David C. Arney

An adaptive mesh-moving and local refinement method for time-dependent partial differential equations

We discuss mesh-moving, static mesh-regeneration, and local mesh-refinement algorithms that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of time-dependent partial differential equations in two space dimensions and time. A coarse base mesh of quadrilateral cells is moved by an algebraic mesh-movement function so as to follow and isolate spatially distinct phenomena. The local mesh-refinement method recursively divides the time step and spatial cells of the moving base mesh in regions where error indicators are high until a prescribed tolerance is satisfied. The static mesh-regeneration procedure is used to create a new base mesh when the existing one becomes too distorted. The adaptive methods have been combined with a MacCormack finite difference scheme for hyperbolic systems and an error indicator based upon estimates of the local discretization error obtained by Richardson extrapolation. Results are presented for several computational examples.

An adaptive local mesh refinement method for time-dependent partial differential equations

Abstract We discuss an adaptive local mesh refinement procedure for solving time-dependent initial boundary value problems for vector systems of partial differential equations on rectangular spatial domains. The method identifies and groups regions having large local error indicators into rectangular clusters. The time step and computational cells within clustered rectangles are recursively divided until a prescribed tolerance is satisfied. The refined meshes are properly nested within coarser mesh boundaries; thus, simplifying the prescription of interface conditions at boundaries between fine and coarse meshes. Our method may be used with several numerical solution strategies and error indicators. The meshes may be nonuniform and either stationary or moving. A code based on our refinement procedure is used with a MacCormack finite difference method to solve some examples involving systems of hyperbolic conservation laws.

A two-dimensional mesh moving technique for time-dependent partial differential equations

Abstract We discuss an adaptive mesh moving technique that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for systems of partial differential equations in two space dimensions and time. The mesh moving technique is based on an algebraic node movement function determined from the geometry and propagation of regions having significant discretization error indicators. Our procedure is designed to be flexible, so that it can be used with many existing finite difference and finite element methods. To test the mesh moving algorithm, we implemented it in a system code with an initial mesh generator and a MacCormack finite difference scheme on quadrilateral cells for hyperbolic vector systems of conservation laws. Results are presented for several computational examples. The moving mesch scheme reduces dispersive errors near shocks and wave fronts and thereby reduces grid requirements necessary to compute accurate solutions while increasing computational efficiency.

Exhibiting chaos and fractals with a microcomputer

Abstract The microcomputer and its graphics capabilities are used to investigate chaos in Newtons method for a complex-valued quartic polynomial. Convergence maps that show fractal geometry are presented.

An Extension of an Old Classical Diophantine Problem

In recent years, a number of articles have appeared in the literature which deal with the problem of finding a set of four numbers such that the product of any two different numbers in the set when incremented by some fixed integer value n is a perfect square.

CORE MATHEMATICS AT THE UNITED STATES MILITARY ACADEMY: LEADING INTO THE 21st CENTURY

ABSTRACT The Department of Mathematical Sciences at the United States Military Academy is prepared to lead the young minds of America into the 21st century with a bold and innovative curriculum coupled with student and faculty growth models and interdisciplinary lively applications. In 1990, the mathematics department began its first iteration of their “7 into 4” core curriculum. Each year improvements have been incorporated into the core mathematics program. In 1992, interdisciplinary applications appeared in the core program as an opportunity to communicate and work with the academic disciplines. Our core curriculum is tied together both vertically and horizontally with threads. These threads tie together both the content within each course as well as among all the courses. Student attitudes are measured through course surveys as we attempt to develop “life long learners”. Student performance is measured or calibrated throughout their four years.

Modeling insurgency, counter-insurgency, and coalition strategies and operations

We model insurgency and counter-insurgency (COIN) operations with a large-scale system of differential equations and a dynamically changing coalition network. We use these structures to analyze the components of leadership, promotion, recruitment, financial resources, operational techniques, network communications, coalition cooperation, logistics, security, intelligence, infrastructure development, humanitarian aid, and psychological warfare, with the goal of informing today’s decision makers of the options available in COIN tactics, operations, and strategy. In modern conflicts, techniques of asymmetric warfare wreak havoc on the inflexible, regardless of technological or numerical advantage. In order to be more effective, the US military must improve its COIN capabilities and flexibility to match the adaptability and rapid time-scales of insurgent networks and terror cells. Our simulation model combines elements of traditional differential equation force-on-force modeling with modern social science modeling of networks, PSYOPs, and coalition cooperation to build a framework that can inform various levels of military decision makers in order to understand and improve COIN strategy. We show a test scenario of eight stages of COIN operation to demonstrate how the model behaves and how it could be used to decide on effective COIN resources and strategies.

Educating for the future: mathematics for understanding the new sciences

ABSTRACT New courses should be designed for the future needs of our students. This paper presents the historical development of required undergraduate core mathematics courses and suggests that a new core course, discrete dynamical modeling, is needed to prepare students for success in the information age. Topics associated with dynamical systems have previously been taught in upper-level, after-calculus courses. Discrete dynamical modeling is accessible to all first-year students, who through this course obtain the foundation for the reasoning, modeling, computation, and language of the new sciences of the 21st century. We propose that now is the time for this course to enter the core program for first-year students.

Recent Developments in Numerical Methods and Software for ODEs/DAEs/PDEs

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A Note on Fundamental Properties of Recurring Series

In this paper we consider certain recurring series and find some new fundamental properties in these series.