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## Über dieses Buch

The proceedings of the 9th conference on "Finite Volumes for Complex Applications" (Bergen, June 2020) are structured in two volumes. The first volume collects the focused invited papers, as well as the reviewed contributions from internationally leading researchers in the field of analysis of finite volume and related methods. Topics covered include convergence and stability analysis, as well as investigations of these methods from the point of view of compatibility with physical principles. Altogether, a rather comprehensive overview is given on the state of the art in the field. The properties of the methods considered in the conference give them distinguished advantages for a number of applications. These include fluid dynamics, magnetohydrodynamics, structural analysis, nuclear physics, semiconductor theory, carbon capture utilization and storage, geothermal energy and further topics. The second volume covers reviewed contributions reporting successful applications of finite volume and related methods in these fields.

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. Many finite volume methods preserve further qualitative or asymptotic properties, including maximum principles, dissipativity, monotone decay of free energy, and asymptotic stability, making the finite volume methods 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

### Interplay Between Diffusion Anisotropy and Mesh Skewness in Hybrid High-Order Schemes

Droniou, J.We explore the effects of mesh skewness on the accuracy of standard Hybrid High-Order (HHO) schemes for anisotropic diffusion equations. After defining a notion of regular skewed mesh sequences, which allows, e.g., for elements that become more and more elongated during mesh refinement, we establish an error estimate in which we precisely track the dependency of the local multiplicative constants in terms of the diffusion tensor and mesh skewness. This dependency makes explicit an interplay between the local diffusion properties and the distortion of the elements. We then provide several numerical results to assess the practical convergence properties of HHO for highly anisotropic diffusion or highly distorted meshes. These tests indicate a more robust behaviour than the theoretical estimate predicts.

J. Droniou

### -Convergence of Finite Volume Solutions of the Euler Equations

WeLukáčová-Medvid’ová, Mária review our recent results on the convergence of invariant domain-preserving finite volume solutions to the Euler equations of gas dynamics. If the classical solution exists we obtain strong convergence of numerical solutions to the classical one applying the weak-strong uniqueness principle. On the other hand, if the classical solution does not exist we adapt the well-known Prokhorov compactness theorem to space-time probability measures that are generated by the sequences of finite volume solutions and show how to obtain the strong convergence in space and time of observable quantities. This can be achieved even in the case of ill-posed Euler equations having possibly many oscillatory solutions.

Mária Lukáčová-Medvid’ová

### Time-Dependent Conservation Laws on Cut Cell Meshes and the Small Cell Problem

May, SandraWhen solving time-dependent conservation laws on cut cell meshes, one has to face the small cell problem: standard explicit schemes are not stable if the time step is chosen based on the size of the background cells. Therefore, special schemes must be developed. The first part of this contribution discusses the small cell problem in detail and summarizes several existing solution approaches in the context of both finite volume (FV) schemes and discontinuous Galerkin (DG) schemes. In the second part, we present our two fundamentally different solution approaches for overcoming the small cell problem: the FV based mixed explicit implicit scheme, developed in collaboration with Berger (J. Sci. Comput. 71, pp. 919–943, 2017), and the DG based Domain-of-Dependence (DoD) stabilization, joint work with Engwer, Nüßing, and Streitbürger ( ArXiv:1906.05642 ).

Sandra May

### Reactive Flow in Fractured Porous Media

Fumagalli, Alessio Scotti, AnnaIn this work we present a model reduction procedure to derive a hybrid-dimensional framework for the mathematical modeling of reactive transport in fractured porous media. Fractures are essential pathways in the underground which allow fast circulation of the fluids present in the rock matrix, often characterized by low permeability. However, due to infilling processes fractures may change their hydraulic properties and become barriers for the flow and creating impervious blocks in the underground. The geometrical as well as the physical properties of the fractures require a special treatment to allow the subsequent numerical discretization to be affordable and accurate. The aim of this work is to introduce a simple yet complete mathematical model to account for such diagenetic effects where chemical reactions will occlude or empty portions of the porous media and, in particular, fractures.

Alessio Fumagalli, Anna Scotti

### Numerical Schemes for Semiconductors Energy-Transport Models

Bessemoulin-Chatard, Marianne Chainais-Hillairet, Claire Mathis, HélèneWe introduce some finite volume schemes for unipolar energy-transport models. Using a reformulation in dual entropy variables, we can show the decay of a discrete entropy with control of the discrete entropy dissipation.

Marianne Bessemoulin-Chatard, Claire Chainais-Hillairet, Hélène Mathis

### Compatible Discrete Operator Schemes for the Steady Incompressible Stokes and Navier–Stokes Equations

Bonelle, Jérôme Ern, Alexandre Milani, RiccardoWe extend the Compatible Discrete Operator (CDO) schemes to the steady incompressible Stokes and Navier–Stokes equations. The main features of the CDO face-based schemes are recalled: a hybrid velocity discretization with degrees of freedom at faces and cells, a stabilized velocity gradient reconstruction defined on the face-based subcell pyramids, and a discrete pressure attached to the mesh cells. We introduce a discrete divergence operator that will account for the velocity-pressure coupling, and a hybrid discretization of the convection term. The results of several benchmark test cases validate the framework.

Jérôme Bonelle, Alexandre Ern, Riccardo Milani

### On the Significance of Pressure-Robustness for the Space Discretization of Incompressible High Reynolds Number Flows

Linke, A. Merdon, C.Only recently, strong gradient fields in the momentum balance of incompressible flows have been identified as a common major source for numerical errors in flow solvers. The novel notion of pressure-robustness denotes those space discretizations that behave in a robust manner with respect to strong gradient fields. This contribution elaborates on certain advantages of pressure-robust solvers versus standard solvers: (i) the asymptotic convergence rate of pressure-robust solvers may be reached on much coarser grids than for standard solvers; (ii) certain preasymptotic convergence-rates may be provably suboptimal for standard solvers; thus, low-order pressure-robust solver can outperform high-order classical solvers on coarse grids. Last but not least, the contribution explains how strong gradient fields develop in complex incompressible flows.

### Well-Balanced Discretisation for the Compressible Stokes Problem by Gradient-Robustness

Linke, Alexander Merdon, ChristianBased on the novel concept of gradient-robustness a well-balanced and provably convergent scheme for the compressible Stokes equations is discussed. Gradient-robustness means that arbitrary gradient fields in the momentum balance are correctly balanced by the discrete pressure gradient if there is enough mass in the system to compensate the force. For low Mach numbers the scheme degenerates to a recent inf-sup stable and pressure-robust discretisation for the incompressible Stokes equations. Numerical examples illustrate the properties for nearly-hydrostatic low Mach number flows also for nonlinear equations of state.

### A Second Order Consistent MAC Scheme for the Shallow Water Equations on Non Uniform Grids

Gallouët, T. Herbin, R. Latché, J.-C. Nasseri, Y.We propose in this paper a formally second order scheme for the numerical simulation of the shallow water equations in two space dimensions, based on the so-called Marker-And-Cell (MAC) staggered discretization on non uniform grids. For the space discretization, we use a MUSCL-like scheme for the convection operators while the pressure gadient is centered; time discretization is performed with the Heun scheme. The scheme preserves the positivity of the water height and “lake at rest” steady states. Its consistency in the Lax-Wendroff sense is proven.

T. Gallouët, R. Herbin, J.-C. Latché, Y. Nasseri

### Post-processing of Fluxes for Finite Volume Methods for Elliptic Problems

Cheng, Hanz MartinWe develop post-processing techniques for fluxes obtained from finite volume methods, which enables us to reconstruct flow densities in an H-div space. These post-processing techniques ensure that mass is conserved locally, which is very important, e.g., in geophysical applications.

Hanz Martin Cheng

### Exponential Decay to Equilibrium of Nonlinear DDFV Schemes for Convection-Diffusion Equations

Chainais-Hillairet, Claire Krell, StellaWe introduce a nonlinear DDFV scheme for an anisotropic linear convec tion-diffusion equation with mixed boundary conditions and we establish the exponential decay of the scheme towards its steady-state.

Claire Chainais-Hillairet, Stella Krell

### Bounds for Numerical Solutions of Noncoercive Convection-Diffusion Equations

Chainais-Hillairet, Claire Herda, MaximeIn this work, we apply an iterative energy method à la de Giorgi in order to establish $$L^{\infty }$$ bounds for numerical solutions of noncoercive convection-diffusion equations with mixed Dirichlet-Neumann boundary conditions.

Claire Chainais-Hillairet, Maxime Herda

### On Four Numerical Schemes for a Unipolar Degenerate Drift-Diffusion Model

We consider a unipolar degenerate drift-diffusion system where the relation between the concentration of the charged species c and the chemical potential h is $$h(c)=\log \frac{c}{1-c}$$. For four different finite volume schemes based on four different formulations of the fluxes of the problem, we discuss stability and existence results. For two of them, we report a convergence proof. Numerical experiments illustrate the behaviour of the different schemes.

Clément Cancès, Claire Chainais Hillairet, Jürgen Fuhrmann, Benoît Gaudeul

### Non-isothermal Scharfetter–Gummel Scheme for Electro-Thermal Transport Simulation in Degenerate Semiconductors

Kantner, Markus Koprucki, ThomasElectro-thermal transport phenomena in semiconductors are described by the non-isothermal drift-diffusion system. The equations take a remarkably simple form when assuming the Kelvin formula for the thermopower. We present a novel, non-isothermal generalization of the Scharfetter–Gummel finite volume discretization for degenerate semiconductors obeying Fermi–Dirac statistics, which preserves numerous structural properties of the continuous model on the discrete level. The approach is demonstrated by 2D simulations of a heterojunction bipolar transistor.

Markus Kantner, Thomas Koprucki

### Entropy Diminishing Finite Volume Approximation of a Cross-Diffusion System

Cancès, Clément Gaudeul, BenoîtWe propose a two-point flux approximation finite volume scheme for the approximation of the solutions of a entropy dissipative cross-diffusion system. The scheme is shown to preserve several key properties of the continuous system, among which positivity and decay of the entropy. Numerical experiments illustrate the behaviour of our scheme.

Clément Cancès, Benoît Gaudeul

### TPFA Finite Volume Approximation of Wasserstein Gradient Flows

Natale, Andrea Todeschi, GabrieleNumerous infinite dimensional dynamical systems arising in different fields have been shown to exhibit a gradient flow structure in the Wasserstein space. We construct Two Point Flux Approximation Finite Volume schemes discretizing such problems which preserve the variational structure and have second order accuracy in space. We propose an interior point method to solve the discrete variational problem, providing an efficient and robust algorithm. We present two applications to test the scheme and show its order of convergence.

Andrea Natale, Gabriele Todeschi

### Free Energy Diminishing Discretization of Darcy-Forchheimer Flow in Poroelastic Media

Both, Jakub W. Nordbotten, Jan M. Radu, Florin A.In this paper, we develop a discretization for the non-linear coupled model of classical Darcy-Forchheimer flow in deformable porous media, an extension of the quasi-static Biot equations. The continuous model exhibits a generalized gradient flow structure, identifying the dissipative character of the physical system. The considered mixed finite element discretization is compatible with this structure, which gives access to a simple proof for the existence, uniqueness, and stability of discrete approximations. Moreover, still within the framework, the discretization allows for the development of finite volume type discretizations by lumping or numerical quadrature, reducing the computational cost of the numerical solution.

Jakub W. Both, Jan M. Nordbotten, Florin A. Radu

### Energy Stable Discretization for Two-Phase Porous Media Flows

Cancès, Clément Nabet, FloreWe propose a $$\mathbb P_1$$ finite-element scheme with mass-lumping for a model of two incompressible and immiscible phases in a porous media flow. We prove the dissipation of the free energy and the existence of a solution to the nonlinear scheme. We also present numerical simulations to illustrate the behavior of the scheme.

Clément Cancès, Flore Nabet

### A Finite-Volume Scheme for a Cross-Diffusion Model Arising from Interacting Many-Particle Population Systems

Jüngel, Ansgar Zurek, AntoineA finite-volume scheme for a cross-diffusion model arising from the mean-field limit of an interacting particle system for multiple population species is studied. The existence of discrete solutions and a discrete entropy production inequality is proved. The proof is based on a weighted quadratic entropy that is not the sum of the entropies of the population species.

Ansgar Jüngel, Antoine Zurek

### Finite Volume Method for a System of Continuity Equations Driven by Nonlocal Interactions

El Keurti, Anissa Rey, ThomasWe present a new finite volume method for computing numerical approximations of a system of nonlocal transport equation modeling interacting species. This method is based on the work [F. Delarue, F. Lagoutière, N. Vauchelet, Convergence analysis of upwind type schemes for the aggregation equation with pointy potential, Ann. Henri. Lebesgue 2019], where the nonlocal continuity equations are treated as conservative transport equations with a nonlocal, nonlinear, rough velocity field. We analyze some properties of the method, and illustrate the results with numerical simulations.

Anissa El Keurti, Thomas Rey

### A Macroscopic Model to Reproduce Self-organization at Bottlenecks

Andreianov, Boris Sylla, AbrahamWe propose a model for self-organized traffic flow at bottlenecks that consists of a scalar conservation law with a nonlocal constraint on the flux. The constraint is a function of an organization marker which evolves through an ODE depending on the upstream traffic density and its variations. We prove well-posedness for the problem, construct and analyze a finite volume scheme, perform numerical simulations and discuss the model and related perspectives.

Boris Andreianov, Abraham Sylla

### A Three-Dimensional Hybrid High-Order Method for Magnetostatics

Chave, Florent Di Pietro, Daniele A. Lemaire, SimonWe introduce a three-dimensional Hybrid High-Order method for magnetostatic problems. The proposed method is easy to implement, supports general polyhedral meshes, and allows for arbitrary orders of approximation.

Florent Chave, Daniele A. Di Pietro, Simon Lemaire

### Hyperbolic Conservation Laws with Stochastic Discontinuous Flux Functions

Brencher, Lukas Barth, AndreaHyperbolic conservation laws are utilized to describe a variety of real-world applications, which require the consideration of the influence of uncertain parameters on the solution to the problem. To extend these models, one is often interested in including discontinuities in the state space to the flux function of the conservation law. This paper studies the solution of a stochastic nonlinear hyperbolic partial differential equation (PDE), whose flux function contains random spatial discontinuities. The first part of the paper defines the corresponding stochastic adapted entropy solution and required properties for existence and uniqueness are addressed. The second part contains the numerical simulation of the nonlinear hyperbolic problem as well as the estimation of the expectation of the problem via the multilevel Monte Carlo method.

Lukas Brencher, Andrea Barth

### Convergence of a Finite-Volume Scheme for a Heat Equation with a Multiplicative Stochastic Force

Bauzet, Caroline Nabet, FloreWe present here the discretization by a finite-volume scheme of a heat equation perturbed by a multiplicative noise of Itô type and under homogeneous Neumann boundary conditions. The idea is to adapt well-known methods in the deterministic case for the approximation of parabolic problems to our stochastic PDE. In this paper, we try to highlight difficulties brought by the stochastic perturbation in the adaptation of these deterministic tools.

Caroline Bauzet, Flore Nabet

### A New Gradient Scheme of a Time Fractional Fokker–Planck Equation with Time Independent Forcing and Its Convergence Analysis

Bradji, AbdallahWe apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the time fractional Fokker–Planck equation with time independent forcing in any space dimension. Using [5] which dealt with GDM for linear advection problems, we develop a new fully discrete implicit GS (Gradient Scheme) for the stated model. We prove new discrete a priori estimates which yield estimates on the discrete solution in $$L^\infty (L^2)$$ and $$L^2(H^1)$$ discrete norms. Thanks to these discrete a priori estimates, we prove new error estimates in the discrete norms of $$L^\infty (L^2)$$ and $$L^2(H^1)$$. The main ingredients in the proof of these error estimates are the use of the stated discrete a priori estimates and a comparison with some well chosen auxiliary schemes. These auxiliary schemes are approximations of convective-diffusive elliptic problems in each time level. We state without proof the convergence analysis of these auxiliary schemes. Such proof uses some adaptations of the [6, Proof of Theorem 2.28] dealt with GDM for the case of elliptic diffusion problems. These results hold for all the schemes within the framework of GDM. This work can be viewed as an extension to our recent one [2].

### The Gradient Discretisation Method for Two-Phase Discrete Fracture Matrix Models in Deformable Porous Media

Bonaldi, F. Brenner, K. Droniou, J. Masson, R.We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is also coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. The model is discretized using the gradient discretization method (Droniou et al. in Mathematics & applications. Springer, 2018, [1]), which covers a large class of conforming and non conforming discretizations. This framework allows a generic convergence analysis of the coupled model using a combination of discrete functional tools. Here, we describe the model together with its numerical discretisation, and we state the convergence result, whose proof will be detailed in a forthcoming paper. This is, to our knowledge, the first convergence result for this type of models taking into account two-phase flows and the nonlinear poro-mechanical coupling. Previous related works consider a linear approximation obtained for a single phase flow by freezing the fracture conductivity (Girault et al. in Math Models Methods Appl Sci 25:4, 2015, [2]).

F. Bonaldi, Konstantin Brenner, J. Droniou, R. Masson

### A New Optimal –Error Estimate of a SUSHI Scheme for the Time Fractional Diffusion Equation

Bradji, AbdallahWe consider a finite volume scheme, using the general mesh of [8], for the TFDE (time fractional diffusion equation) in any space dimension. The time discretization is performed using a uniform mesh. We prove a new discrete $$L^\infty (H^1)$$–a priori estimate. Such a priori estimate is proved thanks to the use of the new tool of the discrete Laplace operator developed recently in [7]. Thanks to this a priori estimate, we prove a new optimal convergence order in the discrete $$L^\infty (H^1)$$–norm. These results improve the ones of [1, 4] which dealt respectively with finite volume and GDM (Gradient Discretization Method) for the TFDE. In [4], we only proved a priori estimate and error estimate in the discrete $$L^\infty (L^2)$$–norm whereas in [1] we proved a priori estimate and error estimate in the discrete $$L^2(H^1)$$–norm. The a priori estimate as well as the error estimate presented here were stated without proof for the first time in [3, Remark 1, p. 443] in the context of the general framework of GDM and [2, Remark 1, p. 205] in the context of finite volume methods. They also were mentioned, as future works, in [1, Remark 4.1].

### Note on the Convergence of a Finite Volume Scheme for a Second Order Hyperbolic Equation with a Time Delay in Any Space Dimension

Benkhaldoun, Fayssal Bradji, AbdallahIn this note, we establish a finite volume scheme for a model of a second order hyperbolic equation with a time delay in any space dimension. This model is considered in [10, 11] where some exponential stability estimates and oscillatory behaviour are proved. The scheme we shall present uses, as space discretization, the general class of nonconforming finite volume meshes of [5]. In addition to the proof of the existence and uniqueness of the discrete solution, we develop a new discrete a priori estimate. Thanks to this a priori estimate, we prove error estimates in discrete seminorms of $$L^\infty (H^1_0)$$, $$L^\infty (L^2)$$, and $$W^{1,\infty }(L^2)$$. This work can be viewed as extension to the previous ones [2, 4] which dealt with the analysis of finite volume methods for respectively semilinear parabolic equations with a time delay and the wave equation.

### A Cell-Centered Finite Volume Method for the Navier–Stokes/Biot Model

Caucao, Sergio Li, Tongtong Yotov, Ivan We develop a cell-centered finite volume method for the Navier–Stokes/Biot model, based on a fully mixed formulation with weakly symmetric stresses. The multipoint stress mixed finite element method is employed for the Navier–Stokes and elasticity equations, while the multipoint flux mixed finite element method is used for Darcy’s flow. These methods allow for local elimination of the fluid and poroelastic stresses, vorticity, and rotation, resulting in a positive definite finite volume scheme for the fluid and structure velocities and the Darcy pressure, coupled via Lagrange multipliers on the interface to impose the transmission conditions.

Sergio Caucao, Tongtong Li, Ivan Yotov

### Convergence Study of a DDFV Scheme for the Navier-Stokes Equations Arising in the Domain Decomposition Setting

Goudon, Thierry Krell, Stella Lissoni, GiuliaWe consider DDFV discretization of the Navier-Stokes equations where the convection fluxes are computed by means of B-schemes, generalizing the classical centered and upwind discretizations. This study is motivated by the analysis of domain decomposition approaches. We investigate on numerical grounds the convergence of the method.

Thierry Goudon, Stella Krell, Giulia Lissoni

### Interface Conditions for Arbitrary Flows in Coupled Porous-Medium and Free-Flow Systems

Eggenweiler, Elissa Rybak, IrynaPhysically consistent interface conditions are important for accurate mathematical modelling and numerical simulation of flow and transport processes in coupled free-flow and porous-medium systems. Traditional coupling concepts are valid for simplified cases only, such as flows parallel to the fluid-porous interface or very specific boundary value problems. This severely limits the range of applications that can be accurately modelled. Evidently, there is a need for more general interface conditions to couple free flow to porous-medium flow. In this paper, we propose new coupling conditions for arbitrary flow directions and periodic porous media. These conditions are derived by the theory of homogenisation and boundary layers and are applicable to general filtration problems. The derived set of coupling conditions are validated by comparison of pore-scale to macroscale numerical simulations.

Elissa Eggenweiler, Iryna Rybak

### On the Convergence Rate of the Dirichlet-Neumann Iteration for Coupled Poisson Problems on Unstructured Grids

Görtz, Morgan Birken, PhilippWe consider thermal fluid structure interaction with a partitioned approach, where typically, a finite volume and a finite element code would be coupled. As a model problem, we consider two coupled Poisson problems with heat conductivities $$\lambda _1$$, $$\lambda _2$$ in one dimension on intervals of length $$l_1$$ and $$l_2$$. Hereby, we consider linear discretizations on arbitrary meshes, such as finite volumes, finite differences, finite elements. For these, we prove that the convergence rate of the Dirichlet-Neumann iteration is given by $$\lambda _1l_2/\lambda _2l_1$$ and is thus independent of discretization and mesh.

Morgan Görtz, Philipp Birken

### Optimized Overlapping DDFV Schwarz Algorithms

Gander, Martin J. Halpern, Laurence Hubert, Florence Krell, StellaWe introduce an overlapping optimized Schwarz methods in the DDFV framework for an anisotropic diffusion equation, and we show that a discrete and bounded domain convergence analysis is important to get best performance for strong anisotropy.

Martin J. Gander, Laurence Halpern, Florence Hubert, Stella Krell

### Model Adaptation of Balance Laws Based on A Posteriori Error Estimates and Surrogate Fluxes

Giesselmann, Jan Joshi, HrishikeshIn this proceeding, we present model adaptation for hyperbolic balance laws based on a posteriori error estimates. The model adaptation is carried out by decomposing the computational domain and choosing to solve either the full system or a simpler reduced system. The decision is made based on error estimates constructed employing the relative entropy framework which allows us to bound the difference between the numerical solution to the reduced system and the exact solution to the full system. Furthermore, the use of surrogate fluxes in the simple model constructed by machine learning is proposed to further reduce the computational expenses.

Jan Giesselmann, Hrishikesh Joshi

### Robust Newton Solver Based on Variable Switch for a Finite Volume Discretization of Richards Equation

Bassetto, Sabrina Cancès, Clément Enchéry, Guillaume Tran, Quang HuyWe propose an efficient nonlinear solver for the resolution of the Richards equation. It is based on variable switching and is easily implemented thanks to a fictitious variable allowing to describe both the saturation and the pressure. Numerical experiments show that our method enables to use Newton’s method with large time steps, reasonable number of iterations and in regions where the pressure-saturation relationship is given by a graph.

Sabrina Bassetto, Clément Cancès, Guillaume Enchéry, Quang Huy Tran

### Acceleration of Newton’s Method Using Nonlinear Jacobi Preconditioning

Brenner, KonstantinFor mildly nonlinear systems, involving concave diagonal nonlinearities, semi-global monotone convergence of Newton’s method is guarantied provided that the Jacobian of the system is an M-matrix. However, regardless this convergence result, the efficiency of Newton’s method becomes poor for stiff nonlinearities. We propose a nonlinear preconditioning procedure inspired by the Jacobi method and resulting in a new system of equations, which can be solved by Newton’s method much more efficiently. The obtained preconditioned method is shown to exhibit semi-global convergence.

Konstantin Brenner

### A Finite Volume Method for a Convection-Diffusion Equation Involving a Joule Term

Calgaro, Caterina Creusé, EmmanuelThis work is devoted to a Finite Volume method to approximate the solution of a convection-diffusion equation involving a Joule term. We propose a way to discretize this so-called “Joule effect” term in a consistent way with the non linear diffusion one, in order to ensure some maximum principle properties on the solution. We then investigate the numerical behavior of the scheme on two original benchmarks.

Caterina Calgaro, Emmanuel Creusé

### On the Stability of Finite Volumes for Stationary First Order Systems

Ndjinga, Michaël Ngwamou, Sédrick Kameni The aim of this paper is two-folds. Firstly we study first order stationary systems of PDEs of the form $$\sum _k A_k\partial _k U + KU=0$$ with $$K\ngtr 0$$ on $$\mathbb {R}^d$$. We prove that the classical assumption $$K>0$$ is not necessary for the well-posedness of the system and is violated in the particular case of the first order Poisson problem. Secondly we prove the $$L^2$$ stability of the finite volume discretisations provided the term KU is appropriately discretised on faces. Our result relies on a discrete Gagliardo-Nirenberg-Sobolev inequality to be submitted [15].

Michaël Ndjinga, Sédrick Kameni Ngwamou

### A New Class of -Stable Schemes for the Isentropic Euler Equations on Staggered Grids

Ndjinga, Michaël Ait-Ameur, KatiaStaggered schemes for compressible flows are highly non linear and the stability analysis has historically been performed with a heuristic approach and the tuning of numerical parameters [12]. We investigate the $$L^2$$-stability of staggered schemes by analysing their numerical diffusion operator. The analysis of the numerical diffusion operator gives new insight into the scheme and is a step towards a proof of linear stability or stability for almost constant initial data. For most classical staggered schemes [9–11, 14], we are able to prove the positivity of the numerical diffusion only in specific cases (constant sign velocities). We then propose a class of linearly $$L^2$$-stable staggered schemes for the isentropic Euler equations based on a carefully chosen numerical diffusion operator. We give an example of such a scheme and present some first numerical results on a Riemann problem.

Michaël Ndjinga, Katia Ait-Ameur

### Convergence of a TPFA Finite Volume Scheme for Mixed-Dimensional Flow Problems

Boon, Wietse M. Nordbotten, Jan M.A two-point flux approximation (TPFA) finite volume method is considered for mixed-dimensional fracture flow problems. Its construction is based on applying a face-based quadrature rule to a conforming, mixed finite element scheme of lowest order. A concise argument shows linear convergence in theory, which we confirm in practice by a numerical experiment.

Wietse M. Boon, Jan M. Nordbotten

### A Relaxation Method for the Simulation of Possibly Non-hyperbolic Polymer Flooding Models with Inaccessible Pore Volume Effect

Dongmo Nguepi, Guissel Lagnol Braconnier, Benjamin Preux, Christophe Tran, Quang-Huy Berthon, ChristophePolymer flooding models used in the simulation of enhanced oil recovery of reservoirs commonly involve a system of conservation laws that may be ill-posed, especially when an inaccessible pore volume (IPV) empirical law is considered. Depending on the IPV law, the flow model is either weakly hyperbolic with resonance or non-hyperbolic with complex eigenvalues. In this paper, we propose a Suliciu-type relaxation, which unconditionally ensures hyperbolicity for any IPV law. This approximation gives rise to a new numerical scheme, which is compared with the classical upwind scheme and the exact solution whenever possible.

Guissel Lagnol Dongmo Nguepi, Benjamin Braconnier, Christophe Preux, Quang-Huy Tran, Christophe Berthon

### The FVC Scheme on Unstructured Meshes for the Two-Dimensional Shallow Water Equations

Ziggaf, Moussa Boubekeur, Mohamed Kissami, Imad Benkhaldoun, Fayssal Mahi, Imad ElThe fluid flow transport and hydrodynamic problems often take the form of hyperbolic systems of conservation laws. In this work we will present a new scheme of finite volume methods for solving these evolution equations. It is a family of finite volume Eulerian–Lagrangian methods for the solution of non-linear problems in two space dimensions on unstructured triangular meshes. The proposed approach belongs to the class of predictor-corrector procedures where the numerical fluxes are reconstructed using the method of characteristics, while an Eulerian method is used to discretize the conservation equation in a finite volume framework. The scheme is accurate, conservative and it combines advantages of the modified method of characteristics to accurately solve the non-linear conservation laws with a finite volume method to discretize the equations. The proposed Finite Volume Characteristics (FVC) scheme is also non-oscillatory and avoids the need to solve a Riemann problem. Several test examples will be presented for the shallow water equations. The results will be compared to those obtained with the Roe.

### Numerical Analysis of a Finite Volume Scheme for the Optimal Control of Groundwater Pollution

Choquet, Catherine Mory Diédhiou, Moussa Nasser El Dine, HousseinThis paper is devoted to an optimal control problem of the underground water contaminated by agricultural pollution, the spatiotemporal objective taking into account the trade-off between the fertilizer used by the farmer to increase profits and the cleaning costs which are necessary to treat the water before it is distributed to users. The constraint is a hydrogeological model for the spread of the pollution in the aquifer which consists in a system of a parabolic partial differential equation and an elliptic equation. Hydrogeological and economic modelling are thus combined in the problem. We propose a finite volume scheme based on a two-point flux approximation with upwind mobilities of an optimal control. Numerical simulations are provided to illustrate the 2D and 3D optimal solutions.

Catherine Choquet, Moussa Mory Diédhiou, Houssein Nasser El Dine

### Space-Time Discontinuous Galerkin Methods for Linear Hyperbolic Systems and the Application to the Forward Problem in Seismic Imaging

Dörfler, Willy Wieners, Christian Ziegler, DanielWe consider a p-adaptive discontinuous Galerkin method in space and time for linear hyperbolic systems. This is applied to the visco-acoustic wave equation in the formulation as first-order system. The method is applied to the forward problem in seismic imaging, and we study the convergence of the fully adaptive parallel method by the numerical evaluation of measurements in form of seismograms. The method is based on a formulation of Generalized Standard Linear Solids as symmetric Friedrichs system and an inf-sup stable variational Petrov–Galerkin setting. With respect to suitable DG norms the discretization is p-robust inf-sup stable, and the approximation of material parameters can be estimated by a Strang type argument. In order to restrict the computation to the domain of interest, an absorbing boundary layer in included. Numerical results for a benchmark configuration in geophysics are obtained with a p-adaptive method based on a dual-primal error estimator with respect to a goal functional corresponding to the seismic measurements. The linear system is solved in parallel with a space-time multigrid method.

Willy Dörfler, Christian Wieners, Daniel Ziegler

### A Hybrid Discontinuous Galerkin Method for Transport Equations on Networks

Egger, Herbert Philippi, NoraWe discuss the mathematical modeling and numerical discretization of transport problems on one-dimensional networks. Suitable coupling conditions are derived that guarantee conservation of mass across network junctions and dissipation of a mathematical energy which allows us to prove existence of unique solutions. We then consider the space discretization by a hybrid discontinuous Galerkin method which provides a suitable upwind mechanism to handle the transport problem and allows to incorporate the coupling conditions in a natural manner. In addition, the method inherits mass conservation and stability of the continuous problem. Order optimal convergence rates are established and illustrated by numerical tests.

Herbert Egger, Nora Philippi

### MUSCL Discretization for the Fluid Flow Convection Operator on Staggered Meshes

Brunel, A. Herbin, R. Latché, J.-C.We propose in this paper a second order discretization of the momentum convection operator for fluid flow simulation on staggered quadrangular or hexahedral meshes. The velocity is approximated by the Rannacher-Turek finite element. The implemented MUSCL-like approach is of algebraic type, in the sense that the limitation procedure does not invoke any slope reconstruction, and is independent from the geometry of the cells. The derived discrete convection operator applies both to constant or variable density flows; we perform here numerical tests for the barotropic and incompressible Navier-Stokes equations.

A. Brunel, R. Herbin, J.-C. Latché

### An Active Flux Method for Cut Cell Grids

Helzel, Christiane Kerkmann, DavidWe present recent work in progress towards the development of a third order accurate Cartesian grid cut cell method for the approximation of hyperbolic conservation laws in complex geometries. Our cut cell method is based on the Active Flux method of Eymann and Roe, a new finite volume method, which evolves both cell average values and point values of the conserved quantities. The evolution of the point values leads to an automatic stabilisation of the cut cell update, i.e. the method is stable for time steps that are appropriate for the regular cells. While most of the existing cut cell stabilisation methods lead to a loss of accuracy, we show that it is possible to obtain third order accurate results. In this contribution we restrict our considerations to the linear transport equation in one and two space dimensions.

Christiane Helzel, David Kerkmann

### Finite Volume Discretisation of Fracture Deformation in Thermo-poroelastic Media

Stefansson, Ivar Berre, Inga Keilegavlen, EirikThis paper presents a model where thermo-hydro-mechanical processes are coupled to a deformation model for preexisting fractures. The model is formulated within a discrete-fracture-matrix framework where the rock matrix and the fractures are considered as individual subdomains, and interaction between them takes place on the matrix-fracture interfaces. A finite volume discretisation implemented in the simulation toolbox PorePy is presented and applied in a simulation showcasing the effects of the different mechanisms on fracture deformation governed by contact mechanics, as well as their different timescales.

Ivar Stefansson, Inga Berre, Eirik Keilegavlen

### A Control Volume Finite Element Formulation with Subcell Reconstruction for Phase-Field Fracture

Sargado, Juan MichaelWe present a control volume finite element formulation for phase-field brittle fracture based on unstructured simplex meshes. Linear finite elements are employed for the linear momentum equation, while a cell-centered finite volume method based on the two-point flux approximation scheme is used to discretize the phase-field equation. Additionally, we perform linear reconstruction of the phase-field variable over subcells of the original control volumes in order to better model gradient discontinuities. This yields a higher order scheme that also gives more conservative predictions of critical loads compared to existing low-order methods.

### A Conservative Phase-Field Model for Reactive Transport

Bringedal, CarinaWe present a phase-field model for single-phase flow and reactive transport where ions take part in mineral precipitation/dissolution reactions. The evolving interface between fluid and mineral is approximated by a diffuse interface, which is modeled using an Allen–Cahn equation. As the original Allen–Cahn equation is not conservative, we apply a reformulation ensuring conservation of the phase-field variable and address the sharp-interface limit of the reformulated model. This model is implemented using a finite volume scheme and the discrete conservation of the reformulated Allen–Cahn equation is shown. Numerical examples show how the discrete phase-field variable is conserved up to the chemical reaction.

Carina Bringedal

### A Fully Conforming Finite Volume Approach to Two-Phase Flow in Fractured Porous Media

Burbulla, Samuel Rohde, ChristianIn many natural and technical applications in porous media fluid’s flow behavior is highly affected by fractures. Many approaches employ mixed-dimensional models that model thin features as dimension-reduced manifolds. Following this idea, we consider porous media where dominant heterogeneities are geometrically represented by sharp interfaces. We model incompressible two-phase flow in porous media both in the bulk porous medium and within the fractures. We present a reliable and geometrically flexible implementation of a fully conforming finite volume approach within the DUNE framework for two and three spatial dimensions. The implementation is based on the new dune-mmesh grid implementation that manages bulk and surface triangulation simultaneously. The model and the implementation are extended to handle fracture junctions. We apply our scheme to benchmark cases with complex fracture networks to show the reliability of the approach.

Samuel Burbulla, Christian Rohde

### Monotone Embedded Discrete Fracture Method for the Two-Phase Flow Model

Nikitin, Kirill D. Yanbarisov, Ruslan M.We propose an application of the new monotone embedded discrete fracture method (mEDFM) [13] to the two-phase flow model. The new method for modelling of flows in fractured media consists in coupling of the embedded discrete fracture method (EDFM) with the nonlinear monotone finite volume (FV) scheme with two-point flux approximation, which preserves non-negativity of the discrete solution. The resulting method combines effectiveness and simplicity of the standard EDFM approach with accuracy and physical relevance of the nonlinear FV schemes for non-orthogonal grids and anisotropic media. Numerical experiments show that the two-phase flow modelling with the mEDFM provides much more accurate solution compared to the conventional EDFM, and is in a good agreement with the discrete fracture method, which directly applies the nonlinear FV method to a grid with fractures explicitly represented by 3D cells.

Kirill D. Nikitin, Ruslan M. Yanbarisov

### A Robust VAG Scheme for a Two-Phase Flow Problem in Heterogeneous Porous Media

Brenner, K. Masson, R. Quenjel, E. H.A positive Vertex Approximate Gradient (VAG) scheme is proposed to discretize the total velocity formulation of two-phase Darcy flow problems in heterogeneous porous media. The discretization is based on the physical variables and allows for multiple rock types with highly contrasted petrophysical and hydrodynamical properties. The numerical experiment shows that, compared to the Phase Potential Upwind (PPU) version of VAG scheme, this new discretization is more robust and efficient in terms of nonlinear convergence.

Konstantin Brenner, R. Masson, E. H. Quenjel

### Design of Coupled Finite Volume Schemes Minimizing the Grid Orientation Effect in Reservoir Simulation

Laurent, Karine Flauraud, Éric Preux, Christophe Tran, Quang Huy Berthon, ChristopheIn this paper, we present and compare two nine-point finite volume schemes to reduce the so-called grid orientation effect (GOE) which occurs in the simulation of unstable two phase flow in porous media. The first scheme is a more classical nine-point scheme with one tuning parameter whereas the second one, more original, uses two parameters (one per direction). A numerical test problem testify to the improvement brought by the new scheme.

Karine Laurent, Éric Flauraud, Christophe Preux, Quang Huy Tran, Christophe Berthon

### A Comparison of Consistent Discretizations for Elliptic Problems on Polyhedral Grids

Klemetsdal, Øystein S. Møyner, Olav Raynaud, Xavier Lie, Knut-AndreasIn this work, we review a set of consistent discretizations for second-order elliptic equations, and compare and contrast them with respect to accuracy, monotonicity, and factors affecting their computational cost (degrees of freedom, sparsity, and condition numbers). Our comparisons include the linear and nonlinear TPFA method, multipoint flux-approximation (MPFA-O), mimetic methods, and virtual element methods. We focus on incompressible flow and study the effects of deformed cell geometries and anisotropic permeability.

Øystein S. Klemetsdal, Olav Møyner, Xavier Raynaud, Knut-Andreas Lie

### Global Implicit Solver for Multiphase Multicomponent Flow in Porous Media with Multiple Gas Phases and General Reactions

Knodel, Markus M. Kräutle, Serge Knabner, PeterMultiphase multicomponent flow processes in porous media have to be considered to study the efficiency of mineral trapping mechanisms for climate killing gas storage in deep layers. Robust predictions ask for the solution of large nonlinear coupled systems of diffusion-advection-reaction (partial differential) equations containing equilibrium reactions. In that we elaborate the fully globally implicit Kräutle-Knabner PDE reduction method (cf. a former paper Kräutle and Knabner in Water Resour Res 43(3):W03429 [8]) for the case of multiple gas phases, we solve the arising Finite Element discretized/Finite Volume stabilized equations by means of a semismooth nested Newton solver. We present preliminary simulation results for the case of mutual injection of CO$$_2$$, CH$$_4$$ and H$$_2$$S into deep layers and investigate the arising mineral trapping scenario. Our methods are applicable also to other fields such as nuclear waste storage or oil recovery.

Markus M. Knodel, Serge Kräutle, Peter Knabner

### Partitioned Coupling Schemes for Free-Flow and Porous-Media Applications with Sharp Interfaces

Jaust, Alexander Weishaupt, Kilian Mehl, Miriam Flemisch, BerndWe investigate a partitioned coupling scheme applied to a system of free flow over a porous medium. The coupling scheme follows a partitioned approach which means that the flow fields in the two domains are solved separately and information is exchanged over the sharp interface that separates the free-flow and the porous-medium domain. Technically, the coupling is realized via the open-source library preCICE, employing a pure black-box approach such that different solver frameworks can be used with highly specialized solvers in each of the flow domains. We investigate the partitioned coupling approach numerically by comparing it to a monolithic coupling scheme with respect to convergence and accuracy. This is the first time a partitioned black-box coupling is used for coupling free flow and porous-media flow. The coupling approach is numerically validated and different partitioned coupling approaches are compared with each other.

Alexander Jaust, Kilian Weishaupt, Miriam Mehl, Bernd Flemisch

### Challenges in Drift-Diffusion Semiconductor Simulations

Farrell, Patricio Peschka, DirkWe study and compare different discretizations of the van Roosbroeck system for charge transport in bulk semiconductor devices that can handle nonlinear diffusion. Three common challenges corrupting the precision of numerical solutions will be discussed: boundary layers, discontinuities in the doping profile, and corner singularities in L-shaped domains. The most problematic of these challenges are boundary layers in the quasi-Fermi potentials near ohmic contacts, which can have a drastic impact on the convergence order.

Patricio Farrell, Dirk Peschka

### Unipolar Drift-Diffusion Simulation of S-Shaped Current-Voltage Relations for Organic Semiconductor Devices

Fuhrmann, Jürgen Hai Doan, Duy Glitzky, Annegret Liero, Matthias Nika, GrigorWe discretize a unipolar electrothermal drift-diffusion model for organic semiconductor devices with Gauss–Fermi statistics and charge carrier mobilities having positive temperature feedback. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and use a finite volume based generalized Scharfetter-Gummel scheme. Applying path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, only recently observed experimentally.

Jürgen Fuhrmann, Duy Hai Doan, Annegret Glitzky, Matthias Liero, Grigor Nika

### A Second Order Numerical Scheme for Large-Eddy Simulation of Compressible Flows

Gamal, B. Gastaldo, L. Latché, J.-C. Veynante, D.In the context of large eddy simulation of turbulent flows, the control of kinetic energy seems to be an essential requirement for a numerical scheme. We propose in this paper a formally second order non-dissipative scheme dedicated to the numerical simulation of the filtered Naviers-Stokes equations for compressible flows. The spatial discretization is staggered and based on the so-called Marker-And-Cell (MAC) scheme. A MUSCL-like technique is used for convection operators of the mass and the internal energy balance equations in order to preserve the positivity of the density and of the internal energy. Time discretization is performed with the Heun scheme. A kinetic energy conservation identity at discrete level is proved. The good behaviour of the scheme is assessed on the simulation of compressible decaying isotropic turbulence.

B. Gamal, L. Gastaldo, J.-C. Latché, D. Veynante

### A Marker-and-Cell Scheme for Viscoelastic Flows on Non Uniform Grids

Mokhtari, O. Davit, Y. Latché, J.-C. De Loubens, R. Quintard, M.In this paper, we develop a numerical scheme for the solution of the coupled Stokes and Navier-Stokes equations with constitutive equations describing the flow of viscoelastic fluids. The space discretization is based on the so-called Marker-And-Cell (MAC) scheme. The time discretization uses a fractional-step algorithm where the solution of the Navier-Stokes equations is first obtained by a projection method and then the transport-reaction equation for the conformation tensor is solved by a finite-volume scheme. In order to obtain consistency, the space discretization of the divergence of the elastic part of the stress tensor in the momentum balance equation is derived using a weak form of the MAC scheme. For stability and accuracy reasons, the solution of the transport-reaction equation for the conformation tensor is split into pure convection steps, with a change of variable from $${\mathbf{c}}$$ to $$\log ({\mathbf{c}})$$, and a reaction step, which consists in solving one ODE per cell via an Euler scheme with local sub-cycling. Numerical computations for the Stokes flow of an Oldroyd-B fluid in the lid-driven cavity at We = 1 confirm the scheme efficiency.

O. Mokhtari, Y. Davit, J.-C. Latché, R. de Loubens, M. Quintard

### A Numerical Convergence Study of Some Open Boundary Conditions for Euler Equations

Colas, C. Ferrand, M. Hérard, J.-M. Hurisse, O. Le Coupanec, E. Quibel, L.We discuss herein the suitability of some open boundary conditions. Considering the Euler system of gas dynamics, we compare approximate solutions of one-dimensional Riemann problems in a bounded sub-domain with the restriction in this sub-domain of the exact solution in the infinite domain. Assuming that no information is known from outside of the domain, some basic open boundary condition specifications are given, and a measure of the $$L^1$$-norm of the error inside the computational domain enables to show consistency errors in situations involving outgoing shock waves, depending on the chosen boundary condition formulation. This investigation has been performed with Finite Volume methods, using approximate Riemann solvers in order to compute numerical fluxes for inner interfaces and boundary interfaces.

C. Colas, M. Ferrand, J.-M. Hérard, Olivier Hurisse, E. Le Coupanec, Lucie Quibel

### Simulation of a Liquid-Vapour Compressible Flow by a Lattice Boltzmann Method

Helluy, Philippe Hurisse, Olivier Quibel, Lucie This work is devoted to the numerical resolution of a compressible three-phase flow with phase transition by a Lattice-Boltzmann Method (LBM). The flow presents complex features and large variations of physical quantities. The LBM is a robust numerical method that is entropy stable and that can be extended to second order accuracy without additional numerical cost. We present preliminary numerical results, which confirm its competitiveness compared to other Finite Volume methods.

Philippe Helluy, Olivier Hurisse, Lucie Quibel

### Discontinuous Galerkin Method for Incompressible Two-Phase Flows

Gerstenberger, Janick Burbulla, Samuel Kröner, DietmarIn this contribution we present a local discontinuous Galerkin (LDG) pressure-correction scheme for the incompressible Navier–Stokes equations. The scheme does not need penalty parameters and satisfies the discrete continuity equation exactly. The scheme is especially suitable for two-phase flow when used with a piecewise-linear interface construction (PLIC) volume-of-fluid (VoF) method and cut-cell quadratures.

Janick Gerstenberger, Samuel Burbulla, Dietmar Kröner

### High-Order Numerical Methods for Compressible Two-Phase Flows

Kozhanova, Ksenia Goncalves, Eric Hoarau, YannickWe study the numerical methods to solve stiff two-phase flow problem which involves strong shock and expansion waves. In particular we focus the present study on high order reconstruction techniques coupled with HLLC and KNP numerical flux formulations associated to a four-equation model. These numerical methods are first tested on 1-D expansion tube case to investigate the accuracy of the schemes. The originality of our project is to construct a high-order numerical tool for solving the 2-D problem of two-phase shock-interface interaction with high density ratio between the phases. This paper presents the intermediate results with tests of low density ratio.

Ksenia Kozhanova, Eric Goncalves, Yannick Hoarau

### A Python Framework for Solving Advection-Diffusion Problems

Dedner, Andreas Klöfkorn, RobertThis paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efficient implementations of the Discontinuous Galerkin (DG) method for solving a wide range of non linear partial differential equations (PDE). Although the C++ interfaces of Dune-Fem-DG are highly flexible and customizable, a solid knowledge of C++ is necessary to make use of this powerful tool. With this work easier user interfaces based on Python and the Unified Form Language are provided to open Dune-Fem-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and first order hyperbolic PDEs.

Andreas Dedner, Robert Klöfkorn

### 3-Dimensional Particulate Flow Modelling Using a Viscous Penalty Combined with a Stable Projection Scheme

Batteux, L. Laminie, J. Latché, J.-C. Poullet, P.We introduce a strategy for the simulation a particulate flow in a 3-dimensional domain. The particles are assumed to be rigid, and the homogeneous fluid flow to be governed by the incompressible Navier–Stokes equations. The system is solved using a predictor-corrector scheme for the Navier–Stokes equations with variable density. The latter scheme is adapted to take into account the solid domain by adopting a volume penalization method. In order to advect efficiently the particles, the approximation of the mass balance equation is carried out by an anti-dissipative scheme similar to the Ultra-Bee scheme. We conclude with numerical tests in the context of particulate flows.

L. Batteux, J. Laminie, J.-C. Latché, P. Poullet

### Data Assimilation for Ocean Drift Trajectories Using Massive Ensembles and GPUs

Holm, Håvard H. Sætra, Martin L. Brodtkorb, André R.In this work, we perform fully nonlinear data assimilation of ocean drift trajectories using multiple GPUs. We use an ensemble of up to 10,000 members and the sequential importance resampling algorithm to assimilate observations of drift trajectories into the underlying shallow-water simulation model. Our results show an improved drift trajectory forecast using data assimilation for a complex and realistic simulation scenario, and the implementation exhibits good weak and strong scaling.

Håvard H. Holm, Martin L. Sætra, André R. Brodtkorb

### Application of an Unstructured Finite Volume Method to the Shallow Water Equations with Porosity for Urban Flood Modelling

Moumna, Abdelhafid Kissami, Imad Elmahi, Imad Benkhaldoun, FayssalWe present a finite volume model for the simulation of floods in urban areas. The model consists of the two-dimensional shallow water equations with variable horizontal porosity which is introduced in order to reflect the effects of obstructions. An extra porosity source term appears in the momentum equations. The main advantage of this model is the significant reduction of the computational cost while preserving an acceptable level of accuracy. The finite volume method uses a modified Roe’s scheme involving the sign of the Jacobian matrix in the system for the discretization of gradient fluxes. The performance of the numerical model is demonstrated by comparing the results obtained using the proposed method to laboratory experiments for a flow problem over an array of obstacles.

### Semi-implicit Two-Speed Well-Balanced Relaxation Scheme for Ripa Model

Franck, Emmanuel Navoret, LaurentIn this paper, we propose a semi-implicit well-balanced scheme for the Ripa model based on a two-speed relaxation. The method both preserves equilibria and has an implicit step that reduces to the inversion of a constant Laplacian. Numerical simulations show that the scheme well capture low-Froude flows.

Emmanuel Franck, Laurent Navoret

### Kinetic Over-Relaxation Method for the Convection Equation with Fourier Solver

Hélie, Romane Helluy, Philippe Franck, Emmanuel Navoret, LaurentIn this paper, we apply the CFL-less kinetic over-relaxation scheme presented in Coulette et al. (Comput Fluids 190:485–502 [1]) to the convection equation in two space dimensions. The method is a succession of free-transport steps and collisions steps. The free transport steps are solved with Fourier discretization. The collision steps are solved with over-relaxation for achieving high order. The method reaches six-order accuracy when using palindromic composition method. We apply the method to the guiding-center model in plasma physics.

Romane Hélie, Philippe Helluy, Emmanuel Franck, Laurent Navoret

### Cell-Centered Finite Volume Method for Regularized Mean Curvature Flow on Polyhedral Meshes

Hahn, Jooyoung Mikula, Karol Frolkovic, Peter Balažovjech , Martin Basara, BranislavA cell-centered finite volume method is used to numerically solve a regularized mean curvature flow equation on polyhedral meshes. It is based on an over-relaxed correction method used previously for linear diffusion problems. An iterative nonlinear Crank-Nicolson method is proposed to obtain the second-order accuracy in time and space. The proposed algorithm is used for three-dimensional domains decomposed for parallel computing for two examples that numerically verify the second order accuracy on polyhedral meshes.

Jooyoung Hahn, Karol Mikula, Peter Frolkovič, Martin Balažovjech, Branislav Basara

### A Fully Eulerian Finite Volume Method for the Simulation of Fluid-Structure Interactions on AMR Enabled Quadtree Grids

Bergmann, Michel Fondanèche, Antoine Iollo, AngeloWe present a versatile fully Eulerian method for the simulation of fluid-structure interactions. The model equations are solved using a finite-volume scheme on a compact and possibly dynamic quadtree stencil. The structure geometry is followed using a level-set model and a distance function. A regularized Heaviside function that allows to discriminate between the fluid and the elastic phases is then defined with respect to the moving structure. The elastic deformation of the structure is described according to the backward characteristics which are in turn used to express the Cauchy stress tensor of a two-parameter Mooney-Rivlin material. The numerical model is validated with respect to the literature and an example of application is detailed.

Michel Bergmann, Antoine Fondanèche, Angelo Iollo

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