Discontinuous Galerkin Methods

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.

Coupling local discontinuous and continuous galerkin methods for flow problems

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in Yuan et al. (SIAM J Sci Comput 38:A2987–A3019, 2016). The DG methods in Yuan et al. (2016) can maintain positivity and high order accuracy, but they rely both on the scaling limiter in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax–Wendroff type theorem is proved in Yuan et al. (2016), guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $$P^k$$ of piecewise polynomials of degree at most k. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu (2010) can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $$P^k$$ or $$Q^k$$ spaces (piecewise kth degree polynomials or piecewise tensor-product kth degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu (2010). Computational results are provided to demonstrate the good performance of our DG schemes.

Cavities are strong scattering parts for an aircraft. A discontinuous Galerkin (DG) method is presented to efficiently optimize cavity scattering. The DG method is shown to have ability of only remeshing the optimized parts and not remeshing the whole cavity in the computing process of scattering from the whole cavity. An effective and simple preconditioner, called distance sparse preconditioner (DSP), is applied to the DG method, different from the conventional block-diagonal preconditioner (BDP) and sparse approximate inverse preconditioner (SAI). The former one is usually recommended in the DG methods and the latter one is generally used in method of moment (MoM). Numerical results show that the DSP is more efficient than the BDP and SAI in the DG method for the cavity analysis. The effects of cavity opening shape, area, and transition shape are studied for scattering from cavities. Some interesting and useful numerical results are presented for cavity design.

Split form ALE discontinuous Galerkin methods with applications to under-resolved turbulent low-Mach number flows

In this paper, a finite element formulation is defined in the framework of the discontinuous Galerkin method. Discontinuous Galerkin (dG) methods are classically used in fluid mechanics, however recently their application in solid mechanics has become more vivid among scientists. Of special interest is their application in elliptic problems with constraints such as incompressibility which leads to volumetric locking phenomenon and also in some structural models of shells, plates and beams with compatibility constraints, which brings about shear locking [1]. While classical standard Galerkin methods must be continuous, dG methods can be applied for discontinuities across element boundaries, where a jump of a value (displacement) can be observed. In the present work, a dG method is applied to a linear elastic bar, where a weak discontinuity is allowed in the bar. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws

Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations

Local discontinuous Galerkin methods for nonlinear dispersive equations

Discontinuous Galerkin (DG) methods support elements of various types, nonmatching grid and varying polynomial order in each element. In DG methods continuity at element interfaces is weakly enforced with the addition of proper penalty terms on the variational formulation commonly referred to as numerical fluxes. An interior penalty approach to derive a DG method for solving the two first order Maxwell's equations in the time domain is presented. The proposed method is explicit and conditionally stable. In addition, a local time-stepping strategy is applied to increase efficiency and reduce the computational time.

On large geographic scales, ocean waves are represented in a spectral sense via the action balance equation, which propagates action density through both geographic and spectral space. In this paper, a new computational spectral wave model is developed by using discontinuous Galerkin (DG) methods in both geographic and spectral space. DG methods allow for the use of unstructured geographic meshes and higher-order approximations for action propagation in both geographic and spectral space, which we show leads to increased accuracy. This DG spectral wave propagation model is verified and validated through comparisons to manufactured and analytic solutions as well as to the Simulating WAves Nearshore (SWAN) model. Coupled wave/circulation models are needed for many applications including for the interaction between waves and currents during daily wind and tide driven flows. We loosely couple the new DG spectral wave model to the DG Shallow Water Equation Model (DG-SWEM), an existing DG circulation model. In addition to formulating the DG method for the coupled wave/circulation model, we derive an a priori error estimate. Preliminary numerical results of the DG coupled wave/circulation model are presented with comparisons to DG-SWEM coupled tightly to SWAN.

A dissipation-rate reserving DG method for wave catching-up phenomena in a nonlinearly elastic composite bar

We introduce the application of an arbitrary high‐order derivative (ADER) discontinuous Galerkin (DG) method to simulate earthquake rupture dynamics. The ADER‐DG method uses triangles as computational cells which simplifies the process of discretization of very complex surfaces and volumes by using external automated tools. Discontinuous Galerkin methods are well suited for solving dynamic rupture problems in the velocity‐stress formulation as the variables are naturally discontinuous at the interface between two elements. Therefore, the fault has to be honored by the computational mesh. The so‐called Riemann problem can be solved to obtain well defined values of the variables at the discontinuity itself. Fault geometries of high complexity can be modeled thanks to the flexibility of unstructured meshes, which solves a major bottleneck of other high‐order numerical methods. Additionally, element refinement and coarsening are easily controlled in the meshing process to better resolve the near‐fault area of the model. The fundamental properties of the method are shown, as well as a series of validating exercises with reference solutions and a comparison with the well‐established finite difference, boundary integral, and spectral element methods, in order to test the accuracy of our formulation. An example of dynamic rupture on a nonplanar fault based upon the Landers 1992 earthquake fault system is presented to illustrate the main potentials of the new method.

In this final chapter we present the discontinuous Galerkin (dG) method. This method is based on finite element spaces that consist of discontinuous piecewise polynomials defined on a partition of the computational domain. Such methods are very flexible, for example, since they allow construction of more general methods and since they allow for simple adaptation. Discontinuous Galerkin methods were originally developed for first order problems and were later extended to second order problems. We cover both categories, and derive basic stability and error estimates. Due to the discontinuous nature of the finite element space additional terms in the weak form are necessary to enforce the proper continuity conditions between adjacent elements. We also consider how to handle these additional terms in the implementation of the method.

An efficient second-order cell-centered Lagrangian discontinuous Galerkin (DG) method for solving two-dimensional (2D) elastic-plastic flows with the hypo-elastic constitutive model and von Mises yield condition is presented. First, starting from the governing equations of conserved quantities in the Euler framework, the integral weak formulation of them in the Lagrangian framework is derived. Next, the DG method is used for spatial discretization of both the weak formulation of conserved quantities and the evolution equation of deviatoric stress tensor. The Taylor basis functions defined in the reference coordinates provide the piecewise polynomial expansion of the variables, including the conserved quantities and the deviatoric stress tensor. The vertex velocities and Cauchy stress tensor on the edges are computed using a nodal solver equipped with a variant of Li's new Harten-Lax-van Leer-contact approximate Riemann solver [Li et al., “An HLLC-type approximate Riemann solver for two-dimensional elastic-perfectly plastic model,” J. Comput. Phys. 448, 110675 (2022)], in which the longitudinal wave velocity in the plastic state is modified. Then the vertex velocities and Cauchy stress tensor on the edges are used to compute numerical fluxes. A second-order total variation diminishing Runge–Kutta scheme is used for time discretization of both the governing equations of conserved quantities and the evolution equation of deviatoric stress tensor. After solving the evolution equation of deviatoric stress tensor, a radial return algorithm is performed at the Gauss points of each element according to the von Mises yield condition. And then the coefficients of the DG expansion for the deviatoric stress tensor on each element are modified by a least squares procedure using the deviatoric stress tensors at these Gauss points. To achieve second-order accuracy, the least squares procedure is used for piecewise linear reconstruction of conserved quantities and the deviatoric stress tensor, and the Barth–Jespersen limiter is used to suppress the nonphysical numerical oscillation near the discontinuities. After that, the coefficients of the DG expansion are modified through L2 projection using the reconstructed polynomials. Finally, a second-order cell-centered Lagrangian DG scheme is established. Several tests demonstrate that the new scheme achieves second-order accuracy with good robustness, and that the DG method of updating the deviatoric stress tensor has comparable accuracy and much higher efficiency with mesh refinement compared with previous works.