Numerical Solutions of Partial Differential Equations

xiaozhouli@uestc.edu.cn School of Mathematical Sciences

University of Electronic Science and Technology of China

Dr. Xiaozhou Li

Introduction

Overview

Precise solutions needed for problems in science, engineering and applied math.

Many of these problems governed by partial differential equations (PDEs).

Analytical solutions to PDEs, few and limited.

Very effective numerical methods are now available.

Powerful computers make it possible to obtain solutions to large, real-world problems.

Algorithms make it happen. They apply to broad classes of PDEs, not to a specific PDE. Learn general classes of algorithms and you can solve broad classes of PDEs.

Broad classes of PDEs of interest:

Elliptic PDEs : Don’t have time variation, convey action at a distance. Examples: Gravitational field, electrostatics.

Parabolic PDEs : Enable information to travel as diffusive processes. Examples: heat transfer, mass diffusion in the ground, diffusion of photons out of the sun.

Hyperbolic PDEs: Enable information to propagate as waves. Examples: Water waves, sound waves, oscillations in a solid structure and electromagnetic radiation.

We first study solution techniques for these PDEs piecemeal and then learn how to assemble them together for more complex PDEs.

The Euler Equations

The Navier-Stokes Equations

Incompressible Flow Equations

The Shallow Water Equations

Maxwell’s Equation

The Magnetohydrodrodyamic Equation

The Equations of Linear Elasticity

…

Purpose of the course:

Presents the fundamentals of modern numerical techniques for a wide range of equations.

The emphasis is on building an understanding of the essential ideas that underlie the development, analysis, and practical use of finite difference methods.

Expect goals:

To understand the fundamental mathematics theory and algorithms of finite difference methods.

To be able to implement finite difference methods for simple 1d and 2d problems as well as to evaluate and to interpret the numerical results.

To be able to solve some engineering problems by using known algorithms.

not covered in the course:

Numerical analysis: interpolation, polynomials, norm, etc.

Numerical algebra: matrix, solvers for linear systems, etc.

PDEs: concept, derivation, etc.

Part III of the textbook.

Textbook

Finite Difference Methods for Ordinary and Partial Differential Equations

Randall J. LeVeque

References

Numerical Solution of Partial Differential Equations

Morton and Mayers

Teaching technique:

Chalkboard.

Experiments demonstrated by using Jupyter notebook (python).

- This lecture is not about programming languages, so feel free to choose your own.

Assessment Method:

Written assignments.

Computer projects.

Final exam.

Other information:

There is no lecture on week May 13 - 17. Therefore, the course is ended one week later than it scheduled (May 28th).

Some resources will be available on my website xiaozhouli.com/teaching/ns.

Finite Difference Methods:

The best known methods, finite difference, consists of replacing each derivative by a difference quotient in the classic formulation.

It is simple to code and economic to compute.

The drawback of the finite difference methods is accuracy and flexibility. Difficulties also arises in imposing boundary conditions.