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

The one dimensional wave equation,

has discriminant so it is classified as hyperbolic. This type of equation arises in many fields ranging from elasticity and acoustics to atmospheric science and hydraulics. Of interest to the computational scientist is the knowledge that solutions to linear hyperbolic equations can be only as smooth as their boundary and initial conditions are. Moreover, any sharp fronts or peaks in the solution are persistent and can reflect off of boundaries. For a nonlinear hyperbolic PDE, even smooth boundary and initial conditions can give rise to nonsmooth or even discontinuous solutions. Of the three types of PDEs discussed in this example, hyperbolic equations are generally the most challenging to the computational scientist. Since explicit time stepping methods are usually called for to numerically solve hyperbolic PDEs, the computational scientist must be aware of important algorithm stability issues. Explicit algorithms give excellent performance rates on vector and SIMD architectures.