arXiv subjects

Andreas Dedner

Publications and source records attributed to Andreas Dedner.

30 records · Page 2Linked to original sources

arXiv2018

The Dune Python Module

In this paper we present the new Dune-Python module which provides Python bindings for the Dune core, which is a C++ environment for solving partial differential equations. The aim of this new module is to firstly provide the general infrastructure for exporting realizations of statically polymorphic interfaces based on just-in-time compilation and secondly to provide bindings for the central interfaces of the dune core modules. In the first release we focus on the grid interface. Our aim is to only introduce a thin layer when passing objects into Python which can be removed when the object is passed back into a C++ algorithm. Thus no efficiency is lost and little additional code maintenance cost is incurred. To make the transition for Dune users to the Python environment straightforward the Python classes provide a very similar interface to their C++ counterparts. In addition, vectorized versions of many interfaces allow for more efficient code on the Python side. The infrastructure for exporting these interfaces and the resulting bindings for a Dune grid are explained in detail in this paper for both experienced Dune users and others interested in a flexible Python environment for implementing grid based schemes for solving partial differential equations.

cs.MS↗

arXiv2018

Python Framework for HP Adaptive Discontinuous Galerkin Method for Two Phase Flow in Porous Media

In this paper we present a framework for solving two phase flow problems in porous media. The discretization is based on a Discontinuous Galerkin method and includes local grid adaptivity and local choice of polynomial degree. The method is implemented using the new Python frontend Dune-FemPy to the open source framework Dune. The code used for the simulations is made available as Jupyter notebook and can be used through a Docker container. We present a number of time stepping approaches ranging from a classical IMPES method to fully coupled implicit scheme. The implementation of the discretization is very flexible allowing for test different formulations of the two phase flow model and adaptation strategies.

cs.CE↗

arXiv2017

Formulation, Implementation and Validation of the Horizontal Coupling Method for 1D/2D Shallow Water Flow Models

One dimensional (1D) simulations of the flow and flooding of open channels are known to be inaccurate as the flow is multi-dimensional in nature, especially at the flooded regions. However, multi-dimensional simulations, even in two dimensions (2D), are computationally expensive, hence the problem of efficiently coupling 2D and 1D simulations for the flow and flooding of open channels has been the subject of much research and is investigated in this paper. We adopt a 1D model with coupling term for the channel flow and the 2D shallow water flow model for the floodplain. The 1D model with coupling term is derived by integrating the 3D Free Surface Euler equations but without imposing any restriction on the channel width variations. %leading to a new coupling term. Finite volume methods are formulated for both the 2D and 1D models including a discrete coupling term in closed form. Coupling is achieved through the discrete coupling term in the 1D model and the lateral numerical fluxes in the 2D model. Since the lateral discharge in the channel cannot be guaranteed to be zero during flooding, we aim to recover the lateral variation by computing two lateral discharges over each cross section and propose to use an ad-hoc model based on the $y$-discharge equation in the 2D model for this purpose. We then propose the numerical scheme for this ad-hoc model following the hydrostatic reconstruction philosophy. Then, we show that the resulting method, named Horizontal Coupling Method (HCM), is well-balanced; we introduce the no-numerical flooding property and also show that the method satisfies the property. Three numerical test cases are used to verify the performance of the method. The results show that the method performs well in both accuracy and efficiency and also approximates the channel lateral discharges with very good accuracy and little computational overhead.

math.NA↗

arXiv2016

Convection in a Single Column -- Modelling, Algorithm and Analysis

The group focused on a model problem of idealised moist air convection in a single column of atmosphere. Height, temperature and moisture variables were chosen to simplify the mathematical representation (along the lines of the Boussinesq approximation in a height variable defined in terms of pressure). This allowed exact simple solutions of the numerical and partial differential equation problems to be found. By examining these, we identify column behaviour, stability issues and explore the feasibility of a more general solution process.

physics.ao-ph↗

arXiv2015

A posteriori analysis of fully discrete method of lines DG schemes for systems of conservation laws

We present reliable a posteriori estimators for some fully discrete schemes applied to nonlinear systems of hyperbolic conservation laws in one space dimension with strictly convex entropy. The schemes are based on a method of lines approach combining discontinuous Galerkin spatial discretization with single- or multi-step methods in time. The construction of the estimators requires a reconstruction in time for which we present a very general framework first for odes and then apply the approach to conservation laws. The reconstruction does not depend on the actual method used for evolving the solution in time. Most importantly it covers in addition to implicit methods also the wide range of explicit methods typically used to solve conservation laws. For the spatial discretization, we allow for standard choices of numerical fluxes. We use reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. We study under which conditions on the numerical flux the estimate is of optimal order pre-shock. While the estimator we derive is computable and valid post-shock for fixed meshsize, it will blow up as the meshsize tends to zero. This is due to a breakdown of the relative entropy framework when discontinuities develop. We conclude with some numerical benchmarking to test the robustness of the derived estimator.

math.NA↗

arXiv2015

The DUNE-ALUGrid Module

In this paper we present the new DUNE-ALUGrid module. This module contains a major overhaul of the sources from the ALUgrid library and the binding to the DUNE software framework. The main changes include user defined load balancing, parallel grid construction, and an redesign of the 2d grid which can now also be used for parallel computations. In addition many improvements have been introduced into the code to increase the parallel efficiency and to decrease the memory footprint. The original ALUGrid library is widely used within the DUNE community due to its good parallel performance for problems requiring local adaptivity and dynamic load balancing. Therefore, this new model will benefit a number of DUNE users. In addition we have added features to increase the range of problems for which the grid manager can be used, for example, introducing a 3d tetrahedral grid using a parallel newest vertex bisection algorithm for conforming grid refinement. In this paper we will discuss the new features, extensions to the DUNE interface, and explain for various examples how the code is used in parallel environments.

cs.MS↗

arXiv2015

Discontinuous Galerkin methods for hyperbolic and advection-dominated problems on surfaces

We extend the discontinuous Galerkin (DG) framework to the analysis of first-order hyperbolic and advection-dominated problems posed on implicitly defined surfaces. The focus will be on the hyperbolic part, which is discretised using a "discrete surface" generalisation of the jump-stabilised upwind flux. A key issue arising in the analysis (which does not appear in the planar setting) is the treatment of the discrete velocity field, choices of which play an important role in the stability of the scheme. We then prove optimal error estimates in an appropriate norm given a number of assumptions on the discrete velocity field, which are then investigated and discussed in more detail. The theoretical results are verified numerically for a number of test problems exhibiting advection-dominated behaviour.

math.NA↗

arXiv2015

Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio

Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. For example, in numerical weather- and climate-prediction an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by Börm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the analysis to three dimensions under slightly weakened assumptions, and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment (DUNE), which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR supercomputer.

math.NA↗

arXiv2014

High order discontinuous Galerkin methods on surfaces

We derive and analyze high order discontinuous Galerkin methods for second-order elliptic problems on implicitely defined surfaces in $\mathbb{R}^{3}$. This is done by carefully adapting the unified discontinuous Galerkin framework of Arnold et al. [2002] on a triangulated surface approximating the smooth surface. We prove optimal error estimates in both a (mesh dependent) energy norm and the $L^2$ norm.

math.NA↗

arXiv2014

Adaptive discontinuous Galerkin methods on surfaces

We present a dual weighted residual-based a posteriori error estimate for a discontinuous Galerkin (DG) approximation of a linear second-order elliptic problem on compact smooth connected and oriented surfaces in $\mathbb{R}^{3}$ which are implicitly represented as level sets of a smooth function. We show that the error in the energy norm may be split into a "residual part" and a higher order "geometric part". Upper and lower bounds for the resulting a posteriori error estimator are proven and we consider a number of challenging test problems to demonstrate the reliability and efficiency of the estimator. We also present a novel "geometric" driven refinement strategy for PDEs on surfaces which considerably improves the performance of the method on complex surfaces.

math.NA↗

arXiv2013

Discontinuous Galerkin methods for nonvariational problems

We extend the finite element method introduced by Lakkis and Pryer [2011] to approximate the solution of second order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the NVFEM as a mixed method whereby the finite element Hessian is an auxiliary variable in the formulation. Representing the finite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic problems. Furthermore, the system matrix is very easy to assemble, Thus this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach. We conduct a stability and consistency analysis making use of the unified framework set out in Arnold et. al. [2001]. We also give an apriori analysis of the method. The analysis applies to any consistent representation of the finite element Hessian, thus is applicable to the previous works making use of continuous Galerkin approximations.

math.NA↗

arXiv2013

Analysis of the discontinuous Galerkin method for elliptic problems on surfaces

We extend the discontinuous Galerkin (DG) framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface. An interior penalty (IP) method is introduced on a discrete surface and we derive a-priori error estimates by relating the latter to the original surface via the lift introduced in Dziuk (1988). The estimates suggest that the geometric error terms arising from the surface discretisation do not affect the overall convergence rate of the IP method when using linear ansatz functions. This is then verified numerically for a number of test problems. An intricate issue is the approximation of the surface conormal required in the IP formulation, choices of which are investigated numerically. Furthermore, we present a generic implementation of test problems on surfaces.

math.NA↗