A multidimensional HLL‐Riemann solver for non‐linear hyperbolic systems

This article presents a numerical model that enables to solve on unstructured triangular meshes and with a high order of accuracy, Riemann problems that appear when solving hyperbolic systems. For this purpose, we use a MUSCL‐like procedure in a ‘cell‐vertex’ finite‐volume framework. In the first part of this procedure, we devise a four‐state bi‐dimensional HLL solver (HLL‐2D). This solver is based upon the Riemann problem generated at the barycenter of a triangular cell, from the surrounding cell‐averages. A new three‐wave model makes it possible to solve this problem, approximately. A first‐order version of the bi‐dimensional Riemann solver is then generated for discretizing the full compressible Euler equations. In the second part of the MUSCL procedure, we develop a polynomial reconstruction that uses all the surrounding numerical data of a given point, to give at best third‐order accuracy. The resulting over determined system is solved by using a least‐square methodology. To enforce monotonicity conditions into the polynomial interpolation, we use and adapt the monotonicity‐preserving limiter, initially devised by Barth (AIAA Paper 90‐0013, 1990). Numerical tests and comparisons with competing numerical methods enable to identify the salient features of the whole model. Copyright © 2010 John Wiley & Sons, Ltd.     

A Riemann solver and upwind methods for a two‐phase flow model in non‐conservative form

We present a theoretical solution for the Riemann problem for the five‐equation two‐phase non‐conservative model of Saurel and Abgrall. This solution is then utilized in the construction of upwind non‐conservative methods to solve the general initial‐boundary value problem for the two‐phase flow model in non‐conservative form. The basic upwind scheme constructed is the non‐conservative analogue of the Godunov first‐order upwind method. Second‐order methods in space and time are then constructed via the MUSCL and ADER approaches. The methods are systematically assessed via a series of test problems with theoretical solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Fifth‐order Hermitian schemes for computational linear aeroacoustics

We develop a class of fifth‐order methods to solve linear acoustics and/or aeroacoustics. Based on local Hermite polynomials, we investigate three competing strategies for solving hyperbolic linear problems with a fifth‐order accuracy. A one‐dimensional (1D) analysis in the Fourier series makes it possible to classify these possibilities. Then, numerical computations based on the 1D scalar advection equation support two possibilities in order to update the discrete variable and its first and second derivatives: the first one uses a procedure similar to that of Cauchy–Kovaleskaya (the ‘Δ‐P5 scheme’); the second one relies on a semi‐discrete form and evolves in time the discrete unknowns by using a five‐stage Runge–Kutta method (the ‘RGK‐P5 scheme’). Although the RGK‐P5 scheme shares the same local spatial interpolator with the Δ‐P5 scheme, it is algebraically simpler. However, it is shown numerically that its loss of compactness reduces its domain of stability. Both schemes are then extended to bi‐dimensional acoustics and aeroacoustics. Following the methodology validated in (J. Comput. Phys. 2005; 210 :133–170; J. Comput. Phys. 2006; 217 :530–562), we build an algorithm in three stages in order to optimize the procedure of discretization. In the ‘reconstruction stage’, we define a fifth‐order local spatial interpolator based on an upwind stencil. In the ‘decomposition stage’, we decompose the time derivatives into simple wave contributions. In the ‘evolution stage’, we use these fluctuations to update either by a Cauchy–Kovaleskaya procedure or by a five‐stage Runge–Kutta algorithm, the discrete variable and its derivatives. In this way, depending on the configuration of the ‘evolution stage’, two fifth‐order upwind Hermitian schemes are constructed. The effectiveness and the exactitude of both schemes are checked by their applications to several 2D problems in acoustics and aeroacoustics. In this aim, we compare the computational cost and the computation memory requirement for each solution. The RGK‐P5 appears as the best compromise between simplicity and accuracy, while the Δ‐P5 scheme is more accurate and less CPU time consuming, despite a greater algebraic complexity. Copyright © 2007 John Wiley & Sons, Ltd.     

The Multidimensional Optimal Order Detection method in the three‐dimensional case: very high‐order finite volume method for hyperbolic systems

The Multidimensional Optimal Order Detection (MOOD) method for two‐dimensional geometries has been introduced by the authors in two recent papers. We present here 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, whereas 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 with existing high‐order Finite Volume methods.Copyright © 2013 John Wiley & Sons, Ltd.