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1945 results in Engineering mathematics and programming

Page 7 of 98

  • 29 - Finite Difference Methods for Hyperbolic Problems

  • from Part V - Boundary and Initial Boundary Value Problems
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book:
    Classical Numerical Analysis
    Published online:
    29 September 2022
    Print publication:
    20 October 2022, pp 811-836

  • Summary

    We present finite difference schemes for hyperbolic problems. We begin with the transport equation and show the necessity of a certain type of upwinding and/or numerical diffusion for stability. This is illustrated by analyzing all the classical schemes: upwind, downwind, centered, Lax–Friedrichs, Lax–Wendroff. Beam–Warming, and Crank–Nicolson. We then focus on the topic of positivity and max-norm dissipativity of finite difference schemes. We present and sketch the proof of Godunov’s theorem. A brief discussion of dispersion relations is then carried out. Next, we study schemes for the wave equation, and show how to properly choose the discrete initial velocity to attain the desired consistency. To show stability we employ energy and negative norm arguments. The last section is dedicated to developing schemes for symmetric hyperbolic systems. The most well-known finite difference schemes are presented, and their matrix valued symbols are introduced. The symbol is then used to develop the von Neumann stability analysis for this case.

  • 12 - Fourier Series

  • from Part II - Constructive Approximation Theory
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book:
    Classical Numerical Analysis
    Published online:
    29 September 2022
    Print publication:
    20 October 2022, pp 320-344

  • Summary

    This chapter is a close companion to the previous one. Here we study the best least squares approximation to periodic functions via trigonometric polynomials. Many of the ideas and results of the previous chapter are repeated in this scenario. They are then expanded to deal with merely square integrable functions. The Fourier transform of periodic functions, and its inverse, is then introduced and studied. Uniform convergence of trigonometric series, under several different smoothness assumptions is then discussed.Trigonometric approximation in periodic Sobolev spaces is then discussed.

Classical Numerical Analysis
  • A Comprehensive Course
  • Abner J. Salgado, Steven M. Wise
  • Published online:
    29 September 2022
    Print publication:
    20 October 2022
  • Numerical Analysis is a broad field, and coming to grips with all of it may seem like a daunting task. This text provides a thorough and comprehensive exposition of all the topics contained in a classical graduate sequence in numerical analysis. With an emphasis on theory and connections with linear algebra and analysis, the book shows all the rigor of numerical analysis. Its high level and exhaustive coverage will prepare students for research in the field and become a valuable reference as they continue their career. Students will appreciate the simple notation, clear assumptions and arguments, as well as the many examples and classroom-tested exercises ranging from simple verification to qualifying exam-level problems. In addition to the many examples with hand calculations, readers will also be able to translate theory into practical computational codes by running sample MATLAB codes as they try out new concepts.

Introduction to Finite Elements in Engineering
  • 5th edition
  • Tirupathi Chandrupatla, Ashok Belegundu
  • Published online:
    08 November 2021
    Print publication:
    21 October 2021
  • Thoroughly updated with improved pedagogy, the fifth edition of this classic textbook continues to provide students with a clear and comprehensive introduction the fundamentals of the finite element method. New features include enhanced coverage of introductory topics in the context of simple 1D problems, providing students with a solid base from which to advance to 2D and 3D problems; expanded coverage of more advanced concepts, to reinforce students' understanding; over 30 additional solved problems; and downloadable MATLAB, Python, C, Javascript, Fortran and Excel VBA code packages, providing students with hands-on experience, and preparing them for commercial software. Accompanied by online solutions for instructors, this is the definitive text for senior undergraduate and graduate students studying a first course in the finite element method and finite element analysis, and for professional engineers keen to shore up their understanding of finite element fundamentals.

Page 7 of 98