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2 Constitutive Equations     continued...

In Eq. 6, is a positive quantity representing the flux of material M in response to a unit gradient in the field variable u. Time dependent conservation law PDEs with flux ``down gradient'' are usually parabolic while steady state conservation laws with such flux are generally elliptic. These parabolic and elliptic conservation law equations will be the vehicles by which a number of important computational issues surrounding PDE problems will be introduced.

If we put the constitutive relationships Eq. 4 and Eq. 6 into the conservation law , the result is the one dimensional parabolic conservation law PDE

where c denotes . If density is a linear function of u,

and if is a constant, then Eq. 7 reduces to the constant coefficient diffusion equation

where .