We propose a new class of fourth-and sixth-order schemes in time for parabolic and hyperbolic equations. The method follows the compact scheme methodology by elaborating implicit relations between the approximations of the function and its derivatives. We produce a series of A-stable methods with low dispersion and high accuracy. Several benchmarks for linear and non-linear Ordinary Differential Equations demonstrate the effectiveness of the method. Then a second set of numerical benchmarks for Partial Differential Equations such as convection-diffusion, Schrödinger equation, wave equation, Bürgers, and Euler system give the numerical evidences of the superior advantage of the method with respect to the traditional Runge-Kutta or multistep methods.

We present an adaptation of the MOOD method, initially introduced in [1,2], for the two-dimensional shallow-water system with varying bathymetry, where the major novelty of the study is the non- conservative term discretization in the framework of the MOOD strategy. We derive a robust sixth-order well-balanced scheme and propose a large panel of numerical tests to assess the accuracy of the method and show that numerical solutions are free of oscillations in the vicinity of discontinuities. We also demonstrate that the MOOD method guarantees the height positivity as long as the first-order scheme does.

This paper extends the MOOD method proposed by the authors in [A high-order finite volume method for hyperbolic systems: Multi-dimensional 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 efectively high-order (up to sixth-order), intrinsincally positivity-preserving on hydrodynamics test cases and computationaly efficient.

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.

We present a new formalism to characterize high-order reconstruction algorithms used in finite volume methods. This formalism provides new tools to examine the properties of these methods. Included in this formalism is the notion of admissible reconstruction methods providing concrete statements regarding the satisfaction of the maximum principle and positivity preservation properties. We demonstrate that the traditional reconstruction limiting algorithms can be recast in our formalism, thus providing new proofs of the maximum principle.

A sixth-order finite volume method is proposed to solve the bidimensional linear steady- state convection–diffusion equation. A new class of polynomial reconstructions is proposed to provide accurate fluxes for the convective and the diffusive operators. The method is also designed to compute accurate approximations even with discontinuous diffusion coeffi- cient or velocity and remains robust for large Peclet numbers. Discontinuous solutions deriving from the linear heat transfer Newton law are also considered where a decompo- sition domain technique is applied to maintain an effective sixth-order approximation. Numerical tests covering a large panel of situations are addressed to assess the perfor- mances of the method.

An arbitrary order finite difference method for solving non-linear convection Diffusion Reaction equations in curved boundary domains with Cartesian grid is proposed. Ghost points' values are determined with the Reconstruction Off-Site Data based on a polynomial interpolation using the least square method with constraints to enforce the boundary conditions. We propose a second-, fourth-, and sixth-order schemes for linear non-constant coefficients problem in both the conservative and non-conservative scalar equations. Extensions to non-linear scalar problems and systems are then implemented while preserving the optimal orders. Numerical simulations are carried out to provide evidence about the convergence order and the stability of the method.

The 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.

Breast reduction is one of the most common procedures in breast surgery. The aim of this work is to develop a computational model allowing one to forecast the final breast geometry according to the incision marking parameters. This model can be used in surgery simulators that provide preoperative planning and training, allowing the study of the errors origin in breast reduction.

In this work we carries out the modelling, the discretization and the numerical simulation of a vacuum breaker. We present a mathematical mpdel of the multi.physical problem incolving mechanical, electrical and electromagnetic phenomena. The finite element method is employed in conjuntion with a technique of domaine decomposition to solve the mechanical probleme and the electrical problem while a direct integration of the Biot-Savart formula allows to compute an approximation of the magnetic field, hence the Lorentz-Laplace forces.