Related papers: Finite Difference Weights, Spectral Differentiatio…
Finite difference schemes, using Backward Differentiation Formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term, of the form $$\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v-…
In this paper, we consider finite difference approximations of the second order wave equation. We use finite difference operators satisfying the summation-by-parts property to discretize the equation in space. Boundary conditions and grid…
We demonstrate that the finite difference grid method (FDM) can be simply modified to satisfy the variational principle and enable calculations of both real and complex poles of the scattering matrix. These complex poles are known as…
We prove effective Nullstellensatz and elimination theorems for difference equations in sequence rings. More precisely, we compute an explicit function of geometric quantities associated to a system of difference equations (and these…
A common approach for minimizing a smooth nonlinear function is to employ finite-difference approximations to the gradient. While this can be easily performed when no error is present within the function evaluations, when the function is…
This paper introduces weighted finite difference methods for numerically solving dispersive evolution equations with solutions that are highly oscillatory in both space and time. We consider a semiclassically scaled cubic nonlinear…
In this work, we describe how to approximate solutions of some partial differential equations using the finite difference method defined on the Minkowski self-similar curve.
In this paper, a class of finite difference numerical techniques is presented to solve the second-order linear inhomogeneous damped wave equation. The consistency, stability, and convergences of these numerical schemes are discussed. The…
Averaging iterations of Stochastic Gradient Descent (SGD) have achieved empirical success in training deep learning models, such as Stochastic Weight Averaging (SWA), Exponential Moving Average (EMA), and LAtest Weight Averaging (LAWA).…
This paper presents a new finite difference algorithm for solving the 2D one-way wave equation with a preliminary approximation of a pseudo-differential operator by a system of partial differential equations. As opposed to the existing…
In this article, we present a simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations by optimally selecting the time step used to advance the numerical solution and…
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…
We study the numerical differentiation formulae for functions given in grids with arbitrary number of nodes. We investigate the case of the infinite number of points in the formulae for the calculation of the first and the second…
We develop summation by parts (SBP) approach for generating high-order finite-difference schemes on the interval and propose new sets of schemes up to the 12th order. The coefficients of the schemes are governed by values of grid spacing…
Modelling radar wave propagation in frequency domain is appealing in full waveform inversion because it allows decreasing the non-linearity of the problem, decreasing the dimension of the data space, better description of attenuation, and…
Calculating dynamical diffraction patterns for X-ray topography and similar x-ray scattering-imaging techniques require the numerical integration of the Takagi-Taupin equations. This is usually performed with a simple second order finite…
In this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified N\'ed\'elec-Raviart-Thomas finite element methods for model problems in $\mathbf{H}(\operatorname{\mathbf{curl}})$ and…
In the given paper we consider finite difference approximations to systems of polynomially-nonlinear partial differential equations whose coefficients are rational functions over rationals in the independent variables. The notion of strong…
In this paper, Cimmino's classical reflection algorithm for solving the $n\times n$ nonsingular linear system $A\bx=\bb$ is analysed through the lens of spectral theory. Reformulating the weighted iteration as…
Solving the three-dimensional (3D) Bratu equation is highly challenging due to the presence of multiple and sharp solutions. Research on this equation began in the late 1990s, but there are no satisfactory results to date. To address this…