Finite Differences

High order finite volume approximations of differential operators on nonuniform grids

Sylvester [7] proposed the following question in 1893. If a finite set of points in a plane is such that on the line determined by any two points of the set there is always a third point of the set, is the set collinear? Equivalently, given a finite planar set of non-collinear points, does there exist a line containing exactly two of the points?

The notion of a centerpoint of a finite set of points in two and higher dimensions is a generalisation of the concept of the median of a (finite) set of points on the real line. In this paper, we present an algorithm for computing a centerpoint of a set of n points in the plane. The algorithm has complexity O(n) which significantly improves the O(n log3 n) complexity of the previously best known algorithm. We use suitable modifications of the ham-sandwich-cut algorithm and the prune-and-search technique to achieve this improvement.

Sixth order compact finite difference schemes for Poisson interface problems with singular sources

The spring-mass conditions are an efficient way to model imperfect contacts between elastic media. These conditions link together the limit values of the elastic stress and of the elastic displacement on both sides of interfaces. To insert these spring-mass conditions in classical finite-difference schemes, we use an interface method, the explicit simplified interface method (ESIM). This insertion is automatic for a wide class of schemes. The interfaces do not need to coincide with the uniform Cartesian grid. The local truncation error analysis and numerical experiments show that the ESIM maintains, with interfaces, properties of the schemes in homogeneous medium.

A finite-difference ghost-point multigrid method for multi-scale modelling of sorption kinetics of a surfactant past an oscillating bubble

We present a finite-difference numerical method to solve systems of hyperbolic conservation laws in domains with arbitrary shapes. The curved boundary is immersed in a uniform Cartesian grid and implicitly defined by a level-set function. The method is based on a semi-implicit discretization of the differential equations coupled with a ghost-point approach to impose the boundary conditions. The method is designed to be straightforwardly extended to higher order accuracy. The semi-implicit approach alleviates the stability restriction on the time step that is associated with acoustic waves in explicit methods, while preventing the numerical dissipation introduced in fully implicit methods. Several numerical tests to solve the Euler equations of gas dynamics past steady obstacles with arbitrary shapes are presented to show the efficiency of the semi-implicit method and the efficacy of the ghost-point approach.

The immersed interface method is modified to compute Schrödinger equation with discontinuous potential. By building the jump conditions of the solution into the finite difference approximation near the interface, this method can give at least second order convergence rate for the numerical solution on uniform cartesian grids. The accuracy of this algorithm is tested via several numerical examples.

We construct a sequence of compact embedded minimal disks in a ball in R 3 , whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the x 3 -axis. This extends a result of Colding and Minicozzi, who constructed a sequence for which the curvature blows up only at the center of the ball, and is a partial affirmative answer to the larger question of the existence of a sequence for which the curvature blows up precisely on a prescribed closed set on the x 3 -axis.

Computing the shape of a planar points set

High order marching schemes for the wave equation in complex geometry

In this article we survey recent results on the explicit construction of finite point sets and infinite sequences with optimal order of $\mathcal{L}_q$ discrepancy. In 1954 Roth proved a lower bound for the $\mathcal{L}_2$ discrepancy of finite point sets in the unit cube of arbitrary dimension. Later various authors extended Roth's result to lower bounds also for the $\mathcal{L}_q$ discrepancy and for infinite sequences. While it was known already from the early 1980s on that Roth's lower bound is best possible in the order of magnitude, it was a longstanding open question to find explicit constructions of point sets and sequences with optimal order of $\mathcal{L}_2$ discrepancy. This problem was solved by Chen and Skriganov in 2002 for finite point sets and recently by the authors of this article for infinite sequences. These constructions can also be extended to give optimal order of the $\mathcal{L}_q$ discrepancy of finite point sets for $q \in (1,\infty)$. The main aim of this article is to give an overview of these constructions and related results.

Given a finite set of points in the projective plane, we use the module of Kähler differentials to investigate the configurations of these points. More precisely, depending on the values of the Hilbert function of the module of Kähler differential 3-forms, we determine whether the set of points lies on a nonsingular conic, on two different lines, or on a single line.

This paper solves the problem of computing the best grasp in a discrete point set based on a popular grasp quality measure, namely the largest origin-centered ball contained in the grasp wrench set. So far, the solution to this problem is very limited. Noticing that the quality measure for a grasp is equal to the minimum value of the support function of its grasp wrench set over all directions and its computation together with force closure test can be fulfilled by evaluating the support function in a sequence of directions, we can quickly determine that a new grasp is worse whenever its support function in the sequence of directions or any other specific direction is less than the quality value of the current best grasp and avoid further computation. Furthermore, we enumerate candidate grasps in the point set in an adaptive way such that grasps that are more likely to outperform the current best grasp will be checked first, which helps find the best grasp earlier and significantly reduce the number of candidate grasps to be fully examined. With the aid of the adaptive enumeration and the quick comparison of grasps, the proposed algorithm takes tens of seconds to several hours on a normal PC to compute the best grasp in tens to hundreds of points on 3-D objects and it is two to several orders of magnitude faster than the brute-force search.

In this paper, we describe a machine learning (ML) approach for estimating interface orientation in multiphase flow using the volume of fluid (VOF) method on a uniform Cartesian mesh. By using complex shapes generated with the parametric radial star formula during training, we significantly improve prediction accuracy without increasing the network's structural complexity or processing cost. Our key contribution is the development of a robust ML model capable of reliably predicting interface orientation angles on uniform Cartesian grids. To enhance performance and robustness, we conducted a parametric study of the ML models' hyperparameters. The proposed method produced two models: a full augmented (9-cell) stencil and a compact (5-cell) stencil. Both were compared against popular finite difference/volume methods commonly used in VOF schemes. The results show that our approach is more accurate while remaining computationally efficient, particularly when employing a small stencil. Numerical studies, including challenging flow scenarios, demonstrate that the technique can predict interface orientation with an absolute mean error of less than 1 degree. Implemented in the OpenFOAM isoAdvector, our technique reliably produces accurate interface tracking with minimal deviation from the exact solution. These findings highlight the potential for incorporating machine learning approaches into classical numerical methods to improve the accuracy and reliability of the VOF method in a variety of challenging applications using uniform Cartesian meshes.