Category Archives: High Order and Spectral Methods

Complex Magic, Part 3: Function Evaluation via Integration

Accurate evaluation of the function is a classic problem in numerical analysis. This function can be seen as an approximation to the first derivative of at , . More generally, is the divided difference of at and and this fact … Continue reading

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Complex Magic, Part 2: Differentiation via Integration

In the previous post, Complex Magic, Part 1, the use of complex variables in approximating the first derivative of a holomorphic function was discussed. This post delves deeper and introduces formulas suitable for approximating the th derivative of a holomorphic … Continue reading

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Complex Magic, Part 1: Differentiation via Function Evaluation at a Single Point

At the heart of conventional methods of derivative approximation based on divided differences lies a sum of the form in which the coefficients have different signs and vary (usually considerably) in magnitude. As a result, when these coefficients of different … Continue reading

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Neumann Boundary Conditions, Decoded

The following function (from L.N. Trefethen, Spectral Methods in MATLAB, with slight modifications) solves the 2nd order wave equation in 2 dimensions () using spectral methods, Fourier for x and Chebyshev for y direction. On its rectangular domain, the equation … Continue reading

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