Numerical Mathematics and Advanced Applications ENUMATH 2019

Table of Contents

1. Frontmatter

2. High Order Whitney Forms on Simplices and the Question of Potentials

   Francesca Rapetti, Ana Alonso Rodríguez

   Abstract

   In the frame of high order finite element approximations of PDEs, we are interested in an explicit and efficient way for constructing finite element functions with assigned gradient, curl or divergence in domains with general topology. Three ingredients, that bear the name of their scientific fathers, are involved: the de Rham’s diagram and theorem, Hodge’s decomposition for vectors, Whitney’s differential forms. Some key images are presented in order to illustrate the mathematical concepts.

3. The Candy Wrapper Problem: A Temporal Multiscale Approach for PDE/PDE Systems

   Thomas Richter, Jeremi Mizerski

   Abstract

   We discuss the application of a multiscale scheme to a medical flow problem, the so called Candy Wrapper problem. This problem describes the re-stenosis of a stented blood vessel, which will take several months but which is governed by the rapidly oscillating dynamics of the blood flow. A long term simulation of this three dimensional free-boundary flow problem resolving the fast dynamics is not feasible. Our multiscale approach which has been recently published is based on capturing the fast dynamics by locally isolated periodic-in-time problems which have to be approximated once in each macro step of the long term process. Numerical results show the accuracy and efficiency of this multiscale approach.

4. Systematisation of Systems Solving Physics Boundary Value Problems

   Tuomo Rossi, Jukka Räbinä, Sanna Mönkölä, Sampsa Kiiskinen, Jonni Lohi, Lauri Kettunen

   Abstract

   A general conservation law that defines a class of physical field theories is constructed. First, the notion of a general field is introduced as a formal sum of differential forms on a Minkowski manifold. By the action principle the conservation law is defined for such a general field. By construction, particular field notions of physics, e.g., magnetic flux, electric field strength, stress, strain etc. become instances of the general field. Hence, the differential equations that constitute physical field theories become also instances of the general conservation law. The general field and the general conservation law together correspond to a large class of relativistic hyperbolic physical field models. The parabolic and elliptic models can thereafter be derived by adding constraints. The approach creates solid foundations for developing software systems for scientific computing; the unifying structure shared by the class of field models makes it possible to implement software systems which are not restricted to certain predefined problems. The versatility of the proposed approach is demonstrated by numerical experiments with moving and deforming domains.

5. On the Convergence of Flow and Mechanics Iterative Coupling Schemes in Fractured Heterogeneous Poro-Elastic Media

   Tameem Almani, Kundan Kumar, Abdulrahman Manea

   Abstract

   In this work we establish the convergence of an adaptation of the fixed-stress split coupling scheme in fractured heterogeneous poro-elastic media. Here, fractures are modeled as possibly non-planar interfaces, and the flow in the fracture is described by a lubrication type system. The flow in the reservoir matrix and in the fracture are coupled to the geomechanics model through a fixed-stress split iteration, in which mass balance equations (for both flow in the matrix and in the fracture) are augmented with fixed-stress split regularization terms. The convergence proof determines the appropriate localized values of these regularization terms.

6. Finite Difference Solutions of 2D Magnetohydrodynamic Channel Flow in a Rectangular Duct

   Sinem Arslan, Münevver Tezer-Sezgin

   Abstract

   The magnetohydrodynamic (MHD) flow of an electrically conducting fluid is considered in a long channel of rectangular cross-section along with the z-axis. The fluid is driven by a pressure gradient along the z-axis. The flow is steady, laminar, fully-developed and is influenced by an external magnetic field applied perpendicular to the channel axis. So, the velocity field V = (0, 0, V ) and the magnetic field B = (0, B ₀, B) have only channel-axis components V and B depending only on the plane coordinates x and y on the cross-section of the channel which is a rectangular duct. The finite difference method (FDM) is devised to solve the problem tackling mixed type of boundary conditions such as no-slip and insulated walls and both slipping and variably conducting walls. Thus, the numerical results show the effects of the Hartmann number Ha, the conductivity parameter c and the slipping length α on both of the velocity and the induced magnetic field, especially near the walls. It is observed that the well-known characteristics of the MHD flow are also caught.

7. Applications of the PRESB Preconditioning Method for OPT-PDE Problems

   Owe Axelsson

   Abstract

   Optimal control problems constrained by partial differential equations arise in a multitude of important applications. They lead mostly to the solution of very large scale algebraic systems to be solved, which must be done by iterative methods. The problems should then be formulated so that they can be solved fast and robust, which requires the construction of an efficient preconditioner. After reduction of a variable, a two-by-two block matrix system with square blocks arises for which such a preconditioner, PRESB is presented, involving the solution of two algebraic systems which are a linear combination of the matrix blocks. These systems can be solved by inner iterations, involving some available classical solvers to some relative, not very demanding tolerance.

8. Model Order Reduction Framework for Problems with Moving Discontinuities

   H. Bansal, S. Rave, L. Iapichino, W. Schilders, N. van de Wouw

   Abstract

   We propose a new model order reduction (MOR) approach to obtain effective reduction for transport-dominated problems or hyperbolic partial differential equations. The main ingredient is a novel decomposition of the solution into a function that tracks the evolving discontinuity and a residual part that is devoid of shock features. This decomposition ansatz is then combined with Proper Orthogonal Decomposition applied to the residual part only to develop an efficient reduced-order model representation for problems with multiple moving and possibly merging discontinuous features. Numerical case-studies show the potential of the approach in terms of computational accuracy compared with standard MOR techniques.

9. Numerical Simulation of a Phase-Field Model for Reactive Transport in Porous Media

   Manuela Bastidas, Carina Bringedal, Iuliu Sorin Pop

   Abstract

   We consider a Darcy-scale model for mineral precipitation and dissolution in a porous medium. This model is obtained by homogenization techniques starting at the scale of pores. The model is based on a phase-field approach to account for the evolution of the pore geometry and the outcome is a multi-scale strongly coupled non-linear system of equations. In this work we discuss a robust numerical scheme dealing with the scale separation in the model as well as the non-linear character of the equations. We combine mesh refinement with stable linearization techniques to illustrate the behaviour of the multi-scale iterative scheme.

10. A Structure-Preserving Approximation of the Discrete Split Rotating Shallow Water Equations

    Werner Bauer, Jörn Behrens, Colin J. Cotter

    Abstract

    We introduce an efficient split finite element (FE) discretization of a y-independent (slice) model of the rotating shallow water equations. The study of this slice model provides insight towards developing schemes for the full 2D case. Using the split Hamiltonian FE framework (Bauer et al., A structure-preserving split finite element discretization of the rotating shallow water equations in split Hamiltonian form (2019). https://hal.inria.fr/hal-02020379), we result in structure-preserving discretizations that are split into topological prognostic and metric-dependent closure equations. This splitting also accounts for the schemes’ properties: the Poisson bracket is responsible for conserving energy (Hamiltonian) as well as mass, potential vorticity and enstrophy (Casimirs), independently from the realizations of the metric closure equations. The latter, in turn, determine accuracy, stability, convergence and discrete dispersion properties. We exploit this splitting to introduce structure-preserving approximations of the mass matrices in the metric equations avoiding to solve linear systems. We obtain a fully structure-preserving scheme with increased efficiency by a factor of two.

11. Iterative Coupling for Fully Dynamic Poroelasticity

    Markus Bause, Jakub W. Both, Florin A. Radu

    Abstract

    We present an iterative coupling scheme for the numerical approximation of the mixed hyperbolic-parabolic system of fully dynamic poroelasticity. We prove its convergence in the Banach space setting for an abstract semi-discretization in time that allows the application of the family of diagonally implicit Runge–Kutta methods. Recasting the semi-discrete solution as the minimizer of a properly defined energy functional, the proof of convergence uses its alternating minimization. The scheme is closely related to the undrained split for the quasi-static Biot system.

12. A Time-Dependent Parametrized Background Data-Weak Approach

    Amina Benaceur

    Abstract

    This paper addresses model reduction with data assimilation by elaborating on the Parametrized Background Data-Weak (PBDW) approach (Maday et al. Internat J Numer Methods Engrg 102(5):933–965, 2015) recently introduced to combine numerical models with experimental measurements. This approach is here extended to a time-dependent framework by means of a POD-greedy reduced basis construction.

13. Comparison of the Influence of Coniferous and Deciduous Trees on Dust Concentration Emitted from Low-Lying Highway by CFD

    Luděk Beneš

    Abstract

    Different types of vegetation barriers are frequently used for reduction of dust and noise levels. The effectivity of the measures depending on the type of used vegetation (decideous, coniferous) is studied in this article. The mathematical model is based on Reynolds—averaged Navier–Stokes (RANS) equations for turbulent fluid flow in Boussinesq approximation completed by the standard k-𝜖 model. Pollutants, considered as passive scalar, were modelled by additional transport equation. An advanced vegetation model was used. The numerical method is based on finite volume formulation. Two fractions of pollutants, PM10 and PM75, emitted from a four–lane highway were numerically simulated. Forty-nine cases of coniferous and deciduous-type forest differing in density, width and height were studied. The main processes that play a role in modelled cases are described. The differences between the effects of coniferous and deciduous trees on pollutants deposition were studied.

14. A Linear Domain Decomposition Method for Non-equilibrium Two-Phase Flow Models

    Stephan Benjamin Lunowa, Iuliu Sorin Pop, Barry Koren

    Abstract

    We consider a model for two-phase flow in a porous medium posed in a domain consisting of two adjacent regions. The model includes dynamic capillarity and hysteresis. At the interface between adjacent subdomains, the continuity of the normal fluxes and pressures is assumed. For finding the semi-discrete solutions after temporal discretization by the θ-scheme, we proposed an iterative scheme. It combines a (fixed-point) linearization scheme and a non-overlapping domain decomposition method. This article describes the scheme, its convergence and a numerical study confirming this result. The convergence of the iteration towards the solution of the semi-discrete equations is proved independently of the initial guesses and of the spatial discretization, and under some mild constraints on the time step. Hence, this scheme is robust and can be easily implemented for realistic applications.

15. An Adaptive Penalty Method for Inequality Constrained Minimization Problems

    W. M. Boon, J. M. Nordbotten

    Abstract

    The primal-dual active set method is observed to be the limit of a sequence of penalty formulations. Using this perspective, we propose a penalty method that adaptively becomes the active set method as the residual of the iterate decreases. The adaptive penalty method (APM) therewith combines the main advantages of both methods, namely the ease of implementation of penalty methods and the exact imposition of inequality constraints inherent to the active set method. The scheme can be considered a quasi-Newton method in which the Jacobian is approximated using a penalty parameter. This spatially varying parameter is chosen at each iteration by solving an auxiliary problem.

16. Multipreconditioning with Application to Two-Phase Incompressible Navier–Stokes Flow

    Niall Bootland, Andrew Wathen

    Abstract

    We consider the use of multipreconditioning to solve linear systems when more than one preconditioner is available but the optimal choice is not known. In particular, we consider a selective multipreconditioned GMRES algorithm where we incorporate a weighting that allows us to prefer one preconditioner over another. Our target application lies in the simulation of incompressible two-phase flow. Since it is not always known if a preconditioner will perform well within all regimes found in a simulation, we also consider robustness of the multipreconditioning to a poorly performing preconditioner. Overall, we obtain promising results with the approach.

17. On the Dirichlet-to-Neumann Coarse Space for Solving the Helmholtz Problem Using Domain Decomposition

    Niall Bootland, Victorita Dolean

    Abstract

    We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used.

18. A Comparison of Boundary Element and Spectral Collocation Approaches to the Thermally Coupled MHD Problem

    Canan Bozkaya, Önder Türk

    Abstract

    The thermally coupled full magnetohydrodynamic (MHD) flow is numerically investigated in a square cavity subject to an externally applied uniform magnetic field. The governing equations given in terms of stream function, vorticity, temperature, magnetic stream function, and current density, are discretized spatially using both the dual reciprocity boundary element method (DRBEM) and the Chebyshev spectral collocation method (CSCM) while an unconditionally stable backward difference scheme is employed for the time integration. Apart from the novelty of the methodology that allows the use of two different methods, the work aims to accommodate various characteristics related to the application of approaches differ in nature and origin. The qualitative and quantitative comparison of the methods are conducted in several test cases. The numerical simulations indicate that the effect of the physical controlling parameters of the MHD problem on the flow and heat transfer can be monitored equally well by both proposed schemes.