Solving partial differential equations (PDEs)

Solving partial differential equations (PDEs)

Hans Fangohr

Engineering and the Environment University of Southampton United Kingdom fangohr@soton.ac.uk

May 3, 2012

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Outline I

1 Introduction: what are PDEs? 2 Computing derivatives using finite differences 3 Diffusion equation 4 Recipe to solve 1d diffusion equation 5 Boundary conditions, numerics, performance 6 Finite elements 7 Summary

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This lecture

tries to compress several years of material into 45 minutes has lecture notes and code available for download at

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What are partial differential equations (PDEs)

Ordinary Differential Equations (ODEs)

one independent variable, for example t in

d2x

k

dt2

=- x m

often the indepent variable t is the time

solution is function x(t)

important for dynamical systems, population growth,

control, moving particles

Partial Differential Equations (ODEs)

multiple independent variables, for example t, x and y in

u

2u 2u

=D t

x2 + y2

solution is function u(t, x, y) important for fluid dynamics, chemistry, electromagnetism, . . . , generally problems with spatial resolution

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2d Diffusion equation

u

2u 2u

=D

+

t

x2 y2

u(t, x, y) is the concentration [mol/m3] t is the time [s] x is the x-coordinate [m] y is the y-coordinate [m] D is the diffusion coefficient [m2/s]

Also known as Fick's second law. The heat equation has the same structure (and u represents the temperature). Example:

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