A HLL_nc (HLL nonconservative) method for the one-dimensional nonconservative Euler system

10 pagesAn adaptation of the original HLL scheme for the nonconservative Euler proble

Stencil and kernel optimisation for mesh-free very high-order generalised finite difference method

Generalised Finite Difference Methods and similar mesh-free methods (Point set method, Multipoint method) are based on three main ingredients: a stencil around the reference node, a polynomial reconstruction and a weighted functional to provide the relation sbetween the derivatives at the reference node and the nodes of the stencil.Very few studies were dedicated to the optimal choice of the stencil together with the other parameters that could reduce the global conditioning of the system and bring more stability and better accuracy. We propose a detailed construction of the very high-order polynomial representation and define a functional that assesses the quality of the reconstruction. We propose and implement several techniques of optimisation and demonstrate the advantages in terms of accuracy and stability.J. Figueiredo acknowledges the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 – Programa Operational Fatores de Competitividade through FCT – Fundação para a Ciência e a Tecnologia, project N° UID/FIS/04650/2019. S. Clain the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional , through COMPETE 2020 – Programa Operational Fatores de Competitividade through FCT – Fundação para a Ciência e a Tecnologia, project N° UIDB/00324/2020

First- and Second-order finite volume methods for the one-dimensional nonconservative Euler system

68 pagesGas flow in porous media with a nonconstant porosity function provides a nonconservative Euler system. We propose a new class of schemes for such a system for the one-dimensional situations based on nonconservative fluxes preserving the steady-state solutions. We derive a second-order scheme using a splitting of the porosity function into a discontinuous and a regular part where the regular part is treated as a source term while the discontinuous part is treated with the nonconservative fluxes. We then present a classification of all the configurations for the Riemann problem solutions. In particularly, we carefully study the resonant situations when two eigenvalues are superposed. Based on the classification, we deal with the inverse Riemann problem and present algorithms to compute the exact solutions. We finally propose new Sod problems to test the schemes for the resonant situations where numerical simulations are performed to compare with the theoretical solutions. The schemes accuracy (first- and second-order) is evaluated comparing with a nontrivial steady-state solution with the numerical approximation and convergence curves are established

The half-planes problem for the level set equation

International audienceThe paper is dedicated to the construction of an analytic solution for the level set equation in $\mathbb R^2$ with an initial condition constituted by two half-planes. Such a problem can be seen as an equivalent Riemann problem in the Hamilton-Jacobi equation context. We first rewrite the level set equation as a non-strictly hyperbolic problem and obtain a Riemann problem where the line sharing the initial discontinuity corresponds to the half-planes junction. Three different solutions corresponding to a shock, a rarefaction and a contact discontinuity are given in function of the two half-planes configuration and we derive the solution for the level set equation. The study provides theoretical examples to test the numerical methods approaching the solution of viscosity of the level set equation. We perform simulations to check the three situations using a classical numerical method on a structured grid

Numerical simulation of electrical problems in a vacuum disjuntor

A vacuum circuit breaker is a device that allows the cutting of electrical power. This device consists essentially of two electrodes, one of them being mobile and is subject to a mechanical force produced by a spring, giving rise to the contact between the two electrodes. The current passing between two electrodes is determined by the extension of the contact zone. Moreover, the passage of current generated Laplace forces in areas bordering the contact, but not yet in contact. Due to the curved geometry of the electrodes, these Laplace forces are opposite and therefore cause the repulsion of the electrodes. This means that for a given power we have to evaluate the electric potential, the magnetic field corresponding to the contact zone

Second-order finite volume mood method for the shallow water with dry/wet interface

The shallow water system is a fundamental work-piece for tsunami or flooding simulations. One of the major difficulties is the correct location of the dry/wet interface to evaluate accurate approximations of the velocity and kinetic energy. On the other hand, the MOOD method has been recently proposed to provide more efficient schemes in the framework of the Euler system. We propose to compare two second-order methods, namely the MUSCL and the MOOD techniques, and draw comparisons on accuracy shock capturing and dry/wet interface.This research was financed by FEDER Funds through Programa Operational Fatores de Competitividade — COMPETE and by Portuguese Funds FCT — Fundaçãoo 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

A very high-order finite volume method for the time-dependent convection-diffusion problem with Butcher tableau extension

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.Fundação para a Ciência e a Tecnologia (FCT

NUMERICAL SIMULATIONS OF COUPLED PROBLEMS IN A VACUUM DISJUNTOR

In this work we carried out the modelling, the discretization and the numerical simulation of a vacuum breaker. We present a mathematical model of the multi-physical problem involving mechanical, electrical and electromagnetic phenomena. The finite element method is employed in conjunction with a technique of domain decomposition to solve the mechanical problem 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

Analyse mathématique et numérique d'un modèle de chauffage par induction

In this work, we deal with a mathematical model of heat induction processes. We first build a model derived from Maxwell and heat equations and, using certain simplifying assumptions, we obtain a system of coupled partial differential equations describing the evolution of thermal and magnetic fields. We show that this evolutive problem has a solution in a weak sense. The study of the problem is carried out using numerical analysis techniques. A numerical scheme is build to be implemented on a computer in order to obtain numerical results. Finally, we present theoretical results for a steady-state problem and prove existence of a solution under assumptions weaker than in the evolutive case