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Using High Order Finite Differences
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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