Related papers: Computing with functions in spherical and polar ge…
We propose a patchwise local Fourier extension method for approximating smooth functions on general two dimensional domains with curved boundaries. The domain is embedded into a Cartesian background grid and decomposed into rectangular…
In geophysical fluid dynamics, the screened Poisson equation appears in the shallow-water, quasi geostrophic equations. Recently, many attempts have been made to solve those equations on the sphere using different numerical methods. These…
We describe the quantum sphere of Podle\'{s} for $c=0$ by means of a stereographic projection which is analogous to that which exhibits the classical sphere as a complex manifold. We show that the algebra of functions and the differential…
The solution of the Poisson equation is a ubiquitous problem in computational astrophysics. Most notably, the treatment of self-gravitating flows involves the Poisson equation for the gravitational field. In hydrodynamics codes using…
We use the Hardy spaces for Fourier integral operators to obtain bounds for spherical maximal functions in $L^{p}(\mathbb{R}^{n})$, $n\geq2$, where the radii of the spheres are restricted to a compact interval in $(0,\infty)$. These bounds…
We propose implicit integrators for solving stiff differential equations on unit spheres. Our approach extends the standard backward Euler and Crank-Nicolson methods in Cartesian space by incorporating the geometric constraint inherent to…
A new scheme is proposed to construct an n-times differentiable function extension of an n-times differentiable function defined on a smooth domain D in d-dimensions. The extension scheme relies on an explicit formula consisting of a linear…
We present an algorithm of finding numerical solutions of pulsar equation. The problem of finding the solutions was reduced to finding expansion coefficients of the source term of the equation in a base of orthogo- nal functions defined on…
Numerical integration on spheres, including the computation of the areas of spherical triangles, is a core computation in geomathematics. The commonly used techniques sometimes suffer from instabilities and significant loss of accuracy. We…
Spherical and polar geometries arise in many important areas of computational science, including weather and climate forecasting, optics, and astrophysics. In these applications, tensor-product grids are often used to represent unknowns.…
In this short note, we present an easy to implement and fast algorithm for the computation of the steady solitary gravity wave solution of the free surface Euler equations in irrotational motion. First, the problem is reformulated in a…
This paper considers filtered polynomial approximations on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$, obtained by truncating smoothly the Fourier series of an integrable function $f$ with the help of a "filter" $h$, which is a…
In this paper, we study the computation of optimal discrete Morse functions on stratifolds. In particular, we present an algorithm that efficiently computes such functions for a broad class of them. Moreover, we characterize the conditions…
We introduce an efficient algorithm, called partition of unity extension or PUX, to construct an extension of desired regularity of a function given on a complex multiply connected domain in $2D$. Function extension plays a fundamental role…
This paper presents a "two-dimensional Fourier Continuation" method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly…
In this paper we provide a reconstruction algorithm for piecewise-smooth functions with a-priori known smoothness and number of discontinuities, from their Fourier coefficients, posessing the maximal possible asymptotic rate of convergence…
We describe computational tools that have been developed to simulate dynamical mass transfer in semi-detached, polytropic binaries that are initially executing synchronous rotation upon circular orbits. Initial equilibrium models are…
Real-space refinement of atomic models in macromolecular crystallography or in cryo electron microscopy fits a model to a map obtained experimentally. This requires generating model maps of a limited resolution which moreover may vary from…
In this work, we study probability functions associated with Gaussian mixture models. Our primary focus is on extending the use of spherical radial decomposition for multivariate Gaussian random vectors to the context of Gaussian mixture…
Blending schemes based on circles provide smooth `fair' interpolations between series of points. Here we demonstrate a simple, robust set of algorithms for performing circle blends for a range of cases. An arbitrary level of G-continuity…