High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case

International audienceOne of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in [89, 90, 22], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in [85], and proves to fit a wide range of numerical schemes in the literature, such as those presented in [19, 64, 15, 82, 84]

Toward a Simple, Accurate Lagrangian Hydrocode.

Lagrangian hydrocodes play an important role in the computation of transient, compressible, multi-material flows. This research was aimed at developing a simply constructed cell-centered Lagrangian method for the Euler equations that respects multidimensional physics while achieving second-order accuracy. Algorithms that can account for the multidimensional physics associated with acoustic wave propagation and vorticity transport are needed in order to increase accuracy and prevent mesh imprinting. Many of the building blocks of traditional finite volume schemes, such as Riemann solvers and spatial gradient limiters, have their foundations in one-dimensional ideas and so were not used here. Instead, multidimensional point estimates of the fluxes were computed with a Lax-Wendroff type procedure and then nonlinearly modified using a temporal flux limiting mechanism. The linear acoustic equations were used as a simplified test environment for the Lagrangian Euler system. Here Lax-Wendroff methods that exactly preserve vorticity were investigated and found to resist mesh imprinting. However, the dispersion properties of the schemes were poor and so third-order accurate vorticity preserving methods were developed to remedy the problem. The third-order methods guided the construction of a temporal limiting mechanism, which was then used in a vorticity preserving flux-corrected transport scheme. While the acoustic work was interesting in its own right, it also proved to be a useful stepping stone to Lagrangian hydrodynamics. The acoustics algorithms were extended to produce the Simple Lagrangian Method (SLaM). Standard test problems have shown that a first-order accurate version of the method is able to resist mesh imprinting and spurious vorticity despite its minimalistic structure. SLaM is capable of second-order accuracy with a simple parameter change and some preliminary work was done to extend the temporal flux limiting ideas from acoustics to the Lagrangian case. The limited SLaM method converges at second-order for smooth data and is able to capture shocks without producing large unphysical oscillations.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113577/1/tblung_1.pd

Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes. High order piecewise polynomials are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Our numerical method belongs to the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method, in which the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed with a second order TVD finite volume scheme. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure

Schémas numériques mimétiques et conservatifs pour la simulation d'écoulements multiphasiques compressibles

In some highly demanding fluid dynamics simulations, it appears necessary tosimulate multiphase flows involving numerous constraints at the same time : large numbers of fluids, both isentropic and strongly shocked compressible evolution, highly variable and contrasted equations of state, large deformations, and transport over large distances. Fulfilling such a challengein a robust and tractable way demands that thermodynamic consistency of the numerical scheme be carefully ensured.In the first chapter, a Lagrange plus remap scheme is proposed for the simulation of two-phase flows with a dissipation-free six-equation bakcbone model. The importance of the property of isentropic flow preservation is highlighted with a comparison with Ransom test results fromthe literature. This chapter also also point out certain limitations of the Lagrange plus remap approach for multiphase simulations.In order to overcome these limitations, a novel derivation procedure is proposed to construct a mimetic scheme for the simulation of unsteady and compressible flows in a direct ALE (ArbitraryLagrangian-Eulerian) formalism. The possibility to choose a priori the degrees of freedom allows to obtain a continuity with historical staggered scheme, while imposing conservativity at discretelevel. The discrete momentum evolution equation is obtained by application of a variational principle, thus natively ensuring the thermodynamic consistency of pressure efforts. This approach is applied to single-fluid flows as a proof of concept in Chapter 3, then it is extended to N-phasecompressible flows in Chapter 4. Single- and multi-phase tests show satisfactory behavior in terms on conservation, versatility to grid motions, and robustness.Dans certaines simulations numériques exigeantes de mécanique des fluides, ilest nécessaire de simuler des écoulements multiphasiques impliquant de nombreuses contraintes simultanées : nombre de fluides important, évolutions compressibles à la fois isentropes et fortement choquées, équations d’états variables et contrastées, déformations importantes et transport surdes longues distances. Afin de remplir ces objectifs de manière robuste, il est nécessaire que la cohérence thermodynamique du schéma numérique soit vérifiée.Dans le premier chapitre, un schéma de type Lagrange plus projection est proposé pour la simulation d’écoulements diphasiques avec un modèle squelette à six équations et sans termes de dissipation. L’importance de la propriété de préservation des écoulements isentropiques est mise en évidence à l’aide d’une comparaison avec des résultats issus de la littérature pour le test deRansom. Ce chapitre souligne aussi certaines limitations de l’approche Lagrange plus projection pour simuler des modèles multiphasiques.Afin de pallier à ces limitations, une nouvelle procédure de dérivation est proposée afin de construire un schéma mimétique pour la simulation d’écoulements instationnaires compressibles dans un formalisme ALE direct (Arbitrary Lagrangian–Eulerian). La possibilité de choisir a prioriles degrés de liberté permet de s’inscrire dans une continuité avec les schémas historiques décalés, tout en imposant les conservations au niveau discret. L’équation de quantité de mouvement discrèteest obtenue par application d’un principe variationnel, assurant par construction la cohérence thermodynamique des efforts de pression. Cette approche est appliquée au cas d’écoulements monofluides comme preuve de concept au Chapitre 3, puis elle est étendue au cas d’écoulements à Nphasescompressibles au Chapitre 4. Des tests mono et multiphasiques montrent un comportement satisfaisant en terme de conservativité, versatilité aux mouvements de grilles et robustesse

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving 2D Voronoi meshes that are regenerated at each time step and which explicitly allow topology changes in time. The Voronoi tessellations are obtained from a set of generator points that move with the local fluid velocity. We employ an AREPO-type approach, which rapidly rebuilds a new high quality mesh rearranging the element shapes and neighbors in order to guarantee a robust mesh evolution even for vortex flows and very long simulation times. The old and new Voronoi elements associated to the same generator are connected to construct closed space--time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also incorporate degenerate space--time sliver elements, needed to fill the space--time holes that arise because of topology changes. The final ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and space--time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed space--time control volumes combined with a fully-discrete space--time conservation formulation of the governing PDE system. In this way the discrete solution is conservative and satisfies the GCL by construction. Numerical convergence studies as well as a large set of benchmarks for hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and robustness of the proposed method. Our numerical results clearly show that the new combination of very high order schemes with regenerated meshes with topology changes lead to substantial improvements compared to direct ALE methods on conforming meshes

High-order variational Lagrangian schemes for compressible fluids

We present high-order variational Lagrangian finite element methods for compressible fluids using a discrete energetic variational approach. Our spatial discretization is mass/momentum/energy conserving and entropy stable. Fully implicit time stepping is used for the temporal discretization, which allows for a much larger time step size for stability compared to explicit methods, especially for low-Mach number flows and/or on highly distorted meshes. Ample numerical results are presented to showcase the good performance of our proposed scheme.Comment: 24 page

Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers

In this paper we use the genuinely multidimensional HLL Riemann solvers recently developed by Balsara et al. to construct a new class of computationally efficient high order Lagrangian ADER-WENO one-step ALE finite volume schemes on unstructured triangular meshes. A nonlinear WENO reconstruction operator allows the algorithm to achieve high order of accuracy in space, while high order of accuracy in time is obtained by the use of an ADER time-stepping technique based on a local space-time Galerkin predictor. The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the grid, considering the entire Voronoi neighborhood of each node and allows for larger time steps than conventional one-dimensional Riemann solvers. The results produced by the multidimensional Riemann solver are then used twice in our one-step ALE algorithm: first, as a node solver that assigns a unique velocity vector to each vertex, in order to preserve the continuity of the computational mesh; second, as a building block for genuinely multidimensional numerical flux evaluation that allows the scheme to run with larger time steps compared to conventional finite volume schemes that use classical one-dimensional Riemann solvers in normal direction. A rezoning step may be necessary in order to overcome element overlapping or crossing-over. We apply the method presented in this article to two systems of hyperbolic conservation laws, namely the Euler equations of compressible gas dynamics and the equations of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to fourth order of accuracy in space and time have been carried out. Several numerical test problems have been solved to validate the new approach