Fixed-distance multipoint formulas for the scattering amplitude from phaseless measurements

We give new formulas for finding the complex (phased) scattering amplitude at fixed frequency and angles from absolute values of the scattering wave function at several points $x_1,..., x_m$. In dimension $d\geq 2$, for $m>2$, we significantly improve previous results in the following two respects. First, geometrical constraints on the points needed in previous results are significantly simplified. Essentially, the measurement points $x_j$ are assumed to be on a ray from the origin with fixed distance $\tau=|x_{j+1}- x_j|$, and high order convergence (linearly related to $m$) is achieved as the points move to infinity with fixed $\tau$. Second, our new asymptotic reconstruction formulas are significantly simpler than previous ones. In particular, we continue studies going back to [Novikov, Bull. Sci. Math. 139(8), 923-936, 2015].

Variable Step Size Adams Methods for BSDEs

For backward stochastic differential equations (BSDEs), we construct variable step size Adams methods by means of Itô–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to Y and Z reach high order. The coefficients in the numerical methods are inferred and bounded under appropriate conditions. A necessary and sufficient condition is given to judge the stability of our numerical schemes. Moreover, the high-order convergence of the schemes is rigorously proved. The numerical illustrations are provided.

Five-point thirty-two optimal order iterative method for solving non-linear equations

A five-point thirty-two convergence order derivative-free iterative method to find simple roots of non-linear equations is constructed. Six function evaluations are performed to achieve optimal convergence order 26-1 = 32 conjectured by Kung and Traub [1]. Secant approximation to the derivative is computed around the initial guess. High order convergence is attained by constructing polynomials of quotients for functional values.

Discontinuities in numerical radiative transfer

Observations and magnetohydrodynamic simulations of solar and stellar atmospheres reveal an intermittent behavior or steep gradients in physical parameters, such as magnetic field, temperature, and bulk velocities. The numerical solution of the stationary radiative transfer equation is particularly challenging in such situations, because standard numerical methods may perform very inefficiently in the absence of local smoothness. However, a rigorous investigation of the numerical treatment of the radiative transfer equation in discontinuous media is still lacking. The aim of this work is to expose the limitations of standard convergence analyses for this problem and to identify the relevant issues. Moreover, specific numerical tests are performed. These show that discontinuities in the atmospheric physical parameters effectively induce first-order discontinuities in the radiative transfer equation, reducing the accuracy of the solution and thwarting high-order convergence. In addition, a survey of the existing numerical schemes for discontinuous ordinary differential systems and interpolation techniques for discontinuous discrete data is given, evaluating their applicability to the radiative transfer problem.

Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the P1 Model

This work deals with the first Trefftz Discontinuous Galerkin (TDG) scheme for a model problem of transport with relaxation. The model problem is written as a {P_{N}} or {S_{N}} model, and we study in more details the {P_{1}} model in dimension 1 and 2. We show that the TDG method provides natural well-balanced and asymptotic preserving discretization since exact solutions are used locally in the basis functions. High-order convergence with respect to the mesh size in two dimensions is proved together with the asymptotic property for {P_{1}} model in dimension one. Numerical results in dimensions 1 and 2 illustrate the theoretical properties.