Numerical Methods for Partial Differential Equations

CS555 :: Spring 2026

• Class Time: Tuesday/Thursday 12:30-1:45pm [Catalog]
• Class Location: 1035 Campus Instructional Facility (CIF)
• Instructor: Paul Fischer
• Office Hours: Tue. 2:15-4:00
• About the Course
• Lectures
• Quizzes
• Homework
• Final Project

About the Course

Are you interested in the numerical approximation of solutions to partial differential equations? Then this course is for you!

The course covers roughly three topics: the fundamentals of finite difference approximations, an introduction to finite volume schemes, and comprehensive look at finite element methods.

Topics

• Introduction

  • Overview
  • FDiff cost examples in 1D/2D/3D
    • derivation
    • eigenvalues
    • error analysis
  • Tensor-products
    • solvers and eigenvalues
  • Cost analysis in d dimensions

• Timestepping considerations

  • Stability/costs
  • Survey of timesteppers: EF/EB, Trap/CN, BDFk, RK, exponential

• Hyperbolic PDEs

  • Advection + FDiff: stability/dispersion
  • FDiff: equivalent differential equation
  • FDiff: upwinding/variable grid spacing
  • FVol: conservation/FV methods
  • FVol: flux limiters/WENO schemes
  • WRTs: finite element methods
  • Multiple space dimensions

• Parabolic PDEs

  • Explicit, implicit, semi-implicit

• Elliptic PDEs

  • Poisson FEM in 1D and 2D with tensor products
  • Poisson FEM in 2D with triangles
  • Stokes FEM in 2D with triangles

• Special topics

  • High-order methods
  • Other equations
  • Reduced Order Models

Prerequites

The expectation is that you have had a course in numerical methods (like CS450), covering conditioning, numerical linear algebra, numerical quadrature and differentiation, and the basics of numerical methods for ordinary and differential equations. If you do not have this background, please check with the instructor.

In addition, you should have familiarity with Python or Matlab (or Octave). If you do not have this background, please check with the instructor.

The course will provide practical approaches to generating (highly accurate!) approximate solutions to partial differential equations and will explore mathematical underpinnings that give insight into the behavior (i.e., stability and convergence) of these methods. Basic linear algebra will be an important tool in this course. If you have questions about your mathematical preparation for this course, please check with the instructor.

Expected work

The course involves several assignments along with midterm and final project presentations based on material developed over the semester. There is also a strong participation grade based on quizzes, handouts, and other in-class tasks. The tentative grade breakdown is:

Work in teams of two for the Homework and Projects.

• Quizzes, in-class work 10
• Homework 30
• Midterm Project 30
• Final Project 30

This will be finalized in the first week of class.

The course assignments and examples in class will be in Matlab/Octave or Python.

Lectures

Lecture	Date	Topic
	01-20	About the course, classifying PDEs, survey of methods cs555-notes-20260120.pdf Book Chapter 1
	01-22	Finite differencing for time dependent problems cs555-notes-20260122.pdf Book Chapter 1
	01-27	Finite differences for parabolic problems in 1D cs555-notes-20260127.pdf Book Chapter 1
	01-29	Intro to Time-steppers for PDEs cs555-notes-20260129.pdf heat30.m Book Chapters 3 and 5
	02-03	Time-steppers/Multidimensional Heat Eqn cs555-notes-20260203.pdf Book Chapters 3 and 5
	02-05	Multidimensional Heat Eqn cs555-notes-20260205.pdf hw1.pdf hw1_1.tar hw1_2.tar Book Chapters 3 and 5
	02-10	Kronecker Products/Alternating Direction Implicit cs555-notes-20260210.pdf poisson2d_notes.pdf ktest.m
	02-12	ADI/Var. Spacing cs555-notes-20260212.pdf Book Sections 1.2, 5.1 DFM chap04.pdf DFM chap08.pdf
	02-17	One-Way Wave Eqn. cs555-notes-20260217.pdf lec9.tar Book Sections 1.2, 5.2
	02-19	Equivalent Differential Eqn. I. cs555-notes-20260219.pdf Book Sections 1.2, 5.2
	02-24	Equivalent Differential Eqn. II. cs555-notes-20260224.pdf demo_fdiff.m Book Sections 1.2, 5.2
	02-26	Equivalent Differential Eqn. III. cs555-notes-20260226.pdf Eq. Diff. Eqn. Notes Book Sections 1.2, 5.2
	03-03	CFL and High-Order Finite Differences cs555-notes-20260303.pdf stab.m Book Sections 1.2, 5.2
	03-05	CFL and High-Order Finite Differences cs555-notes-20260305.pdf hw2.pdf burgers_86.pdf
	03-10	CFL and High-Order Finite Differences cs555-notes-20260310.pdf Book Sections 5.3
	03-12	Finite Volumes cs555-notes-20260312.pdf fv.m Book Chapters 6 and 9
	03-24	WENO cs555-notes-20260324.pdf hdr.m fw.m ww.m Book Chapters 6 and 9
	03-26	FEM cs555-notes-20260326.pdf hdr.m weno_almost.m poly_glk.m Book Chapters 4 and 8
	04-02	FEM II cs555-notes-20260402.pdf poly_glk.m hw4.pdf Book Chapters 4 and 8
	04-07	FEM III cs555-notes-20260407.pdf Book Chapters 4 and 8
	04-09	FEM IV cs555-notes-20260409.pdf adv_diff_hw4.pdf adv_example.pdf Book Chapters 4 and 8
	04-14	FEM IV cs555-notes-20260414.pdf abqfem.m Book Chapters 4 and 8
	04-16	FEM IV cs555-notes-20260416.pdf triangle_files.tar Book Chapters 4 and 8
	04-21	FEM IV cs555-notes-20260421.pdf malm13.pdf tri_files.tar Book Chapters 4 and 8