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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