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A new solver for the Stokes equations based on the finite volume method is proposed using very accurate polynomial reconstruction to provide a 6th-order scheme. We face two main difficulties: the gradient-divergence duality where the divergence free condition will impose the pressure gradient, and on the other hand, we assume that the domain has a regular curved boundary. The last point implies that a simple approximation of the boundary using piecewise segment lines dramatically reduces the scheme accuracy to at most a second-order one. We propose a new and simple technology which enables to restore the full scheme accuracy based on a specific polynomial reconstruction only using the Gauss points of the curved boundary and does not require any geometrical transformation., Fundação para a Ciência e a Tecnologia (FCT), 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

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)

We present a new class of very-high-order finite volume schemes for the time dependent convection-diffusion problem providing a sixth-order accuracy both in space and time. The method is based on local polynomial reconstructions dedicated to the convective or the diffusive part of the operator where the Dirichlet boundary condition is explicitly integrated in the polynomial reconstruction. The study focus on the time discretization problem where both explicit and implicit accurate time schemes are considered whereas no operator splitting is performed to easily preserve the sixth-order accuracy in time. Numerical tests are proposed to demonstrate the efficiency of the method.

The paper is dedicated to a new very high-order finite volume method based on polynomial reconstruction operators to solve the one-dimensional problem of beams subjected to a combination of tensile and bending forces. The model involves the classical linear elasticity model coupled with a shell model for the bending. The proceeding is ended with the application of very high-order scheme where an intramedullary nail is simulated to determine the stress critical point.

We present a new very high-order finite volume scheme for the two-dimensional convection-diffusion problem system based on the local Polynomial Reconstruction Operator (PRO-Scheme). We detail the design of the schemes and provide an example with regular solution. The solution is presented for Patankar scheme and for PRO scheme. In this last case we present the solution using a polynomial reconstruction of degree 1 and 2.

We present a fourth-order in space, second-order in time finite volume scheme for transient convection diffusion problem based on the Polynomial Reconstruction Operator and a Crank-Nicholson method. A detailed description of the scheme is provided and we perform numerical tests to highlight the performance of the method in comparison with the classical Patankar method., FEDER Funds through Programa Operacional Factores de Competitividade — COMPETE, Fundação para a Ciência e a Tecnologia (FCT) - Projeto PEst-C/MAT/UI0013/2011.

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 based on the mean-value to provide the best order. We give simple numerical examples that illustrate the effectiveness of the method in attaining the expected order of convergence., This research was financed by FEDER Funds through Programa Operacional Factores de Competitividade — COMPETE and by Portuguese Funds through FCT — Fundação para a Ciência e a Tecnologia, within the Project PEst- C/MAT/UI0013/2011.