Conservativity and Weak Consistency of a Class of Staggered Finite Volume Methods for the Euler Equations

We address a class of schemes for the Euler equations with the following features: the space discretization is staggered, possible upwinding is performed with respect to the material velocity only and the internal energy balance is solved, with a correction term designed on consistency arguments. These schemes have been shown in previous works to preserve the convex of admissible states and have been extensively tested numerically. The aim of the present paper is twofold: we derive a local total energy equation satisfied by the solutions, so that the schemes are in fact conservative, and we prove that they are consistent in the Lax-Wendroff sense

An all-regime Lagrange-Projection like scheme for the gas dynamics equations on unstructured meshes

We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations. By all regime, we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization, i.e. a mesh size and time step much bigger than the Mach number M. The key idea is to decouple acoustic and transport phenomenon and then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error in term of M. This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy. A discrete entropy inequality under a condition on the modification is obtained thanks to a reinterpretation of the modified scheme in the Harten Lax and van Leer formalism. A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed. Then a simple and efficient semi implicit scheme is also proposed. The resulting scheme is stable under a CFL condition driven by the (slow) material waves and not by the (fast) acoustic waves and so verifies the all regime property. Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values

Consistent Internal Energy Based Schemes for the Compressible Euler Equations

La deuxieme partie de ce document reprend un travail déjà exposé dans le dépot hal-01553699Numerical schemes for the solution of the Euler equations have recently been developed, which involve the discretisation of the internal energy equation, with corrective terms to ensure the correct capture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. These schemes may be staggered or colocated, using either struc-tured meshes or general simplicial or tetrahedral/hexahedral meshes. The time discretization is performed by fractional-step algorithms; these may be either based on semi-implicit pressure correction techniques or segregated in such a way that only explicit steps are involved (referred to hereafter as "explicit" variants). In order to ensure the positivity of the density, the internal energy and the pressure, the discrete convection operators for the mass and internal energy balance equations are carefully designed; they use an upwind technique with respect to the material velocity only. The construction of the fluxes thus does not need any Rie-mann or approximate Riemann solver, and yields easily implementable algorithms. The stability is obtained without restriction on the time step for the pressure correction scheme and under a CFL-like condition for explicit variants: preservation of the integral of the total energy over the computational domain, and positivity of the density and the internal energy. The semi-implicit first-order upwind scheme satisfies a local discrete entropy inequality. If a MUSCL-like scheme is used in order to limit the scheme diffusion, then a weaker property holds: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L ∞ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned MUSCL-like scheme, the same result only holds if the ratio of the time to the space step tends to zero

A Standard Test Case Suite for Two-Dimensional Linear Transport on the Sphere: Results from a Collection of State-of-the-Art Schemes

Recently, a standard test case suite for 2-D linear transport on the sphere was proposed to assess important aspects of accuracy in geophysical fluid dynamics with a minimal set of idealized model configurations/runs/diagnostics. Here we present results from 19 state-of-the-art transport scheme formulations based on finite-difference/finite-volume methods as well as emerging (in the context of atmospheric/oceanographic sciences) Galerkin methods. Discretization grids range from traditional regular latitude–longitude grids to more isotropic domain discretizations such as icosahedral and cubed-sphere tessellations of the sphere. The schemes are evaluated using a wide range of diagnostics in idealized flow environments. Accuracy is assessed in single- and two-tracer configurations using conventional error norms as well as novel diagnostics designed for climate and climate–chemistry applications. In addition, algorithmic considerations that may be important for computational efficiency are reported on. The latter is inevitably computing platform dependent. The ensemble of results from a wide variety of schemes presented here helps shed light on the ability of the test case suite diagnostics and flow settings to discriminate between algorithms and provide insights into accuracy in the context of global atmospheric/ocean modeling. A library of benchmark results is provided to facilitate scheme intercomparison and model development. Simple software and data sets are made available to facilitate the process of model evaluation and scheme intercomparison

Simulation of Richtmyer-Meshkov Flows for Elastic-Plastic Solids in Planar and Converging Geometries Using an Eulerian Framework

This thesis presents a numerical and analytical study of two problems of interest involving shock waves propagating through elastic-plastic media: the motion of converging (imploding) shocks and the Richtmyer-Meshkov (RM) instability. Since the stress conditions encountered in these cases normally produce large deformations in the materials, an Eulerian description, in which the spatial coordinates are fixed, is employed. This formulation enables a direct comparison of similarities and differences between the present study of phenomena driven by shock-loading in elastic-plastic solids, and in fluids, where they have been studied extensively. In the first application, Whitham's shock dynamics (WSD) theory is employed for obtaining an approximate description of the motion of an elastic-plastic material processed by a cylindrically/spherically converging shock. Comparison with numerical simulations of the full set of equations of motion reveal that WSD is an accurate tool for characterizing the evolution of converging shocks at all stages. The study of the Richtmyer-Meshkov flow (i.e., interaction between the interface separating two materials of different density and a shock wave incoming at an angle) in solids is performed by means of analytical models for purely elastic solids and numerical simulations when plasticity is included in the material model. To this effect, an updated version of a previously developed multi-material, level-set-based, Eulerian framework for solid mechanics is employed. The revised code includes the use of a multi-material HLLD Riemann problem for imposing material boundary conditions, and a new formulation of the equations of motion that makes use of the stretch tensor while avoiding the degeneracy of the stress tensor under rotation. Results reveal that the interface separating two elastic solids always behaves in a stable oscillatory or decaying oscillatory manner due to the existence of shear waves, which are able to transport the initial vorticity away from the interface. In the case of elastic-plastic materials, the interface behaves at first in an unstable manner similar to a fluid. Ejecta formation is appreciated under certain initial conditions while in other conditions, after an initial period of growth, the interface displays a quasi-stationary long-term behavior due to stress relaxation. The effect of secondary shock-interface interactions (re-shocks) in converging geometries is also studied. A turbulent mixing zone, similar to what is observed in gas--gas interfaces, is created, especially when materials with low strength driven by moderate to strong shocks are considered

Probabilistic state estimation in regimes of nonlinear error growth

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences, 2005.Includes bibliographical references (p. 273-286).State estimation, or data assimilation as it is often called, is a key component of numerical weather prediction (NWP). Nearly all implementable methods of state estimation suitable for NWP are forced to assume that errors remain in regimes of linear error growth and retain distributions of Gaussian uncertainty, yet nonlinear systems like the atmosphere can readily allow regimes of nonlinear error growth and, in turn, produce distributions of non- Gaussian uncertainty. State-of-the-art, ensemble-based methods of state estimation suitable for NWP are examined to gauge the consequences and relevance of violating the linear error growth assumption. For quite generic sources of non-Gaussian uncertainty, the methods are observed to fail, as they must, and the obtained analyses become probabilistically unreliable before becoming inaccurate. The mispositioning of coherent features is identified as a specific, geophysically relevant source of non-Gaussian uncertainty that can easily cause the state-of-the-art methods of state estimation to fail. However, an understanding of relevant phenomenology sometimes allows these same methods to remain successful owing to an available redefinition of the involved errors. The redefinition is phrased as an alternative error model. It is recognized and exploited that non-Gaussian additive Eulerian errors can come from Gaussian Lagrangian position errors. A two-step, augmented state vector approach is developed that is suitable for use with coherent features and that relies only on implementable methods of state estimation.(cont.) By combining the dual Eulerian and Lagrangian state information into one vector, an ensemble can approximate their covariance, thus allowing each component's uncertainty to be reduced. The first step of the two-step approach reduces the feature position errors in an effort to render the residual additive errors Gaussian, thereby allowing the second step of an implementable state estimation method to proceed successfully. Philosophically, the two-step approach uses physical knowledge of the problem (as phrased by the error model) to compensate for neglected important non-Gaussian uncertainty structure in the state estimation process. The proposed two-step approach successfully allows use of implementable methods of state estimation to obtain probabilistically reliable analyses in regimes of nonlinear error growth, something unavailable using current standards.by W. Gregory Lawson.Ph.D

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal