Frieder Lörcher

A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes

In this paper, we consider numerical approximations of diffusion terms for finite volume as well as discontinuous Galerkin schemes. Both classes of numerical schemes are quite successful for advection equations capturing strong gradients or even discontinuities, because they allow their approximate solutions to be discontinuous at the grid cell interfaces. But, this property may lead to inconsistencies with a proper definition of a diffusion flux. Starting with the finite volume formulation, we propose a numerical diffusion flux which is based on the exact solution of the diffusion equation with piecewise polynomial initial data. This flux may also be used by discontinuous Galerkin schemes and gives a physical motivation for the Symmetric Interior Penalty discontinuous Galerkin scheme. The flux proposed leads to a one-step finite volume or discontinuous Galerkin scheme for diffusion, which is arbitrary order accurate simultaneously in space and time. This strategy is extended to define suitable numerical fluxes for nonlinear diffusion problems.

A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions

Abstract In part I of these two papers we introduced for inviscid flow in one space dimension a discontinuous Galerkin scheme of arbitrary order of accuracy in space and time. In the second part we extend the scheme to the compressible Navier-Stokes equations in multi dimensions. It is based on a space-time Taylor expansion at the old time level in which all time or mixed space-time derivatives are replaced by space derivatives using the Cauchy-Kovalevskaya procedure. The surface and volume integrals in the variational formulation are approximated by Gaussian quadrature with the values of the space-time approximate solution. The numerical fluxes at grid cell interfaces are based on the approximate solution of generalized Riemann problems for both, the inviscid and viscous part. The presented scheme has to satisfy a stability restriction similar to all other explicit DG schemes which becomes more restrictive for higher orders. The loss of efficiency, especially in the case of strongly varying sizes of grid cells is circumvented by use of different time steps in different grid cells. The presented time accurate numerical simulations run with local time steps adopted to the local stability restriction in each grid cell. In numerical simulations for the two-dimensional compressible Navier-Stokes equations we show the efficiency and the optimal order of convergence being p+1, when a polynomial approximation of degree p is used.

A Discontinuous Galerkin Scheme Based on a Space—Time Expansion. I. Inviscid Compressible Flow in One Space Dimension

In this paper, we propose an explicit discontinuous Galerkin scheme for conservation laws which is of arbitrary order of accuracy in space and time. The basic idea is to use a Taylor expansion in space and time to define a space–time polynomial in each space–time element. The space derivatives are given by the approximate solution at the old time level, the time derivatives and the mixed space–time derivatives are computed from these space derivatives using the so-called Cauchy–Kovalevskaya procedure. The space–time volume integral is approximated by Gauss quadrature with values at the space–time Gaussian points obtained from the Taylor expansion. The flux in the surface integral is approximated by a numerical flux with arguments given by the Taylor expansions from the left and from the right-hand side of the element interface. The locality of the presented method together with the space–time expansion gives the attractive feature that the time steps may be different in each grid cell. Hence, we drop the common global time levels and propose that every grid zone runs with its own time step which is determined by the local stability restriction. In spite of the local time steps the scheme is locally conservative, fully explicit, and arbitrary order accurate in space and time for transient calculations. Numerical results are shown for the one-dimensional Euler equations with orders of accuracy one up to six in space and time.

Polymorphic nodal elements and their application in discontinuous Galerkin methods

In this work, we discuss two different but related aspects of the development of efficient discontinuous Galerkin methods on hybrid element grids for the computational modeling of gas dynamics in complex geometries or with adapted grids. In the first part, a recursive construction of different nodal sets for hp finite elements is presented. They share the property that the nodes along the sides of the two-dimensional elements and along the edges of the three-dimensional elements are the Legendre-Gauss-Lobatto points. The different nodal elements are evaluated by computing the Lebesgue constants of the corresponding Vandermonde matrix. In the second part, these nodal elements are applied within the modal discontinuous Galerkin framework. We still use a modal based formulation, but introduce a nodal based integration technique to reduce computational cost in the spirit of pseudospectral methods. We illustrate the performance of the scheme on several large scale applications and discuss its use in a recently developed space-time expansion discontinuous Galerkin scheme.

An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations

In this paper we propose a discontinuous Galerkin scheme for the numerical approximation of unsteady heat conduction and diffusion problems in multi dimensions. The scheme is based on a discrete space-time variational formulation and uses an explicit approximative solution as predictor. This predictor is obtained by a Taylor expansion about the barycenter of each grid cell at the old time level in which all time or mixed space-time derivatives are replaced by space derivatives using the differential equation several times. The heat flux between adjacent grid cells is approximated by a local analytical solution. It takes into account that the approximate solution may be discontinuous at grid cell interfaces and allows the approximation of discontinuities in the heat conduction coefficient. The presented explicit scheme has to satisfy a typical parabolic stability restriction. The loss of efficiency, especially in the case of strongly varying sizes of cells in unstructured grids, is circumvented by allowing different time steps in each grid cell which are adopted to the local stability restrictions. We discuss the linear stability properties in this case of varying diffusion coefficients, varying space increments and local time steps and extent these considerations also to a modified symmetric interior penalization scheme. In numerical simulations we show the efficiency and the optimal order of convergence in space and time.

Arbitrary high order finite-volume methods for electromagnetic wave propagation

Problems in electromagnetic wave propagation often require high accuracy approximations with low resolution computational grids. For non-stationary problems such schemes should possess the same approximation order in space and time. In the present article we propose for electromagnetic applications an explicit class of robust finite-volume (FV) schemes for the Maxwell equations. To achieve high accuracy we combine the FV method with the so-called ADER approach resulting in schemes which are arbitrary high order accurate in space and time. Numerical results and convergence investigations are shown for two and three-dimensional test cases on Cartesian grids, where the used FV-ADER schemes are up to 8th order accurate in both space and time.

Aeroacoustic Simulations for Complex Geometries based on Hybrid Meshes

This paper advances the idea of a heterogeneous domain decomposition for Computational Aeroacoustics (CAA). Direct simulations of aeroacoustic problems are accelerated by sub-dividing the computational domain into smaller domains. In each of these subdomains the equations, the discretization, the mesh and the time step may be different and are adapted to the local behavior of the solution. High order methods such as ADER-Finite Volume and ADER-Discontinuous Galerkin methods are used to reduce the total number of elements and to ensure good wave propagation properties. Here, we add a high order Finite Difference method to the acoustic solvers and integrate it into the coupling framework. The new scheme shows good performance properties for the convective transport of a Gaussian pulse in density. In the examples section, convergence rates for the coupling procedure in 3D show that high order of accuracy is maintained globally also for partitioned domains. A numerical example that involves multiple domains underlines the flexibility of the approach. Another example shows that the proposed domain decomposition also holds for the coupling of the Navier-Stokes equations with the Linearized Euler Equations.

An Explicit Space-Time Discontinuous Galerkin Scheme with Local Time-Stepping for Unsteady Flows

The objective of our project is the development of high-order methods for the unsteady Euler and Navier Stokes equations. For this, we consider an explicit DG scheme formulated in a space-time context called the Space-Time Expansion DG scheme (STE-DG). Our focus lies on the improvement of two main aspects: Increase of efficiency in the temporal and spatial discretization by giving up the assumption that all grid cells run with the same time step and introducing local time-stepping and the shock capturing property, where we have adopted the artificial viscosity approach as described by Persson and Peraire to our STE-DG scheme. Thus, we try to resolve the shock within a few relatively large grid cells forming a narrow viscous profile by locally adding some amount of artificial viscosity.

A Numerical Diffusion Flux Based on the Diffusive Riemannproblem

In finite volume (FV) or discontinuous Galerkin (DG) schemes the approximate solution may jump at the grid cell interface. Any physical phenomena which can not be resolved on the given grid will result in such a jump. If the time evolution of these jumps can be approximated in a stable and consistent way, then the numerical scheme does not generate spurious oscillations and gives meaningful mean values of under-resolved phenomena. In his pioneering work Godunov proposed to approximate the convection flux between grid cells by solving the break down of the jump into different waves. Approximations of this approach are called Godunov-type schemes and are described, e.g., in the book of [TOR99].

A space-time expansion discontinuous Galerkin scheme with local time-stepping for the ideal and viscous MHD equations

In this paper, we present the extension of the space-time expansion discontinuous Galerkin to handle ideal and viscous magnetohydrodynamics (MHD) equations. The local time-stepping strategy that this scheme is capable of allows each cell to have its own time step whereas the high order of accuracy in time is retained. This may significantly speed up calculations. The diffusive flux is evaluated through a so-called diffusive generalized Riemann problem. The divergence constraint of the MHD equations is addressed, and a hyperbolic cleaning method is shown that can be enhanced by utilizing the local time-stepping framework. MHD problems such as the Orszag-Tang vortex or the magnetic blast problem are performed to challenge the capabilities of the proposed space-time expansion scheme.