 By Stan Skaistis

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8) The entries aii of A(U) are partial derivatives of the components /i of the vector F with respect to the components u; of the vector of conserved variables U, that is aii = afijau;. 6)-(2. 6), namely aF(U) aFau au ax. 2). 5) can be expressed as conservation laws, namely au at af(u) +a;;-= 0' f(u) =au' au af(u) = 0 f( ) = ~ 2 at+ ax ' u 2u . 3 (Eigenvalues). 11) where I is the identity matrix. 2} are called the eigenvalues of the system. 44 2. Notions on Hyperbolic Partial Differential Equations Physically, eigenvalues represent speeds of propagation of information.

Moreover, as the eigenvalues are also distinct this system is strictly hyperbolic. 2 (The Cauchy-Riemann Equations). 21) 46 2. Notions on Hyperbolic Partial Differential Equations where u 1 = u(x,y) and u 2 = v(x,y). These equations arise in the study of analytic functions in Complex Analysis . X2 real solutions for A and thus the system is elliptic. 3 (The Small Perturbation Equations). In Sect. 2 of Chap. 25) 00 Moo= canstant denotes the free-stream Mach number and u(x,y), v(x,y) are small perturbations of the x and y velocity components respectively.

166) for the three unknowns u, v,p, the primitive variables. In principle, given a domain along with initial and boundary conditions for the equations one should be able to solve this problem using the primitive variable formulation. The stream-function vorticity formulation is another way of expressing the incompressible Navier-Stokes equations. This formulation is attractive for the two-dimensional case but not so much in three dimensions, in which the role of a stream function is replaced by that of a vector potential.