Related papers: Spatial discretization of partial differential equ…

Nowadays, fractional differential equations are a well established tool to model phenomena from the real world. Since the analytical solution is rarely available, there is a great effort in constructing efficient numerical methods for their…

In this paper, we explore the finite difference approximation of the fractional Laplace operator in conjunction with a neural network method for solving it. We discretized the fractional Laplace operator using the Riemann-Liouville formula…

In this paper, we study temporal splitting algorithms for multiscale problems. The exact fine-grid spatial problems typically require some reduction in degrees of freedom. Multiscale algorithms are designed to represent the fine-scale…

This paper shows how to build a formal analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the…

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the…

Mathematical descriptions of flow phenomena usually come in the form of partial differential equations. The differential operators used in these equations may have properties such as symmetry, skew-symmetry, positive or negative…

We present a finite element scheme for fractional diffusion problems with varying diffusivity and fractional order. We consider a symmetric integral form of these nonlocal equations defined on general geometries and in arbitrary bounded…

We consider discrete analogue of model pseudo-differential equations in discrete plane sector using discrete variant of Sobolev--Slobodetskii spaces. Starting from the concept of wave factorization for elliptic periodic symbol we describe…

In this paper, we propose a decomposition approach for eigenvalue problems with spatial symmetries, including the formulation, discretization as well as implementation. This approach can handle eigenvalue problems with either Abelian or…

We develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner. The use of a covariant basis decomposition leads to a rotationally invariant scheme that…

Eigenvalue problems for semidefinite operators with infinite dimensional kernels appear for instance in electromagnetics. Variational discretizations with edge elements have long been analyzed in terms of a discrete compactness property. As…

Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be non-robust in the presence of Neumann boundary conditions. In this paper we overcome this issue by formulating the RBF-generated finite…

In the finite difference approximation of the fractional Laplacian the stiffness matrix is typically dense and needs to be approximated numerically. The effect of the accuracy in approximating the stiffness matrix on the accuracy in the…

The variational grid generation method is a powerful tool for generating structured convex grids on irregular simply connected domains whose boundary is a polygonal Jordan curve. Several examples that show the accuracy of a difference…

Imposition methods of interface conditions for the second-order wave equation with non-conforming grids is considered. The spatial discretization is based on high order finite differences with summation-by-parts properties. Previously…

I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of…

In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby modified equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the…

This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess 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…

By employing non-equispaced grid points near boundaries, boundary-optimized upwind finite-difference operators of orders up to nine are developed. The boundary closures are constructed within a diagonal-norm summation-by-parts (SBP)…