A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on ℝ, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed

An improved Gregory-like method for 1-D quadrature

The quadrature formulas described by James Gregory (1638--1675) improve the accuracy of the trapezoidal rule by adjusting the weights near the ends of the integration interval. In contrast to the Newton--Cotes formulas, their weights are constant across the main part of the interval. However, for both of these approaches, the polynomial Runge phenomenon limits the orders of accuracy that are practical. For the algorithm presented here, this limitation is greatly reduced. In particular, quadrature formulas on equispaced 1-D node sets can be of high order (tested here up through order 20) without featuring any negative weights

Atmospheric Propagation of High Energy Lasers: Thermal Blooming Simulation

High Energy Laser (HEL) propagation through turbulent atmosphere is examined via numerical simulation. The beam propagation is modeled with the paraxial equation, which in turn is written as a system of equations for a quantum fluid, via the Madelung transform. A finite volume solver is applied to the quantum fluid equations, which supports sharp gradients in beam intensity. The atmosphere is modeled via a coupled advection-diffusion equation whose initial data have Kolmogorov spectrum. In this model the combined effects of thermal blooming, beam slewing, and deep turbulence are simulated

Implementing Conditional Inequality Constraints for Optimal Collision Avoidance

Current Federal Aviation Administration regulations require that passing aircraft must either meet a specified horizontal or vertical separation distance. However, solving for optimal avoidance trajectories with conditional inequality path constraints is problematic for gradient-based numerical nonlinear programming solvers since conditional constraints typically possess non-differentiable points. Further, the literature is silent on robust treatment of approximation methods to implement conditional inequality path constraints for gradient-based numerical nonlinear programming solvers. This paper proposes two efficient methods to enforce conditional inequality path constraints in the optimal control problem formulation and compares and contrasts these approaches on representative airborne avoidance scenarios. The first approach is based on a minimum area enclosing superellipse function and the second is based on use of sigmoid functions. These proposed methods are not only robust, but also conservative, that is, their construction is such that the approximate feasible region is a subset of the true feasible region. Further, these methods admit analytically derived bounds for the over-estimation of the infeasible region with a demonstrated maximum error of no greater than 0.3% using the superellipse method, which is less than the resolution of typical sensors used to calculate aircraft position or altitude. However, the superellipse method is not practical in all cases to enforce conditional inequality path constraints that may arise in the nonlinear airborne collision avoidance problem. Therefore, this paper also highlights by example when the use of sigmoid functions are more appropriate

A Hybrid Technique applied to the Intermediate-Target Optimal Control Problem

The DoD has introduced the concept of Manned-Unmanned Teaming, a subset of which is the loyal wingman. Optimal control techniques have been proposed as a method for rapidly solving the intermediate-target (mid-point constraint) optimal control problem. Initial results using direct orthogonal collocation and a gradient-based method for solving the resulting nonlinear program reveals a tendency to converge to or to get `stuck’ in locally optimal solutions. The literature suggested a hybrid technique in which a particle swarm optimization is used to quickly find a neighborhood of a more globally minimal solution, at which point the algorithm switches to a gradient-based nonlinear programming solver to converge on the globally optimal solution. The work herein applies the hybrid optimization technique to rapidly solve the loyal wingman optimal control problem. After establishing the background and describing the loyal wingman particle swarm optimization algorithm, the problem is solved first using the gradient-based direct orthogonal collocation method, then re-solved using a hybrid approach in which the results of the particle swarm optimization algorithm are used as the initial guess for the gradient-based direct orthogonal collocation method. Results comparing the final trajectory and convergence time, demonstrate the hybrid technique as a reliable method for producing rapid, autonomous, and feasible solutions to the loyal wingman optimal control problem

Approximate Integrals Over Bounded Volumes with Smooth Boundaries

A Radial Basis Function Generated Finite-Differences (RBF-FD) inspired technique for evaluating definite integrals over bounded volumes that have smooth boundaries in three dimensions is described. A key aspect of this approach is that it allows the user to approximate the value of the integral without explicit knowledge of an expression for the boundary surface. Instead, a tesselation of the node set is utilized to inform the algorithm of the domain geometry. Further, the method applies to node sets featuring spatially varying density, facilitating its use in Applied Mathematics, Mathematical Physics and myriad other application areas, where the locations of the nodes might be fixed by experiment or previous simulation. By using the RBF-FD-like approach, the proposed algorithm computes quadrature weights for N arbitrarily scattered nodes in only O(N logN) operations with tunable orders of accuracy

A Computational Study of the Fourth Painleve Equation and a Discussion of Adams Predictor-Corrector Methods

This thesis explores two unrelated research topics. The first is a numerical study of the fourth Painleve equation, while the second is a characterization of the stability domains of Adams predictor-corrector methods. First, the six Painleve equations were introduced over a century ago, motivated by theoretical considerations. Over the last several decades these equations and their solutions have been found to play an increasingly central role in numerous areas of mathematical physics. Due to extensive dense pole fields in the complex plane, their numerical evaluation remained challenging until the recent introduction of a fast `pole field solver\u27 (Fornberg and Weideman, J. Comp. Phys. 230 (2011), 5957-5973). This study adapts this numerical method to allow for either extended precision or faster numerical solutions to explore the solution space of the fourth Painleve (PIV) equation. This equation has two free parameters in its coefficients, as well as two free initial conditions. After summarizing key analytical results for PIV , the present study applies this new computational tool to the fundamental domain and a surrounding region of the parameter space. We confirm existing analytic and asymptotic knowledge about the equation, and also explore solution regimes which have not been described in the previous literature. In particular, solutions with the special characteristic of having adjacent pole-free sectors, but with no closed form, are identified. Second, the extent that the stability domain of a numerical method reaches along the imaginary axis indicates the utility of the method for approximating solutions to certain differential equations. This maximum value is called the imaginary stability boundary (ISB). It has previously been shown that exactly half of Adams-Bashforth (AB), Adams-Moulton (AM), and staggered Adams-Bashforth methods have nonzero stability ordinates. In the last chapter of this thesis, two categories of Adams predictor-corrector methods are considered, and it is shown that they have a nonzero ISB when (for a method of order p) p = 1,2, 5,6, 9,10,... for ABp-AMp and p = 3,4, 7,8, 11,12,... in the case of and AB(p-1)-AMp

Smooth_Closed_Surface_Quadrature_RBF-julia: v1.0

Julia implementation of the algorithm described in J. A. Reeger, B. Fornberg, and M. L. Watts "Numerical quadrature over smooth, closed surfaces", which can be used to compute quadrature weights for computing surface integral

Numerical Quadrature over Smooth Surfaces with Boundaries

This paper describes a high order accurate method to calculate integrals over curved surfaces with boundaries. Given data locations that are arbitrarily distributed over the surface, together with some functional description of the surface and its boundary, the algorithm produces matching quadrature weights. This extends on the authors\u27 earlier methods for integrating over the surface of a sphere and over arbitrarily shaped smooth closed surfaces by also considering domain boundaries. The core approach consists again of combining RBF-FD (radial basis function-generated finite difference) approximations for curved surface triangles, which together make up the full surface. The provided examples include both curved and flat domains. In the highly special case of equi-spaced nodes over a regular interval in 1-D, the method provides a new opportunity for improving on the classical Gregory enhancements of the trapezoidal rule