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

mailto:xiaozhouli@uestc.edu.cn

Mixed Equations

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. 


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.

We have studid solution techniques for these PDEs piecemeal during previous lectures.

Broad classes of PDEs of interest:

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

Examples: Chapters 2-3.

Initial Value Problems for ODEs : time-dependent differential equations

Examples: Chapters 5-8.

Parabolic PDEs : Enable information to travel as diffusive processes.

Examples: Chapter 9, heat equation.

Hyperbolic PDEs: Enable information to propagate as waves. Examples: Chapter 10, advection equation, Water waves.

Next step: learn how to assemble them together for more complex PDEs. Chapter 11.

Mixed EquationsIn practice several processes may be happening simultaneously, and the PDE model will not be a pure equation of any of the types already discussed but rather will be a mixture.

In this chapter we discuss several approaches to handling more complicated equations.

We restrict our attention to time-dependent PDEs of the form

ut = A1(u) +A2(u) + · · ·+AN (u)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

For simplicity, most of our discussion will be further restricted to only two terms, which we will write as

ut = A(u) + B(u)AAACDHicdVDLSgMxFM3UV62vqks3wSJUhGGmD6sLodaNywr2Ae1QMmmmDc08SDJCGeYD3Pgrblwo4tYPcOffmGlHUdEDgXPPuZfce+yAUSEN413LLCwuLa9kV3Nr6xubW/ntnbbwQ45JC/vM510bCcKoR1qSSka6ASfItRnp2JOLxO/cEC6o713LaUAsF4086lCMpJIG+UI4iGR81neRHGPEovO4GB4efZWNpFRdhn5qVE2zBhUxj6uVlJTNCjR1Y4YCSNEc5N/6Qx+HLvEkZkiInmkE0ooQlxQzEuf6oSABwhM0Ij1FPeQSYUWzY2J4oJQhdHyunifhTP0+ESFXiKlrq85kS/HbS8S/vF4onRMrol4QSuLh+UdOyKD0YZIMHFJOsGRTRRDmVO0K8RhxhKXKL6dC+LwU/k/aJd0s66WrSqHeSOPIgj2wD4rABDVQB5egCVoAg1twDx7Bk3anPWjP2su8NaOlM7vgB7TXD8KEm28=

each of the right-hand side terms are functions or differential operators involving only spatial derivatives of u.

Reaction-diffusion equations

ut = uxx +R(u)AAACAnicdVDLSgMxFM3UV62vUVfiJliEijDM9GF1IRTduKxibaEdhkyatqGZB0lGWobBjb/ixoUibv0Kd/6NaTuCih64cHLOveTe44aMCmmaH1pmbn5hcSm7nFtZXVvf0De3bkQQcUwaOGABb7lIEEZ90pBUMtIKOUGey0jTHZ5P/OYt4YIG/rUch8T2UN+nPYqRVJKj70ROLJPTzhCFIYLqMYKj5PCqEB04et40TsyKZVWhItZRpZySklWGlmFOkQcp6o7+3ukGOPKILzFDQrQtM5R2jLikmJEk14kECREeoj5pK+ojjwg7np6QwH2ldGEv4Kp8Cafq94kYeUKMPVd1ekgOxG9vIv7ltSPZO7Zj6oeRJD6efdSLGJQBnOQBu5QTLNlYEYQ5VbtCPEAcYalSy6kQvi6F/5ObomGVjOJlOV87S+PIgl2wBwrAAlVQAxegDhoAgzvwAJ7As3avPWov2uusNaOlM9vgB7S3T+eXlx0=

The reaction terms might or might not be stiff.

the diffusion term is stiff and requires appropriate methods.

Advection-diffusion equations


the advection term can be handled explicitly

the diffusion term is stiff and requires an appropriate solver

ut + aux = κuxx

Nonlinear hyperbolic equations 


modeling fluid dynamics, for example, the Navier–Stokes equations for compressible gas dynamics have this general form.

ut + f(u)x = κuxx

Advection-diffusion-reaction equations

ut + f(u)x = κuxx + R(u)

ut + uux = vuxxx

The Korteweg–de Vries (KdV) equation


Many approaches can be used for problems that involve two or more different terms, and a huge number of specialized methods have been developed for particular equations. 


Fully coupled method of lines

Fully coupled Taylor series methods

Implicit-explicit methods

Other Numerical Methods for PDEs

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.

Finite element methods

Finite volume methods

Discontinuous Galerkin methods

Spectral methods

…