Convergence and Sensitivity Analysis of Repair Algorithms in 1D

International audienceWe prove the convergence of some repair algorithms for linear advection in dimension one. The convergence depends on the size of the box where the distribution of the mass excess is performed. Various numerical examples illustrate the theoretical results. Applications to gas dynamics in dimension one is also discussed

Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity

The aim of this paper is to compare a hyperelastic with a hypoelastic model describing the Eulerian dynamics of solids in the context of non-linear elastoplastic deformations. Specifically, we consider the well-known hypoelastic Wilkins model, which is compared against a hyperelastic model based on the work of Godunov and Romenski. First, we discuss some general conceptual differences between the two approaches. Second, a detailed study of both models is proposed, where differences are made evident at the aid of deriving a hypoelastic-type model corresponding to the hyperelastic model and a particular equation of state used in this paper. Third, using the same high order ADER Finite Volume and Discontinuous Galerkin methods on fixed and moving unstructured meshes for both models, a wide range of numerical benchmark test problems has been solved. The numerical solutions obtained for the two different models are directly compared with each other. For small elastic deformations, the two models produce very similar solutions that are close to each other. However, if large elastic or elastoplastic deformations occur, the solutions present larger differences.Comment: 14 figure

Multi-dimensional Optimal Order Detection (MOOD) — A very high-order Finite Volume Scheme for conservation laws on unstructured meshes.

Preprint for Finite Volume for Complex Applications 6 (FVCA6)The Multi-dimensional Optimal Order Detection (MOOD) method is an original Very High-Order Finite Volume (FV) method for conservation laws on unstructured meshes. The method is based on an a posteriori degree reduction of local polynomial reconstructions on cells where prescribed stability conditions are not fulfilled. Numerical experiments on advection and Euler equations problems are drawn to prove the efficiency and competitiveness of the MOOD method

A high-order finite volume method for hyperbolic systems: Multi-dimensional Optimal Order Detection (MOOD).

International audienceIn this paper, we investigate an original way to deal with the problems generated by the limitation process of high-order finite volume methods based on polynomial reconstructions. Multi-dimensional Optimal Order Detection (MOOD) breaks away from classical limitations employed in high-order methods. The proposed method consists of detecting problematic situations after each time update of the solution and of reducing the local polynomial degree before recomputing the solution. As multi-dimensional MUSCL methods, the concept is simple and independent of mesh structure. Moreover MOOD is able to take physical constraints such as density and pressure positivity into account through an “a posteriori” detection. Numer- ical results on classical and demanding test cases for advection and Euler system are presented on quadrangular meshes to support the promising potential of this approach

The MOOD method in the three-dimensional case: Very-High-Order Finite Volume Method for Hyperbolic Systems.

The Multi-dimensional Optimal Order Detection (MOOD) method for two-dimensional geometries has been introduced in "A high-order finite volume method for hyperbolic systems: Multi-dimensional Optimal Order Detection (MOOD)", J. Comput. Phys. 230 (2011), and enhanced in "Improved Detection Criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials", Comput. & Fluids 64 (2012). We present in this paper the extension to 3D mixed meshes composed of tetrahedra, hexahedra, pyramids and prisms. In addition, we simplify the u2 detection process previously developed and show on a relevant set of numerical tests for both the convection equation and the Euler system that the optimal high-order of accuracy is reached on smooth solutions while spurious oscillations near singularities are prevented. At last, the intrinsic positivity-preserving property of the MOOD method is confirmed in 3D and we provide simple optimizations to reduce the computational cost such that the MOOD method is very competitive compared to existing high-order Finite Volume methods

Three-dimensional preliminary results of the MOOD method: A Very High-Order Finite Volume method for Conservation Laws.

The Multi-dimensional Optimal Order Detection (MOOD) method has been designed by authors in [5] and extended in [7] to reach Very-High-Order of accuracy for systems of Conservation Laws in a Finite Volume (FV) framework on 2D unstructured meshes. In this paper we focus on the extension of this method to 3D unstructured meshes. We present preliminary results for the three-dimensional advection equation which confirm the good behaviour of the MOOD method. More precisely, we show that the scheme yields up to sixth-order accuracy on smooth solutions while preventing oscillations from appearing on discontinuous profiles

STUDY OF A NEW ASYMPTOTIC PRESERVING SCHEME FOR THE EULER SYSTEM IN THE LOW MACH NUMBER LIMIT

International audienceThis article deals with the discretization of the compressible Euler system for all Mach numbers regimes. For highly subsonic flows, since acoustic waves are very fast compared to the velocity of the fluid, the gas can be considered as incompressible. From the numerical point of view, when the Mach number tends to zero, the classical Godunov type schemes present two main drawbacks: they lose consistency and they suffer of severe numerical constraints for stability to be guaranteed since the time step must follow the acoustic waves speed. In this work, we propose and analyze a new unconditionally stable an consistent scheme for all Mach number flows, from compressible to incompressible regimes, stability being only related to the flow speed. A stability analysis and several one and two dimensional simulations confirm that the proposed method possesses the sought characteristics. 1. Introduction. Almost all fluids can be said to be compressible. However, there are many situations in which the changes in density are so small to be considered negligible. We refer to these situations saying that the fluid is in an incompressible regime. From the mathematical point of view, the difference between compressible and incompressible situations is that, in the second case, the equation for the conservation of mass is replaced by the constraint that the divergence of the velocity should be zero. This is due to the fact that when the Mach number tends to zero, the pressure waves can be considered to travel at infinite speed. From the theoretical point of view, researchers try to fill the gap between those two different descriptions by determining in which sense compressible equations tend to incompressible ones [2, 20, 21, 22, 33]. In this article we are interested in the numerical solution of the Euler system when used to describe fluid flows where the Mach number strongly varies. This causes the gas to pass from compressible to almost incompressible situations and consequently it causes most of the numerical methods build for solving compressible Euler equations to fail. In fact, when the Mach number tends to zero, it is well known that classical Godunov type schemes do not work anymore. Indeed, they lose consistency in the incompressible limit. This means that when close to the limit, the accuracy of theses schemes is not sufficient to describe the flow. Many efforts have been done in the recent past in order to correct this main drawback of Godunov schemes, for instance by using preconditioning methods [34] or by splitting and correcting the pressure on the collocated meshes [5], [9, 10], [12], [13, 14, 30], [15], [23, 24], [26, 27, 28], or instead by using staggered grids like in the famous MAC scheme, see for instance [3], [16], [17], [18], [19], [31]. Unfortunately, even if these approaches permit to bypass the consistency problem of Godunov methods, they all need to resolve the scale of the acoustic waves in the fluid in order to remain stable. This means that they suffer from a restrictive CFL (Courant-Frierichs-Levy) condition which is inversely proportional to the Mach number value. In this work, we derive a method whic

A totally Eulerian Finite Volume solver for multi-material fluid flows: Enhanced Natural Interface Positioning (ENIP)

28 pagesThis work concerns the simulation of compressible multi-material fluid flows and follows the method FVCF-NIP described in the former paper Braeunig et al (Eur. J. Mech. B/Fluids, 2009). This Cell-centered Finite Volume method is totally Eulerian since the mesh is not moving and a sharp interface, separating two materials, evolves through the grid. A sliding boundary condition is enforced at the interface and mass, momentum and total energy are conserved. Although this former method performs well on 1D test cases, the interface reconstruction suffers of poor accuracy in conserving shapes for instance in linear advection. This situation leads to spurious instabilities of the interface. The method Enhanced-NIP presented in the present paper cures an inconsistency in the former NIP method that improves strikingly the results. It takes advantage of a more consistent description of the interface in the numerical scheme. Results for linear advection and compressible Euler equations for inviscid fluids are presented to assess the benefits of this new method