Related papers: Instabilities of explicit finite difference scheme…
The unstable nature of the material point method is widely documented and is a barrier to the method being used for routine engineering analyses of large deformation problems. The vast majority of papers concerning this issue are focused on…
In this paper a fourth order finite difference ghost point method for the Poisson equation on regular Cartesian mesh is presented. The method can be considered the high order extension of the second ghost method introduced earlier by the…
We introduce an unfitted Nitsche finite element method with a new ghost-penalty stabilization based on local projection of the solution gradient. The proposed ghost-penalty operator is straightforward to implement, ensures algebraic…
This manuscript contains some thoughts on the discretization of the classical heat equation. Namely, we discuss the advantages and disadvantages of explicit and implicit schemes. Then, we show how to overcome some disadvantages while…
Unfitted boundary methods are widely used to numerically solve partial differential equations (PDEs) on irregular domains, avoiding the computational burden of generating boundary-conforming grids. In the finite-difference framework,…
Recent work has shown that non-local modifications of the Einstein equations can have interesting cosmological consequences and can provide a dynamical origin for dark energy, consistent with existing data. At first sight these theories are…
Taking place naturally in a gas subject to a given wall temperature distribution [Maxwell1879], the ``ghost effect'' exhibits a rare kinetic effect beyond the prediction of classical fluid theory and Fourier law in such a classical problem…
The paper introduces a finite element method for an Eulerian formulation of partial differential equations governing the transport and diffusion of a scalar quantity in a time-dependent domain. The method follows the idea from Lehrenfeld &…
We develop a stable finite difference method for the elastic wave equation in bounded media, where the material properties can be discontinuous at curved interfaces. The governing equation is discretized in second order form by a fourth or…
We present a finite-difference scheme which solves the Stokes problem in the presence of curvilinear non-conforming interfaces and provides second-order accuracy on physical field (velocity, vorticity) and especially on pressure. The gist…
The Black-Scholes option pricing model remains a cornerstone in financial mathematics, yet its application is often challenged by the need for accurate hedging strategies, especially in dynamic market environments. This paper presents a…
Partial differential equations sometimes have critical points where the solution or some of its derivatives are discontinuous. The simplest example is a discontinuity in the initial condition. It is well known that those decrease the…
Close to a saddle-node bifurcation, when two invariant solutions collide and disappear, the behavior of a dynamical system can closely resemble that of a solution which is no longer present at the chosen parameter value. For bifurcating…
In this letter, we discuss how thermal fluctuations can effect the stability of (generally) charged black holes when close to certain critical points. Our novel treatment utilizes the black hole area spectrum (which is, for definiteness,…
In this paper, a weak formulation of the discontinuous variable coefficient Poisson equation with interfacial jumps is studied. The existence, uniqueness and regularity of solutions of this problem are obtained. It is shown that the…
We introduce a novel class of finite difference approximations, termed zigzag schemes, that employ a hybrid stencil that is neither symmetrical, nor fully one-sided. These zigzag schemes often enjoy more permissive stability constraints and…
In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second order accuracy for the first derivative is obtained as well. The method is…
The aim of these notes is to provide a self-contained review of why it is generically a problem when a solution of a theory possesses ghost fields among the perturbation modes. We define what a ghost field is and we show that its presence…
We investigate the stability against inhomogeneous perturbations and the appearance of ghost modes in Gauss-Bonnet gravitational theories with a non-minimally coupled scalar field, which can be regarded as either the dilaton or a…
Finite-difference simulations of fluid dynamics and magnetohydrodynamics generally require an explicit diffusion operator, either to maintain stability by attenuating grid-scale structure, or to implement physical diffusivities such as…