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3 Classification of Linear PDEs in Two Independent Variables     continued...

The diffusion equation,

for is parabolic since the discriminant, . The diffusion equation arises in diverse settings, but most often in connection with a transient flow problem in which the flow is down gradient of some state variable u. In the setting of heat flow, the diffusion equation (sometimes called the heat equation) could be used to model a thermodynamics problem in which transient heat flow is occurring in one space dimension. Similar to the elliptic case, parabolic equations generally have very smooth solutions. However, parabolic equations often exhibit solutions with evolving regions of high gradient. Most numerical methods for dealing with parabolic equations involve approximating the solution at successive time steps, with each approximation requiring the solution of a system of linear equations. For these types of computational problems, it is often useful to employ some matrix factorization method in conjunction with a dynamic gridding algorithm. Multispace generalizations of this example problem can be solved efficiently on vector architectures using ADI methods or on parallel architectures with some divide and conquer strategies.