6th-order finite volume approximation for the steady-state burger and euler equations: the mood approach

We propose an innovative method based on the MOOD technology (Multi-dimensional Optimal Order Detection) to provide a 6th-order finite volume approximation for the one-dimensional steady-state Burger and Euler equations. The main ingredient consists in using an 'a posteriori' limiting strategy to eliminate non physical oscillations deriving from the Gibbs phenomenon while keeping a high accuracy for the smooth part. A short overview of the MOOD method will be presented and numerical tests with regular or discontinuous solutions will assess the method capacity to produce excellent approximations. In the latter situation, the numerical results enable to detect the zone where it is necessary to reduce the degree of the polynomial reconstructions to preserve the scheme robustness.Fundação para a Ciência e a Tecnologia (FCT

Improved Detection Criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials

This paper extends the MOOD method proposed by the authors in ["A high-order finite volume method for hyperbolic systems: Multidimensional Optimal Order Detection (MOOD)", J. Comput. Phys. 230, pp 4028-4050, (2011)], along two complementary axes: extension to very high-order polynomial reconstruction on non-conformal unstructured meshes and new Detection Criteria. The former is a natural extension of the previous cited work which confirms the good behavior of the MOOD method. The latter is a necessary brick to overcome limitations of the Discrete Maximum Principle used in the previous work. Numerical results on advection problems and hydrodynamics Euler equations are presented to show that the MOOD method is effectively high-order (up to sixth-order), intrinsically positivity-preserving on hydrodynamics test cases and computationally efficient

A Very High-Order Accurate Staggered Finite Volume Scheme for the Stationary Incompressible Navier–Stokes and Euler Equations on Unstructured Meshes

International audienceWe propose a sixth-order staggered finite volume scheme based on polynomial reconstructions to achieve high accurate numerical solutions for the incompressible Navier-Stokes and Euler equations. The scheme is equipped with a fixed-point algorithm with solution relaxation to speed-up the convergence and reduce the computation time. Numerical tests are provided to assess the effectiveness of the method to achieve up to sixth-order con-2 Ricardo Costa et al. vergence rates. Simulations for the benchmark lid-driven cavity problem are also provided to highlight the benefit of the proposed high-order scheme

A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes

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Enhancement of Lagrangian slide lines as a combined force and velocity boundary condition

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A multi-scale residual-based anti-hourglass control for compatible staggered Lagrangian hydrodynamics

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Limiter-free discontinuity-capturing scheme for compressible gas dynamics withreactive fronts

International audienceThis work proposes a new spatial reconstruction scheme in finite volume frameworks. Different from long-lastingreconstruction processes which employ high order polynomials enforced with some carefully designed limiting pro-jections to seek stable solutions around discontinuities, the current discretized scheme employs THINC (Tangentof Hyperbola for INterface Capturing) functions with adaptive sharpness to solve both smooth and discontinuoussolutions. Due to the essentially monotone and bounded properties of THINC function, difficulties to solve sharpdiscontinuous solutions and complexities associated with designing limiting projections can be prevented. A newsimplified BVD (Boundary Variations Diminishing) algorithm, so-called adaptive THINC-BVD, is devised to reducenumerical dissipations through minimizing the total boundary variations for each cell. Verified through numericaltests, the present method is able to capture both smooth and discontinuous solutions in Euler equations for com-pressible gas dynamics with excellent solution quality competitive to other existing schemes. More profoundly, itprovides an accurate and reliable solver for a class of reactive compressible gas flows with stiff source terms, such asthe gaseous detonation waves, which are quite challenging to other high-resolution schemes. The stiff C-J detonationbenchmark test reveals that the adaptive THINC-BVD scheme can accurately capture the reacting front of the gaseousdetonation, while the WENO scheme with the same grid resolution generates unacceptable results. Owing also to itsalgorithmic simplicity, the proposed method can become as a practical and promising numerical solver for compress-ible gas dynamics, particularly for simulations involving strong discontinuities and reacting fronts with stiff sourceterm

Volume consistency in a staggered grid Lagrangian hydrodynamics scheme

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