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. Different numerical examples over triangular, cartesian, quadrangular and locally refined meshes are led in order to illustrate the possibilities and weaknesses of the method.

Sarah Delcourte, Pascal Omnes

### Benchmark Session: The 2D Discrete Duality Finite Volume Method

In this paper, weBoyer, FranckproposeNabet, Flore a contributionKrell, Stella to the FVCA8 benchmark on numerical methods for the Stokes and Navier–Stokes equations. We present some results obtained with the Discrete Duality Finite Volume (DDFV).

Franck Boyer, Stella Krell, Flore Nabet

### FVCA8 Benchmark for the Stokes and Navier–Stokes Equations with the TrioCFD Code—Benchmark Session

This paperPuscas, Maria Adela is devotedAlain, Cartalade to the studyPierre-Emmanuel, Angeli of convergenceFauchet, Gauthier orders of several numerical methods that are implemented in the TrioCFD code dedicated to the simulation of turbulent flows and heat transfer in nuclear engineering applications. The spatial discretization is based on Finite Difference-Volume or Finite Element-Volume methods. A projection method is applied to update the velocity and the pressure. The time scheme can be either explicit or implicit, and hexahedral or tetrahedral meshes can be used for simulations. In this paper, the test cases are relative to steady Stokes problems, steady and unsteady Navier–Stokes problems, and finally the well-known lid-driven cavity flow case. The latter proposes several comparisons between our simulations and numerical data already published in the literature, while the other cases yield the values of convergence orders by using the analytical solutions. The accuracy of the results obtained with TrioCFD differs according to the types of mesh used for simulations, the viscosity values or the source terms in the equations.

P.-E. Angeli, M.-A. Puscas, G. Fauchet, A. Cartalade

### Analysis of a Positive CVFE Scheme for Simulating Breast Cancer Development, Local Treatment and Recurrence

In this paper, a positiveFoucher, Françoise CVFE scheme forSaad, MazensimulatingIbrahim, Moustafa an anisotropic breast cancer development is analyzed. The mathematical model includes reaction–diffusion-convection terms with an anisotropic heterogeneous diffusion tensor. The diffusion term is discretized using a finite element method combined with the use of Godunov scheme over a primal triangular mesh. The convective term is discretized using a nonclassical upwind finite volume scheme over a barycentric dual mesh. The scheme ensures the validity of the discrete positivity preserving and other discrete properties without any restriction on the transmissibility coefficients. Finally, a numerical simulation is provided to simulate the spread of tumor cells before and after applying a local treatment using the surgery.

Françoise Foucher, Moustafa Ibrahim, Mazen Saad

### Céa-Type Quasi-Optimality and Convergence Rates for (Adaptive) Vertex-Centered FVM

For a generalErath, Christoph second order linear elliptic PDE, we showPraetorius, Dirk a generalized Céa lemma for a vertex-centered finite volume method (FVM). The latter implies, in particular, a comparison result between the solutions of FVM and the finite element method (FEM). Furthermore, for a symmetric PDE, i.e., no convection is present, we prove linear convergence with generically optimal algebraic rates for an adaptive FVM algorithm.

Christoph Erath, Dirk Praetorius

### Numerical Convergence for a Diffusive Limit of the Goldstein–Taylor System on Bounded Domain

This paper dealsTherme, Nicolas with the diffusiveMathis, Hélène limit of the scaled Goldstein–Taylor model and its approximation by an Asymptotic Preserving Finite Volume scheme. The problem is set in some bounded interval with non-homogeneous boundary conditions depending on time. We obtain a uniform estimate in the small parameter $$\varepsilon$$ using a relative entropy of the discrete solution with respect to a suitable profile which satisfies the boundary conditions expected to hold as $$\varepsilon$$ goes to 0.

Hélène Mathis, Nicolas Therme

### Lagrange-Flux Schemes and the Entropy Property

The Lagrange-FluxDe Vuyst, Florian schemes are Eulerian finite volume schemes that make use of an approximate Riemann solver in Lagrangian description with particular upwind convective fluxes. They have been recently designed as variant formulations of Lagrange-remap schemes that provide better HPC performance on modern multicore processors, see [De Vuyst et al., OGST 71(6), 2016]. Actually Lagrange-Flux schemes show several advantages compared to Lagrange-remap schemes, especially for multidimensional problems: they do not require the computation of deformed Lagrangian cells or mesh intersections as usually done in the remapping process. The paper focuses on the entropy property of Lagrange-Flux schemes in their semi-discrete in space form, for one-dimensional problems and for the compressible Euler equations as example. We provide pseudo-viscosity pressure terms that ensure entropy production of order $$O(|\varDelta u|^3)$$, where $$|\varDelta u|$$ represents a velocity jump at a cell interface. Pseudo-viscosity terms are also designed to vanish into expansion regions as it is the case for rarefaction waves.

Florian De Vuyst

### -Stability of IMEX-BDF2 Finite Volume Scheme for Convection-Diffusion Equation

In this paper, weCalgaro, Caterina propose a finiteEzzoug, Meriem volume scheme for solving a two-dimensional convection-diffusion equation on general meshes. This work is based on a implicit-explicit (IMEX) second order method and it is issued from the seminal paper [5]. In the framework of MUSCL methods, we will prove that the local maximum property is guaranteed under an explicit Courant–Friedrichs–Levy condition and the classical hypothesis for the triangulation of the domain.

Caterina Calgaro, Meriem Ezzoug

### Low Mach Number Limit of a Pressure Correction MAC Scheme for Compressible Barotropic Flows

We study theHerbin, RaphaèleincompressibleLatché, Jean-ClaudelimitSaleh, Khaled of a pressure correction MAC scheme (Herbin et al., Math. Model. Numer. Anal. 48, 1807–1857, 2013) [3] for the unstationary compressible barotropic Navier–Stokes equations. Provided the initial data are well-prepared, the solution of the numerical scheme converges, as the Mach number tends to zero, towards the solution of the classical pressure correction inf-sup stable MAC scheme for the incompressible Navier–Stokes equations.

Raphaèle Herbin, Jean-Claude Latché, Khaled Saleh

### Convergence of the MAC Scheme for Variable Density Flows

We prove in thisGallouet, Thierry paper theMallem, KhadidjaconvergenceHerbin, Raphaèle of an semi-implicit MAC schemeLatché, Jean-Claude for the time-dependent variable density Navier–Stokes equations.

T. Gallouët, R. Herbin, J.-C. Latché, K. Mallem

### Uniform-in-Time Convergence of Numerical Schemes for a Two-Phase Discrete Fracture Model

Flow and transportHennicker, Julian in fracturedDroniou, JeromeporousMasson, Roland media are of paramount importance for many applications such as petroleum exploration and production, geological storage of carbon dioxide, hydrogeology, or geothermal energy. We consider here the two-phase discrete fracture model introduced in [3] which represents explicitly the fractures as codimension one surfaces immersed in the surrounding matrix domain. Then, the two-phase Darcy flow in the matrix is coupled with the two-phase Darcy flow in the fractures using transmission conditions accounting for fractures acting either as drains or barriers. The model takes into account complex networks of fractures, discontinuous capillary pressure curves at the matrix fracture interfaces and can be easily extended to account for gravity including in the width of the fractures. It also includes a layer of damaged rock at the matrix fracture interface with its own mobility and capillary pressure functions. In this work, the convergence analysis carried out in [3] in the framework of gradient discretizations [2] is extended to obtain the uniform-in-time convergence of the discrete solutions to a weak solution of the model.

J. Droniou, J. Hennicker, R. Masson

### Design and Analysis of a Finite Volume Scheme for a Concrete Carbonation Model

In this paperChainais-Hillairet, Claire we introduceMerlet, Benoît a finiteZurek, Antoine volume scheme for a concrete carbonation model proposed by Aiki and Muntean in [1]. It consists in a Euler discretisation in time and a Scharfetter–Gummel discretisation in space. We give here some hints for the proof of the convergence of the scheme and show numerical experiments.

Claire Chainais-Hillairet, Benoît Merlet, Antoine Zurek

### Equilibrated Stress Reconstructions for Linear Elasticity Problems with Application to a Posteriori Error Analysis

We presentErn, Alexandre an a posterioriDi Pietro, DanieleerrorRiedlbeck, Rita estimate for the linear elasticity problem. The estimate is based on an equilibrated reconstruction of the Cauchy stress tensor, which is obtained from mixed finite element solutions of local Neumann problems. We propose two different reconstructions: one using Arnold–Winther mixed finite element spaces providing a symmetric stress tensor, and one using Arnold–Falk–Winther mixed finite element spaces with a weak symmetry constraint. The performance of the estimate is illustrated on a numerical test with analytical solution.

Rita Riedlbeck, Daniele A. Di Pietro, Alexandre Ern

### Uniform Second Order Convergence of a Complete Flux Scheme on Nonuniform 1D Grids

The accurateFarrell, Patricio and efficientLinke, Alexander discretization of singularly perturbed advection-diffusion equations on arbitrary domains remains an open problem. An interesting approach to tackle this problem is the complete flux scheme (CFS) proposed by G.D. Thiart and further investigated by J. ten Thije Boonkkamp. For the CFS, uniform second order convergence has been proven on structured grids. We extend a version of the CFS to nonuniform grids, predict an error bound and numerically verify it for a solution to an ODE with a boundary layer.

### The Asymmetric Gradient Discretisation Method

An asymmetric version of the gradient discretisation method is developed for linear anisotropic elliptic equations. Error estimates and convergence are proved for this method, which is showed to cover all finite volume methods.

J. Droniou, R. Eymard

### DGM, an Item of GDM

We show that a versionEymard, Robert of the Discontinuous Galerkin Method (DGM) canGuichard, Cindy be included in the Gradient Discretisation Method (GDM) framework. We prove that it meets the main mathematical gradient discretisation properties on any kind of polytopal mesh, and that it is identical to the Symmetric Interior Penalty Galerkin (SIPG) method in the case of first order polynomials. A numerical study shows the effect of the numerical parameter included in the scheme.

Robert Eymard, Cindy Guichard

### Positive Lower Bound for the Numerical Solution of a Convection-Diffusion Equation

In this work, weChainais-Hillairet, Claire apply a methodVasseur, Alexis due to De Giorgi [3] in orderMerlet, Benoît to establish a positive lower bound for the numerical solution of a stationary convection-diffusion equation.

Claire Chainais-Hillairet, Benoît Merlet, Alexis F. Vasseur

### Raviart Thomas Petrov–Galerkin Finite Elements

The generalGreff, IsabelletheoryDubois, Francois of Babuška ensuresPierre, Charles necessary and sufficient conditions for a mixed problem in classical or Petrov–Galerkin form to be well posed in the sense of Hadamard. Moreover, the mixed method of Raviart-Thomas with low-level elements can be interpreted as a finite volume method with a non-local gradient. In this contribution, we propose a variant of type Petrov–Galerkin to ensure a local computation of the gradient at the interfaces of the elements. The in-depth study of stability leads to a specific choice of the test functions. With this choice, we show on the one hand that the mixed Petrov–Galerkin obtained is identical to the finite volumes scheme “volumes finis à 4 points” (“VF4”) of Faille, Galloüet and Herbin and to the condensation of mass approach developed by Baranger, Maitre and Oudin. On the other hand, we show the stability via an inf-sup condition and finally the convergence with the usual methods of mixed finite elements.

François Dubois, Isabelle Greff, Charles Pierre

### Towards Pressure-Robust Mixed Methods for the Incompressible Navier–Stokes Equations

When mixedLinke, Alexander methods forAhmed, NaveedtheMerdon, Christian incompressible Navier–Stokes were introduced in the early 70ies, it was claimed that the divergence-constraint could be relaxed without danger. Recently, this claim has been challenged. Therefore, we review the numerical error analysis of mixed methods and show that divergence-free/pressure-robust mixed methods behave in a provably much more robust way.

Naveed Ahmed, Alexander Linke, Christian Merdon

### Numerical Analysis of the DDFV Method for the Stokes Problem with Mixed Neumann/Dirichlet Boundary Conditions

The aim of this workGoudon, Thierry is to analyze “Discrete DualityKrell, StellaFiniteLissoni, Giulia Volume” schemes (DDFV for short) on general meshes by adapting the theory known for the linear Stokes problem with Dirichlet boundary conditions to the case of Neumann boundary conditions on a fraction of the boundary. We prove well-posedness for stabilized schemes and we derive some error estimates. Finally, we illustrate some numerical results in which we compare stabilized and unstabilized schemes.

Thierry Goudon, Stella Krell, Giulia Lissoni

### An Error Estimate for the Approximation of Linear Parabolic Equations by the Gradient Discretization Method

We establishDroniou, Jerome an errorGallouet, Thierry estimate for fully discreteEymard, Robert time-space gradientGuichard, Cindy schemes on a simpleHerbin, Raphaèle linear parabolic equation. This error estimate holds for all the schemes within the framework of the gradient discretisation method: conforming and non conforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approximation finite volume scheme and some discontinuous Galerkin schemes.

J. Droniou, R. Eymard, T. Gallouët, C. Guichard, R. Herbin

### Uniform Estimates for Approximate Solutions of the Bipolar Drift-Diffusion System

We establish uniform $$L^\infty$$boundsBessemoulin, Marianne for approximateChainais-Hillairet, Claire solutions of the drift-diffusionJüngel, Ansgar system for electrons and holes in semiconductor devices, computed with the Scharfetter–Gummel finite-volume scheme. The proof is based on a Moser iteration technique adapted to the discrete case.

M. Bessemoulin-Chatard, C. Chainais-Hillairet, A. Jüngel

### Some Convergence Results of a Multi-dimensional Finite Volume Scheme for a Time-Fractional Diffusion-Wave Equation

We present an implicitBradji, Abdallah finite volume scheme for a linear time-fractional diffusion-wave equation using the discrete gradient introduced in Eymard et al. (IMA J Numer Anal 30:1009–1043, 2010, [2]). A convergence order for the error between the gradient of the exact solution and the discrete gradient of the approximate solution is proved. This yields an $$L^\infty (L^2)$$–error estimate.

### Optimal Order of Convergence for the Upwind Scheme for the Linear Advection on a Bounded Domain

This proceeding presents anAguillon, NinaoptimalBoyer, Franck error estimate in the $$L^{1}$$-norm of order 1 / 2 between the exact solution of an initial and boundary value problem for the linear advection equation and its approximation by the explicit upwind scheme. The space domain is bounded and a Dirichlet condition is thus imposed on the entering part of the boundary. This result extends the analysis given in Merlet and Vovelle (Numer. Math. 106(1), 129–155 (2007), [10]) that concerns the case where the equation is posed on the whole space. One of the key point of the proof is the analysis of a suitable regularization by convolution of the exact (weak) solution. Compared to Merlet and Vovelle (Numer. Math. 106(1), 129–155 (2007), [10]) we also relax some hypothesis on the velocity field, which in particular is allowed to be somehow discontinuous in time. This proceeding is a short version of Aguillon and Boyer (IMA J. Numer. Anal. (2017), [1]), aiming to present the steps of the proof and the new intermediate results.

Nina Aguillon, Franck Boyer

### Numerical Scheme for Regularised Riemannian Mean Curvature Flow Equation

Finite volume schemeTibensky, Matus for regularised Riemannian mean curvature flowHandlovicová, Angela equation is discussed. Stability estimates and the uniqueness of the numerical solution are listed. Convergence of the numerical scheme to the discrete solution is listed as well. Numerical results are presented in the final section.

Matúš Tibenský, Angela Handlovičová

### A Finite Volume Scheme for a Seawater Intrusion Model with Cross-Diffusion

We consider a finiteAit Hammou Oulhaj, Ahmed volume scheme for a seawater intrusion model. It is based on a two-point flux approximation with upwind mobilities. The scheme preserves at the discrete level the main features of the continuous problem: the nonnegativity of the solutions, the decay of the energy and the control of the entropy and its dissipation. Moreover the scheme converges towards a weak solution to the problem. Numerical results are provided to illustrate the behavior of the model and of the scheme.

Ahmed Ait Hammou Oulhaj

### Finite Volume Approximation of a Degenerate Immiscible Two-Phase Flow Model of Cahn–Hilliard Type

We propose a two-pointCancès, Clément flux approximation Finite VolumeNabet, Flore scheme for a model of incompressible and immiscible two-phase flow of Cahn–Hilliard type with degenerate mobility. This model was derived from a variational principle and can be interpreted as the Wasserstein gradient flow of the free energy. The fundamental properties of the continuous model, namely the positivity of the concentrations, the decay of the free energy, and the boundedness of the Boltzmann entropy, are preserved by the numerical scheme. Numerical simulations are provided to illustrate the behavior of the model and of the numerical scheme.

Clément Cancès, Flore Nabet

### A Nonlinear Discrete Duality Finite Volume Scheme for Convection-Diffusion Equations

We introduceChainais-Hillairet, Claire a nonlinear DDFV scheme for a convection-diffusion equation. The scheme conserves the mass, satisfies an energy-dissipation inequality and provides positiveKrell, Stella approximate solutions even on very general grids. Numerical experiments illustrate these properties.

Clément Cancès, Claire Chainais-Hillairet, Stella Krell

### Stationarity and Vorticity Preservation for the Linearized Euler Equations in Multiple Spatial Dimensions

StationaryBarsukow, Wasilij solutions are a prominent subset of solutions to hyperbolic systems of PDEs. Failure of numerical methods to maintain stationarity is easily visible which makes these solutions an important class. Consider finite volume schemes solving multi-d linearized Euler equations on equidistant Cartesian grids. We formulate conditions for a scheme to have stationary states that are discretizations of all analytic stationary states. Such schemes are termed stationarity preserving. Stationarity preservation for the linearized Euler equations is shown to be equivalent to vorticity preservation.

Wasilij Barsukow

### Goal-Oriented Error Analysis of a DG Scheme for a Second Gradient Elastodynamics Model

In this notePryer, Tristan we proposeGiesselmann, Jan a discontinuous Galerkin in space, continuous Galerkin in time method for a problem arising in elastodynamics with phase transition. We make use of a dispersion operator from (Bona et al., Math. Comput. 82(283), 1401–1432, 2013) [3] allowing us to construct a consistent scheme. We derive goal-oriented a posteriori error estimators for this scheme based on dual weighted residuals. We conclude by summarising extensive numerical experiments.

Jan Giesselmann, Tristan Pryer

### Simplified Model for the Clarinet and Numerical Schemes

A very simplifiedPrignet, Alain model for the clarinet consists in a system of two coupled 1D PDE’s of fluid mechanics-acoustics, expressing the wave equation with the unknowns velocity and pressure with reflection at the boundaries. No damping is taken into account and the reed model is highly simplified. Two numerical schemes are considered. The first one based on staggered grids does not dissipate with the maximal time step satisfying the CFL condition. The second one, using only one grid, is dissipating due to upstream weighting. An energy estimate is proved because of the numerical formulation which is suited to the boundary conditions.

Alain Prignet

### Backmatter

Weitere Informationen