AMATH 585: Numerical Analysis of Boundary-Value Problems

Course Description

Learning objectives

Syllabus

• Finite differencing; order of accuracy
• Finite difference solution of 2-point one-dimensional ODE boundary-value problems (BVPs) (such as the steady-state heat equation). Concepts of local truncation error, consistency, stability and convergence.
• Extensions to nonlinear problems and nonuniform grids.
• Finite element methods for 1D BVPs
• Fourier and Chebyshev spectral methods for 1D and multidimensional BVPs.
• Solution of multidimensional elliptic PDE BVPs
• Iterative methods for solving the linear systems that arise from discretizing elliptic PDE BVPs - SOR, conjugate gradient, multigrid

Special days

• Mo 21 Jan: No class - MLK Day
• We 9 Jan: TA will teach class
• Mo 11 Feb: In-class midterm (bring paper and one double-sided page of hand-written notes)
• Mo 18 Feb: No class - Presidents Day
• We 20 Feb: TA will teach class
• Mo-Fr 25 Feb-1 Mar: TA will teach class
• Fr 15 Mar: Last day of class
• We 20 Mar 11:59pm: Take-home Final Exam due via Canvas

Supplemental reading

Grading

• Homework (50%), posted to class web page and assigned each Friday, due the following Friday. Assignments will alternate between solo (do without consulting without others) and discussable (discussion with your fellow students is encouraged). In either case you must write your own scripts and solutions in your own words. Assignments will involve algorithm implementation and plotting; you may use Matlab or Python for this, but my scripts are all in Matlab. Please submit assignments via Canvas. Late homework (after 11:59pm on due date) must be submitted by 11:59pm on the Monday after the due date. The first late homework will be accepted without penalty. Thereafter, there will be a 20% deduction on the grade for each late assignment.
  
• Midterm (20%), in class, Mo 11 Feb. Bring your own paper. One double-sided page of hand-written notes allowed.
• Final Exam (30%), open book and note, no consultation with any human. Posted We 13 Mar, due by 11:59pm on We 20 Mar.

  Grades will be recorded on the class Canvas page, which I will also use for assignments, class announcements and discussion forums.

Lectures

References to the 585 part of RJL's book are given by 'RJL'. Dates subject to mid-course correction.

1. Finite difference (FD) methods

• Mo 7 Jan: Finite differencing I (RJL Chapter 1.0-1.1)
• We 9 Jan: Finite differencing II (RJL 1.2-1.5).

2. 2-point 1D ODE BVPs

• We 9 Jan: Finite difference approximation (FDA) for a simple problem (RJL 2.1-2.4)
• Fr 11 Jan: Global and local truncation error, stability, consistency, convergence (RJL 2.5-2.9)
• Mo 14 Jan: 2-norm stability and relation to spectral radius of A and eigenvalues of the continuous BVP (RJL 2.10).
• We 16 Jan: Green's function and interpretation of A^-1; use for estimating max-norm stability of FDA (RJL 2.11).
• Fr 18 Jan: Neumann BCs (RJL L2.12).
• Mo 21 Jan: No class - MLK Day.
• We 23 Jan: Issues with FDA solution when underlying BVP is ill-posed (RJ L2.13); 'flux-form' differencing for ODEs with non-constant coeffs (RJL 2.15); a nonlinear pendulum BVP (RJL 2.16).
• Fr 25 Jan: Solution of nonlinear pendulum BVP via Newton's method; uniqueness issues; accuracy (RJL 2.16.2-3)
• Mo 28 Jan: Shooting method for nonlinear BVPs (See 1-slide description) and pendulum_shoot Matlab scripts.)
• We 30 Jan: Discretization considerations at boundary layers and sharp gradients; use of nonuniformly spaced grids (RJL 2.16-2.18).
• Fr 1 Feb: Finite element method (FEM) for a 2-point BVP (1998 RJL FEM notes).
• Mo-We 4-6 Feb: The DFT and Fourier spectral (FS) methods for periodic 1D BVPs (Lecture notes).
• Fr 8 Feb: Chebyshev spectral method for 1D BVPs (Lecture notes).
• Mo 11 Feb: In-class midterm (one double-sided handwritten note sheet)
• We 13 Feb: Midterm post-mortem and wrap up Chebyshev method

3. Solution of multidimensional elliptic PDE BVPs

• Fr 15 Feb: FDA applied to Laplace and Poisson equation on a square domain; efficiency, accuracy and stability (RJL 3.1-3.4)
• Mo 18 Feb: Presidents Day - no class
• We 20 Feb: Fourier spectral method for Poisson's equation in a periodic rectangular domain; DFT-based direct FDA Poisson solver in a rectangular domain with inhomogeneous BCs (Lecture notes).
• Fr 22 Feb: Complications with irregular domains and grids; direct vs. iterative solvers (RJL 3.6-3.7)

4. Iterative methods for sparse linear systems useful for multidimensional BVPs

• Fr 22 Feb: Jacobi and Gauss-Seidel (RJL 4.1)
• Mo 25 Feb: Convergence of Jacobi and GS (RJL 4.2)
• We 27 Feb: Understanding Jacobi and GS using Von Neumann analysis; SOR. (Lecture notes and RJL 4.2.2)
• Fr 1 Mar: Steepest descents (RJL 4.3.1, steepest_descents.m)
• Mo-We 4-6 Mar: Conjugate gradient method (Lecture notes and RJL 4.3.3-4.3.4, CG.m).
• Fr 8 Mar: Matlab CG vs. steepest-descents example (gradient_methods_1D.m). Preconditioned CG (RJL 4.3.5-6, cholesky_pois2d.m)
• Mo-Fr 9-13 Mar: Multigrid method (RJL 4.6)

Matlab scripts

Lectures

• RJL Chapter 2 example scripts/functions
• c = fdcoeffF(k,xbar,x): Function from the above link for computing coefficients c of a finite-difference approximation to the k'th derivative at a point xbar based on a vector x of n distinct nearby points x(j), j=1,..,n. The resulting formula d^kf/dx^k = c dot f(x) will be at least n-k'th order accurate. This is particularly useful for generating FDAs for non-uniform grids or for near boundaries where an asymmetric stencil is needed.
• BVP_FDA_SimpleExample.m: Set up finite difference approximation for 2 point BVP u" = sin(pi*x), u(0)=1, u(1) = 2, calculate truncation and solution error, plot vs. exact solution, and show relationship of inv(A) and its boundary-augmented analogue to the Green's function of exact problem.
• pendulum_Newton.m: Matrix Newton method applied to nonlinear pendulum BVP u" + sin(u) = 0, u(0) = a, u(T) = b. Plots the iterates and the converged solution for a=b=0.7, T=2*pi, and 40-1 interior gridpoints.
• pendulum_shoot.m Shooting method applied to nonlinear pendulum problem. Uses secant method and pendulum_dydt.m.
• pendulum_shoot2.m Shooting method applied to nonlinear pendulum problem. Uses Matlab root finder, pendulum_dydt.m, and pendulum_shooterr.m.
• FS_1DBVP.m: Implements Fourier spectral method on the 1D BVP u'' - u = 1/(1-0.6*cos(x)), first with periodic BCs at x = 0,2*pi, and second with BCs u(0)=u(pi)=0.
• Chebyshev spectral method scripts from Trefethen's book: cheb.m differentiation matrix function; p11.m - accuracy of Chebyshev differentiation of exp(-x)*sin(5x); p13.m - BVP y'' = exp(4x), y(-1)= y(1)= 0; p33.m - Same ODE, but with y'(-1)=y(1)=0.
• pois2Dper.m: 2D Fourier spectral Poisson solver on a square domain with periodic BCs.
• pois_FD_FFT_2D.m: 2D DFT-based solver for FDA of 2D Poisson equation with inhomogeneous Dirichlet BCs.
• gradient_methods_1D.m: Iterative solution of FDA of u'' = 6*x, u(0) = 0, u(1) = 1 using steepest descents and conjugate gradient methods. Uses steepest_descents.m and CG.m
• cholesky_pois2d.m: Condition number of A = FDA of d2/dx2 + d2dy2 vs. h with vs. without incomplete Cholesky preconditioning.
• multigrid.m: Iterative solution of FDA of u'' = 6*x, u(0) = 0, u(1) = 1 using multigrid V-cycles. Uses vcycle.m.
• full_multigrid.m: Solution of FDA of u'' = 6*x, u(0) = 0, u(1) = 1 using one full multigrid V-cycle iteration.