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

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

La méthode MOOD Multi-dimensional Optimal Order Detection : la première approche a posteriori aux méthodes volumes finis d'ordre très élevé

Nous introduisons et développons dans cette thèse un nouveau type de méthodes Volumes Finis d'ordre très élevé pour les systèmes hyperboliques de lois de conservations. Appelée MOOD pour Multidimensional Optimal Order Detection, elle permet de réaliser des simulations très précises en dimensions deux et trois sur maillages non-structurés. La conception d'une telle méthode est rendue délicate par l'apparition de singularités dans la solution (chocs, discontinuités de contact) pour lesquelles des phenomènes parasites (oscillations, création de valeurs non physiques...) sont générés par l'approximation d'ordre élevé. L'originalité de cette thèse réside dans le traitement de ces problèmes. A l'opposé des méthodes classiques qui essaient de contrôler ces phénomènes indésirables par une limitation a priori, nous proposons une approche de traitement a posteriori basée sur une décrémentation locale de l'ordre du schéma. Nous montrons en particulier que ce concept permet très simplement d'obtenir des propriétés qui sont habituellement difficiles à prouver dans le cadre multi-dimensionel non-structuré (préservation de la positité par exemple). La robustesse et la qualité de la méthode MOOD ont été prouvées sur de nombreux tests numériques en 2D et 3D. Une amélioration significative des ressources informatiques (CPU et stockage mémoire) nécessaires à l'obtention de résultats équivalents aux méthodes actuelles a été démontrée.We introduce and develop in this thesis a new type of very high-order Finite Volume methods for hyperbolic systems of conservation laws. This method, named MOOD for Multidimensional Optimal Order Detection, provides very accurate simulations for two- and three-dimensional unstructured meshes. The design of such a method is made delicate by the emergence of solution singularities (shocks, contact discontinuities) for which spurious phenomena (oscillations, non-physical values creation, etc.) are generated by the high-order approximation. The originality of this work lies in a new treatment for theses problems. Contrary to classical methods which try to control such undesirable phenomena through an a priori limitation, we propose an a posteriori treatment approach based on a local scheme order decrementing. In particular, we show that this concept easily provides properties that are usually difficult to prove in a multidimensional unstructured framework (positivity-preserving for instance). The robustness and quality of the MOOD method have been numerically proved through numerous test cases in 2D and 3D, and a significant reduction of computational resources (CPU and memory storage) needed to get state-of-the-art results has been shown

An overview on the multidimensional optimal order detection method

Finite volume method is the usual framework to deal with numerical approximations for hyperbolic systems such as Shallow-Water or Euler equations due to its natural built-in conservation property. Since the first-order method produces too much numerical diffusion, popular second-order techniques, based on the MUSCL methodology, have been widely developed in the ’80s to provide both accurate solutions and robust schemes, avoiding non-physical oscillations in the vicinity of the discontinuities. Although second-order schemes are accurate enough for the major industrial applications, they still generate too much numerical diffusion for particular situations (acoustic, aeronautic, long time simulation for Tsunami) and very high-order methods i.e. larger than third-order, are required to provide an excellent approximation for local smooth solution as well as an efficient control on the spurious oscillations deriving from the Gibbs’ phenomenon. During the ’90s and up to nowadays, two main techniques have been developed to tackle the accuracy issue. The ENO/WENO which can cast in the finite volume context mainly concerns structured grids since the unstructured case turns to be very complex with a huge computational cost. The Discontinuous Galerkin method handles very well accurate approximations but the computational cost and implementation effort are also very high. In 2010 was published a seminal paper that proposed a radically different method. The philosophy consists to use an a posteriori approach to prevent from creating oscillations whereas the traditional methods employ an a priori method which dramatically cuts the accuracy order. In this document, I shall briefly present the MOOD method, show its main advantages and give an overview of the current applications.This research was financed by FEDER Funds through Programa Operational Fatores de Competitividade — COMPETE and by Portuguese Funds FCT — Fundação para a Ciência e a Tecnologia, within the Projects PEst-C/MAT/UI0013/2014, PTDC/MAT/121185/2010 and FCT-ANR/MAT-NAN/0122/2012

Very high-order finite volume method for one-dimensional convection diffusion problems

We propose a new finite volume method to provide very high-order accuracy for the convection diffusion problem. The main tool is a polynomial reconstruction baesd on the mean values to provide the optimal order. Numerical examples are proposed to show the method efficiency

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

International audienceIn this paper, we investigate the coupling of the Multi-dimensional Optimal Order De- tection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decre- menting processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic par- tial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements

The mood method, multi-dimensional optimal older detection ( afirts a posteriori approach to very high-order finite volume methods)

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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