Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)

This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well

Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations

In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations

A direct-hybrid CFD/CAA method based on lattice Boltzmann and acoustic perturbation equations

The accuracy of two direct coupled two-step CFD/CAA methods is discussed. For the flow field either a finite-volume (FV) method for the solution of the Navier–Stokes equations or a lattice Boltzmann (LB) method is coupled to a discontinuous Galerkin (DG) method for the solution of the acoustic perturbation equations. The coupling takes advantage of a joint Cartesian mesh allowing for the exchange of the acoustic sources without MPI communication. An immersed boundary treatment of the acoustic scattering from solid bodies by a novel solid wall formulation is implemented and validated in the DG method. Results for the case of a spinning vortex pair and the low Reynolds number unsteady flow around a circular cylinder show that a solution with comparable accuracy is obtained for the two direct-hybrid methods when using identical mesh resolution

Asymptotically complexity diminishing schemes (ACDS) for kinetic equations in the diffusive scaling

In this work, we develop a new class of numerical schemes for collisional kinetic equations in the diffusive regime. The first step consists in reformulating the problem by decomposing the solution in the time evolution of an equilibrium state plus a perturbation. Then, the scheme combines a Monte Carlo solver for the perturbation with an Eulerian method for the equilibrium part, and is designed in such a way to be uniformly stable with respect to the diffusive scaling and to be consistent with the asymptotic diffusion equation. Moreover, since particles are only used to describe the perturbation part of the solution, the scheme becomes computationally less expensive – and is thus an asymptotically complexity diminishing scheme (ACDS) – as the solution approaches the equilibrium state due to the fact that the number of particles diminishes accordingly. This contrasts with standard methods for kinetic equations where the computational cost increases (or at least does not decrease) with the number of interactions. At the same time, the statistical error due to the Monte Carlo part of the solution decreases as the system approaches the equilibrium state: the method automatically degenerates to a solution of the macroscopic diffusion equation in the limit of infinite number of interactions. After a detailed description of the method, we perform several numerical tests and compare this new approach with classical numerical methods on various problems up to the full three dimensional case

Discontinuous Galerkin Methods for Extended Hydrodynamics.

This dissertation presents a step towards high-order methods for continuum-transition flows. In order to achieve maximum accuracy and efficiency for numerical methods on a distorted mesh, it is desirable that both governing equations and corresponding numerical methods are in some sense compact. We argue our preference for a physical model described solely by first-order partial differential equations called hyperbolic-relaxation equations, and, among various numerical methods, for the discontinuous Galerkin method. Hyperbolic-relaxation equations can be generated as moments of the Boltzmann equation and can describe continuum-transition flows. Two challenging properties of hyperbolic-relaxation equations are the presence of a stiff source term, which drives the system towards equilibrium, and the accompanying change of eigenstructure. The first issue can be solved by an implicit treatment of the source term. To cope with the second difficulty, we develop a space-time discontinuous Galerkin method, based on Huynh’s “upwind moment scheme.” It is called the DG(1)–Hancock method. The DG(1)–Hancock method for one- and two-dimensional meshes is described, and Fourier analyses for both linear advection and linear hyperbolic-relaxation equations are conducted. The analyses show that the DG(1)–Hancock method is not only accurate but efficient in terms of turnaround time in comparison to other semiand fully discrete finite-volume and discontinuous Galerkin methods. Numerical tests confirm the analyses, and also show the properties are preserved for nonlinear equations; the efficiency is superior by an order of magnitude. Subsequently, discontinuous Galerkin and finite-volume spatial discretizations are applied to more practical equations, in particular, to the set of 10-moment equations, which are gas dynamics equations that include a full pressure/temperature tensor among the flow variables. Results for flow around a micro-airfoil are compared to experimental data and to solutions obtained with a Navier–Stokes code, and with particle-based methods. While numerical solutions in the continuum regime for both the 10-moment and Navier–Stokes equations are similar, clear differences are found in the continuum-transition regime, especially near the stagnation point, where the Navier–Stokes code, even when implemented with wall-slip, overestimates the density.Ph.D.Aerospace Engineering and Scientific ComputingUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58411/4/ysuzuki_1.pd

Continuous finite element subgrid basis functions for Discontinuous Galerkin schemes on unstructured polygonal Voronoi meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree N inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor. The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme

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