Using High Order Finite Differences

Table of contents

• 1 Definitions and Basics
  • 1.1 Vector Norms
    • 1.1.1 Definition of a Vector Norm
    • 1.1.2 Common Vector Norms
    • 1.1.3 Inner Product of Vectors
    • 1.1.4 Inequalities Involving Norms
  • 1.2 Matrix Norms
    • 1.2.1 Definition of a Matrix Norm
    • 1.2.2 Common Matrix Norms
    • 1.2.3 Positive definite matrices
    • 1.2.4 Consistency of Norms
• 2 Finite Difference Operators
  • 2.1 approximation of a first derivative
  • 2.2 approximation of a second derivative
  • 2.3 approximation of higher derivatives
  • 2.4 effect of the placement of points
  • 2.5 centered difference operators
• 3 Shift Operators
  • 3.1 Trigometric polynomials
  • 3.2 Definition of a shift operator
  • 3.3 Approximation by trigometric polynomials
  • 3.4 Fundamental Property
  • 3.5 Error Estimation
  • 3.6 Simple Functions
• 4 Specialized Norms
  • 4.1 Energy and Heat Norms
  • 4.2 Discrete Heat Norms
    • 4.2.1 appendix a)
    • 4.2.2 appendix b)
    • 4.2.3 appendix c)
• 5 Special Formulas