Coupling local discontinuous and continuous galerkin methods for flow problems

The aim of this work is to provide a solver for viscoelastic multi-phase flows within the Bounded Support Spectral Solver (BoSSS) currently under development at the Chair of Fluid Dynamics at the Technical University of Darmstadt. The discretisation in BoSSS consists of a high-order Discontinuous Galerkin (DG) method for single-phase flow and a high-order eXtended Discontinuous Galerkin (XDG) method for the multi-phase purpose. The solver shall be used to investigate numerically the behaviour of viscoelastic droplets. The macroscopic Oldroyd B model which is used in a wide range of applications is chosen as the constitutive model. A detailed derivation of the system of equations including the modeling principles for the Oldroyd B model is presented. A DG discretisation of the system of equations including the Local Discontinuous Galerkin (LDG) method is presented after introducing the field of the DG method. The derivation of appropriate flux functions for the constitutive equations and the extra stress tensor are one of the key derivations of this scientific work. Difficulties arising in the numerical solution of viscoelastic flow problems for higher Weissenberg numbers for different discretisation methods are due to the convection dominated, mixed hyperbolic-elliptic-parabolic nature of the system of equations. Several strategies are presented which overcome these problems and are known from the literature. A key achievement of this scientific work is the application of the LDG method, originally developed for a hyperbolic system of equations for a Newtonian fluid, on the viscoelastic system of equations which renders methods for preserving ellipticity unnecessary. Furthermore, various strategiesn to enhance and to support convergence of the solution of the DG discretised system are presented. These are the Newton method with different approaches determining the Jacobian of the system, a homotopy continuation method based on the Weissenberg number for a better initial guess for the Newton method, and a troubled cell indicator combined with an artificial diffusion approach or an adaptive mesh refinement strategy, respectively. For the completeness of this work the XDG method is presented using a sharp interface approach with a signed distance level-set function as it is implemented in BoSSS. The single-phase solver is combined with these methods and appropriate flux functions for the interface are implemented to enable multi-phase applications for viscoelastic fluid. Several numerical experiments are conducted to verify and to validate the viscoelastic singlephase solver and to show the capability of the viscoelastic multi-phase solver to simulate viscoelastic droplets. Advantages and disadvantages of the implementation and an outlook for future research can be found in the conclusion.

In this paper, a local discontinuous Galerkin (LDG) method is is designed for solving the BBM-Burgers equation with four-order spatial derivatives. Our schemes extend the previous work of Xu and Shu [8] solving the nonlinear Schrodinger equations on LDG method. The L2 stability of the LDG methods is proved for general solutions. Numerical results are shown to illustrate the accuracy and capability of the LDG method.

An h-adaptive local discontinuous Galerkin method for second order wave equation: Applications for the underwater explosion shock hydrodynamics

Local discontinuous Galerkin (LDG) methods are popular for convection–diffusion equations. In LDG methods, we introduce an auxiliary variable p to represent the derivative of the primary variable u, and solve them on the same mesh. In this paper, we will introduce a new LDG method, and solve u and p on different meshes. The stability and error estimates will be investigated. The new algorithm is more flexible and flux-free for pure diffusion equations without introducing additional computational cost compared with the original LDG methods, since it is not necessary to solve each equation twice. Moreover, it is possible to construct third-order maximum-principle-preserving schemes based on the new algorithm. However, one cannot anticipate optimal accuracy in some special cases. In this paper, we will find out the reason for accuracy degeneration which further leads to several alternatives to obtain optimal convergence rates. Finally, several numerical experiments will be given to demonstrate the stability and optimal accuracy of the new algorithm.

An Adaptive Strategy for the Local Discontinuous Galerkin Method Applied to Porous Media Problems

Local discontinuous Galerkin methods for nonlinear dispersive equations

A semi-Lagrangian discontinuous Galerkin (DG) – local DG method for solving convection-diffusion equations

The discontinuous Galerkin (dG) method was introduced by Reed and Hill [73] in 1973 for steady-state neutron transport as an hyperbolic problem. This was followed by other studies; by Bassi and Rebay [12] for the compressible Navier-Stokes equations, Cockburn and Shu [31] developed the local discontinuous Galerkin (ldG) method for advection-diffusion equations, and Peraire and Persson [69] introduced the compact discontinuous Galerkin (cdG) method. Independent of the dG methods, interior penalty (IP) methods have been developed for elliptic and parabolic problems by Douglas and Dupont [40] and Wheeler [95]. Then, in the 1980’s, Arnold et al. [6] proposed a unified classification and analysis of various kinds of dG methods. Later on, the dG methods were developed for elliptic problems [8, 24, 77] and for problems with advection [7, 14, 51, 56].

In the present work the Local Discontinuous Galerkin (LDG) method with polynomial adaptivity is applied to the Large Eddy Simulation (LES) of the parallel blade-vortex interaction (BVI). The BVI phenomenon occurs on helicopter and drone rotors in manoeuvring conditions and it produces impulsive changes in the pressure distributions, vibrations and noise. To deeply understand the mechanism of load generation related to the pressure field and three dimensional perturbations growth, to focus on the interaction between the vortex and the three dimensional structures in boundary layer and wake, accurate 3D unsteady numerical simulations of turbulent flows are necessary. For this reason, it is very important the use of a numerical code based on high order schemes such as LDG. Moreover, in the LDG approach, the numerical resolution can be varied on each element and in time, adapting to the requirement of the simulated flow and saving a large amount of computing resources. In the used numerical code the criterion for variation of the polynomial order is based on a refinement indicator especially suited for LES and based on the structure function. The local polynomial representation directly provides a means to separate large from small scale modes, thus providing the starting point for the definition of the subgrid scale models. In the present simulations, the subgrid scales contribution is represented with a sophisticated dynamic anisotropic subgrid model, suitable and well tested for wall resolved LES and complex separated turbulent flows. The BVI is simulated highlighting the effect of the vortex on the pressure distributions, on the boundary layer separation and on the resulting forces.

Stability analysis and error estimates of fully-discrete local discontinuous Galerkin methods for simulating wormhole propagation with Darcy–Forchheimer model

An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions

In this paper, an implicit-explicit (IMEX) local discontinuous Galerkin (LDG) method is proposed to analyze the organic electrochemical transistors (OECTs). The LDG method is used to spatially discretize the coupled system of Poisson and drift-diffusion (DD) equations. For the time integration, the IMEX scheme uses implicit and explicit methods for the linear diffusion and the nonlinearly coupled drift terms, respectively. This results in a time marching scheme where the time-step size is upper-bounded by a positive constant but can be selected independently from spatial mesh size. Numerical results demonstrate that the proposed IMEX-LDG scheme allows for a significantly larger time-step size than its fully-explicit counterpart, and therefore much faster in simulation of OECTs.

We construct and analyze conservative local discontinuous Galerkin (LDG) methods for the Generalized Korteweg-de-Vries equation. LDG methods are designed by writing the equation as a system and performing separate approximations to the spatial derivatives. The main focus is on the development of conservative methods which can preserve discrete versions of the first two invariants of the continuous solution, and a posteriori error estimates for a fully discrete approximation that is based on the idea of dispersive reconstruction. Numerical experiments are provided to verify the theoretical estimates.

A numerical method for analyzing eddy current magnetic field in high-speed moving conductors using local discontinuous Galerkin (LDG) method is presented. A typical numerical example is chosen to illustrate the accuracy and nonoscillatory nature of the method for a convection dominated case. The simulation results using the proposed algorithm are validated using standard finite element method (FEM). As an application of LDG method in engineering problems, the TEAM workshop problem 9 is studied and solved by the proposed method with second order rectangular discontinuous elements. Numerical results show that the method can solve eddy current problems with high speed moving conductors accurately.

We present a new numerical method for solving nonlinear reaction‐diffusion systems with cross‐diffusion which are often taken as mathematical models for many applications in the biological, physical, and chemical sciences. The two‐dimensional system is discretized by the local discontinuous Galerkin (LDG) method on unstructured triangular meshes associated with the piecewise linear finite element spaces, which can derive not only numerical solutions but also approximations for fluxes at the same time comparing with most of study work up to now which has derived numerical solutions only. Considering the stability requirement for the explicit scheme with strict time step restriction (), the implicit integration factor (IIF) method is employed for the temporal discretization so that the time step can be relaxed as Δt = O(hmin). Moreover, the method allows us to compute element by element and avoids solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical simulations about the system with exact solution and the Brusselator model, which is a theoretical model for a type of autocatalytic chemical reaction, are conducted to confirm the expected accuracy, efficiency, and advantages of the proposed schemes.