AMath 585 Notebooks

AMath 585, Winter Quarter 2020 at the University of Washington. Developed by R.J. LeVeque and distributed under the BSD license. You are free to modify and use as you please, with attribution.

These notebooks are all available on Github.

Introductory notebooks

• Demo1.html
• Debugging_hints.html
• Numpy_array_hints.html Some notes on indexing and slicing, and mutable objects

Finite difference approximations

• fdstencil.html Computing finite difference coefficients
• fdstencil_errors.html Testing two approximations to $u''(\bar x)$

2-point boundary values problems

• Banded.html Tips on working with banded matrices
• BVP_stability.html Compute banded matrices for $u''(x)=f(x)$ and investigate inverse matrix and max-norm stability.
• BVP1.html Solve $u''(x) = f(x)$ with Dirichlet BCs and test on a manufactured solution
• BVP_nonlinear.html Solve $\epsilon u''(x) + u(x)(u'(x) - 1) = f(x)$ using Newton iteration
• BVP_layer.html Solve $\epsilon u''(x) + u(x)(u'(x) - 1) = f(x)$ in the singularly perturbed case
• BVP_scipy.html Illustrate use of scipy.integrate.solve_bvp

Function space methods

• FourierSpectral.html Introduction to Discrete Fourier Transform and use in solving a BVP with periodic boundary conditions.
• ChebPolyOnCircle.html Illustrates the relation between Chebyshev polynomials and Fourier series, with some discussion of aliasing in both cases.
• ChebyshevSpectral.html Illustrates Chebyshev interpolation and using to to obtain a spectrally accurate approximation to a derivative.

Sparse matrix solvers

• Grids2D.html Defining and plotting functions on a 2D grid.
• LaplacianMatrix.html Converting 2D grid functions to vectors,creating 5-point Laplacian matrix using Kronecker products.
• IterativeMethods.html Illustrates the classic matrix spitting methods: Jacobi, Gauss-Seidel, and SOR.
• ConjugateGradient.html Illustration of C-G algorithm on 1D BVP.
• ConjugateGradient2D.html Illustration of C-G algorithm on a 2D Poisson problem.
• ConjugateGradient2D_vcoeff.html Illustration of C-G algorithm on a 2D variable coefficient elliptic equation.
• PCG.html Preconditioned conjugate gradients, with corrections to the text.
• DarcyFlow.html An example application - flow through porous media solved with PCG.
• Multigrid.html Illustration of some key ideas used in multigrid methods.