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To cite this version:Stéphane Clain, Raphaël Loubère, Gaspar Machado. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations. 2016. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equationsWe propose a new family of finite volume high-accurate numerical schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which control the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a chain detector to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers', and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations.

Accuracy may be dramatically reduced when the boundary domain is curved and numerical schemes require a specific treatment of the boundary condition to preserve the optimal order. In the finite volume context, Ollivier-Gooch and Van Altena (2002) has proposed a technique to overcome such limitation and restore the very high-order accuracy which consists in specific restrictions considered in the least-squares minimization associated to the polynomial reconstruction. The method suffers from several drawbacks, particularly, the use of curved elements that requires sophisticated meshing algorithms. We propose a new method where the physical domain and the computational domain are distinct and we introduce the Reconstruction for Off-site Data (ROD) where polynomial reconstructions are carried out on the mesh using data localized outside of the computational domain, namely the Dirichlet condition situated on the physical domain. A series of numerical tests assess the accuracy, convergence rates, robustness, and efficiency of the new method and show that the boundary condition is fully integrated in the scheme with a very high-order accuracy and the optimal convergence rate is achieved.

Summary Obtaining very high‐order accurate solutions in curved domains is a challenging task as the accuracy of discretization methods may dramatically reduce without an appropriate treatment of boundary conditions. The classical techniques to preserve the nominal convergence order of accuracy, proposed in the context of finite element and finite volume methods, rely on curved mesh elements, which fit curved boundaries. Such techniques often demand sophisticated meshing algorithms, cumbersome quadrature rules for integration, and complex nonlinear transformations to map the curved mesh elements onto the reference polygonal ones. In this regard, the reconstruction for off‐site data method, proposed in the work of Costa et al, provides very high‐order accurate polynomial reconstructions on arbitrary smooth curved boundaries, enabling integration of the governing equations on polygonal mesh elements, and therefore, avoiding the use of complex integration quadrature rules or nonlinear transformations. The method was introduced for Dirichlet boundary conditions and the present article proposes an extension for general boundary conditions, which represents an important advance for real context applications. A generic framework to compute polynomial reconstructions is also developed based on the least‐squares method, which handles general constraints and further improves the algorithm. The proposed methods are applied to solve the convection‐diffusion equation with a finite volume discretization in unstructured meshes. A comprehensive numerical benchmark test suite is provided to verify and assess the accuracy, convergence orders, robustness, and efficiency, which proves that boundary conditions on arbitrary smooth curved boundaries are properly fulfilled and the nominal very high‐order convergence orders are effectively achieved.

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a b s t r a c tThe design of efficient, simple, and easy to code, second-order finite volume methods is an important challenge to solve practical problems in physics and in engineering where complex and very accurate techniques are not required. We propose an extension of the original Frink's approach based on a cell-to-vertex interpolation to compute vertex values with neighbour cell values. We also design a specific scheme which enables to use whatever collocation point we want in the cells to overcome the mass centre point restrictive choice. The method is proposed for two-and three-dimensional geometries and a second-order extension time-discretization is given for time-dependent equation. A large number of numerical simulations are carried out to highlight the performance of the new method.

a b s t r a c tThe time discretization of a very high-order finite volume method may give rise to new numerical difficulties resulting into accuracy degradations. Indeed, for the simple one-dimensional unstationary convection-diffusion equation for instance, a conflicting situation between the source term time discretization and the boundary conditions may arise when using the standard Runge-Kutta method. We propose an alternative procedure by extending the Butcher Tableau to overcome this specific difficulty and achieve fourth-, sixth-or eighth-order of accuracy schemes in space and time. To this end, a new finite volume method is designed based on specific polynomial reconstructions for the space discretization, while we use the Extended Butcher Tableau to perform the time discretization. A large set of numerical tests has been carried out to validate the proposed method.

Thermoplastic extrusion processes use a dry calibration/cooling system, composed by one or several calibrators in series. One of the major difficulties in the modelling is to prescribe adequate values for the heat transfer coefficients between the polymer and the cooler or the air. We present an optimization procedure coupled with a finite volume method to evaluate such coefficients from experimental data. The tecnhique is based in a new conservative second-order finite volume scheme using a cell-to-vertex interpolation to solve the thermal problem involving a specific method to take the discontinuities (temperature, velocity, and thermal conductivity) into account. We have also performed a sensitive analysis of the different parameters of the problem to evaluate the variation of the thermal transfer coefficient with respect to parameters such as velocity or inflow temperature.