A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation

We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard equation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The latter permits us to show that the discrete free-energy is bounded, and as a result, the scheme is provably stable. The scheme and the stability proof are presented for general curvilinear three-dimensional hexahedral meshes. We use the Bassi-Rebay 1 (BR1) scheme to compute interface fluxes, and an IMplicit-EXplicit (IMEX) scheme to integrate in time. Lastly, we test the theoretical findings numerically and present examples for two and three-dimensional problems

EXPLICIT AND IMPLICIT LARGE EDDY SIMULATION FOR A NACA0012 USING A HIGH ORDER DISCONTINUOUS GALERKIN SOLVER

We perform Large Eddy Simulation on a 3D NACA0012 airfoil using a high order discontiuous Galerkin spectral element solver at moderate Reynolds numbers, Re = 1.0 · 105. The aim is to compare the results of classic explicit Sub Grid Scale models, e.g. the WALE and VREMAN models, with the ones get by using implicit LES, where no explicit SGS is used, but we use stabilising split-form energy-stable formulations, with two-point Pirozzoli and KennedyGruber fluxes. We compare general qualitative variables such as the Q-criteria as well as local quantitative variables, as boundary layer velocity profiles. We show that both methods achive similar predictions for the integrated global variables, but differences can be appreciated in the local velocity profiles

Entropy-stable discontinuous Galerkin approximation with summation-by-parts property for the incompressible Navier-Stokes equations with variable density and artificial compressibility

We present a provably stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes equations with artificial compressibility and variable density. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. We define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressiblity, and derive its associated entropy conservation law. The latter allows us to construct a provably stable split-form nodal Discontinuous Galerkin (DG) approximation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The scheme and the stability proof are presented for general curvilinear three-dimensional hexahedral meshes. We use the exact Riemann solver and the Bassi-Rebay 1 (BR1) scheme at the inter-element boundaries for inviscid and viscous fluxes respectively, and an explicit low storage Runge-Kutta RK3 scheme to integrate in time. We assess the accuracy and robustness of the method by solving the Kovasznay flow, the inviscid Taylor-Green vortex, and the Rayleigh-Taylor instability

Efficient Solvers for Space-Time Discontinuous Galerkin Spectral Element Methods

In this thesis we study efficient solvers for space-time discontinuous Galerkin spectral element methods (DG-SEM). These discretizations result in fully implicit schemes of variable order in both spatial and temporal directions. The popularity of space-time DG methods has increased in recent years and entropy stable space-time DG-SEM have been constructed for conservation laws, making them interesting for these applications. The size of the nonlinear system resulting from differential equations discretized with space-time DG-SEM is dependent on the order of the method, and the corresponding Jacobian is of block form with dense blocks. Thus, the problem arises to efficiently solve these huge nonlinear systems with regards to CPU time as well as memory consumption. The lack of good solvers for three-dimensional DG applications has been identified as one of the major obstacles before high order methods can be adapted for industrial applications.It has been proven that DG-SEM in time and Lobatto IIIC Runge-Kutta methods are equivalent, in that both methods lead to the same discrete solution. This allows to implement space-time DG-SEM in two ways: Either as a full space-time system or by decoupling the temporal elements and using implicit time-stepping with Lobatto IIIC methods. We compare theoretical properties and discuss practical aspects of the respective implementations.When considering the full space-time system, multigrid can be used as solver. We analyze this solver with the local Fourier analysis, which gives more insight into the efficiency of the space-time multigrid method. The other option is to decouple the temporal elements and use implicit Runge-Kutta time-stepping methods. We suggest to use Jacobian-free Newton-Krylov (JFNK) solvers since they are advantageous memory-wise. An efficient preconditioner for the Krylov sub-solver is needed to improve the convergence speed. However, we want to avoid constructing or storing the Jacobian, otherwise the favorable memory consumption of the JFNK approach would be obsolete. We present a preconditioner based on an auxiliary first order finite volume replacement operator. Based on the replacement operator we construct an agglomeration multigrid preconditioner with efficient smoothers using pseudo time integrators. Then only the Jacobian of the replacement operator needs to be constructed and the DG method is still Jacobian-free. Numerical experiments for hyperbolic test problems as the advection, advection-diffusion and Euler equations in several dimensions demonstrate the potential of the new approach

Hybrid multigrid methods for high-order discontinuous Galerkin discretizations

The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE solver. The multigrid hierarchy exploits all possibilities of geometric, polynomial, and algebraic coarsening, targeting engineering applications on complex geometries. Additionally, a transfer from discontinuous to continuous function spaces is performed within the multigrid hierarchy. This does not only further reduce the problem size of the coarse-grid problem, but also leads to a discretization most suitable for state-of-the-art algebraic multigrid methods applied as coarse-grid solver. The relevant design choices regarding the selection of optimal multigrid coarsening strategies among the various possibilities are discussed with the metric of computational costs as the driving force for algorithmic selections. We find that a transfer to a continuous function space at highest polynomial degree (or on the finest mesh), followed by polynomial and geometric coarsening, shows the best overall performance. The success of this particular multigrid strategy is due to a significant reduction in iteration counts as compared to a transfer from discontinuous to continuous function spaces at lowest polynomial degree (or on the coarsest mesh). The coarsening strategy with transfer to a continuous function space on the finest level leads to a multigrid algorithm that is robust with respect to the penalty parameter of the SIPG method. Detailed numerical investigations are conducted for a series of examples ranging from academic test cases to more complex, practically relevant geometries. Performance comparisons to state-of-the-art methods from the literature demonstrate the versatility and computational efficiency of the proposed multigrid algorithms

A discontinuous Galerkin method for the solution of compressible flows

This thesis presents a methodology for the numerical solution of one-dimensional (1D) and two-dimensional (2D) compressible flows via a discontinuous Galerkin (DG) formulation. The 1D Euler equations are used to assess the performance and stability of the discretisation. The explicit time restriction is derived and it is established that the optimal polynomial degree, p, in terms of efficiency and accuracy of the simulation is p = 5. Since the method is characterised by minimal diffusion, it is particularly well suited for the simulation of the pressure wave generated by train entering a tunnel. A novel treatment of the area-averaged Euler equations is proposed to eliminate oscillations generated by the projection of a moving area on a fixed mesh and the computational results are validated against experimental data. Attention is then focussed on the development of a 2D DG method implemented using the high-order library Nektar++. An Euler and a laminar Navier–Stokes solvers are presented and benchmark tests are used to assess their accuracy and performance. An artificial diffusion term is implemented to stabilise the solution of the Euler equations in transonic flow with discontinuities. To speed up the convergence of the explicit method, a new automatic polynomial adaptive procedure (p-adaption) and a new zonal solver are proposed. The p-adaptive procedure uses a discontinuity sensor, originally developed as an artificial diffusion sensor, to assign appropriate polynomial degrees to each element of the domain. The zonal solver uses a modification of a method for matching viscous subdomains to set the interface conditions between viscous and inviscid subdomains that ensures stability of the flow computation. Both the p-adaption and the zonal solver maintain the high-order accuracy of the DG method while reducing the computational cost of the simulation

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions