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3 Classification of Linear PDEs in Two Independent Variables

In addition to the distinction between linear and nonlinear PDEs, it is important for the computational scientist to know that there are different classes of PDEs. Just as different solution techniques are called for in the linear versus the nonlinear case, different numerical methods are required for the different classes of PDEs, whether the PDE is linear or nonlinear. The need for this specialization in numerical approach is rooted in the physics from which the different classes of PDEs arise. By analogy with the conic sections ( ellipse, parabola and hyperbola) partial differential equations have been classified as elliptic, parabolic and hyperbolic. Just as an ellipse is a smooth, rounded object, solutions to elliptic equations tend to be quite smooth. Elliptic equations generally arise from a physical problem that involves a diffusion process that has reached equilibrium, a steady state temperature distribution, for example. The hyperbola is the disconnected conic section. By analogy, hyperbolic equations are able to support solutions with discontinuities, for example a shock wave. Hyperbolic PDEs usually arise in connection with mechanical oscillators, such as a vibrating string, or in convection driven transport problems. Mathematically, parabolic PDEs serve as a transition from the hyperbolic PDEs to the elliptic PDEs. Physically, parabolic PDEs tend to arise in time dependent diffusion problems, such as the transient flow of heat in accordance with Fourier's law of heat conduction.