3 Classification of Linear PDEs in Two Independent Variables

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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.

In the linear PDE of second order in two variables,

if is formally replaced by , by , by , by and by , then associated with equation (7) is a polynomial of degree two in and

The mathematical nature of the solutions of equation (7) are largely determined by the algebraic properties of the polynomial . In turn, the computational strategy that one selects to numerically solve (7) is strongly influenced by the mathematical nature of the solution. Thus, before embarking on a quantitative analysis of a partial differential equation of the form (7), it is important that a computational scientist have an idea of the qualitative nature of the solution. Much of this qualitative understanding of the solution can be obtained via the following classification scheme. P(a,b) and along with it, the PDE (7) is classified as hyperbolic, parabolic, or elliptic according as its discriminant, , is positive, zero, or negative. Note that the type of equation (7) is determined solely by its principal part (the terms involving the highest-order derivatives of ) and that the type will generally change with position in the -plane unless ,, and are constants.

Laplace's equation,

is elliptic since the discriminant, , is negative. Laplace's equation occurs in numerous physically based simulation models and is usually associated with a diffusive or dispersive process in which the state variable, is in an equilibrium condition. For example, could represent an equilibrium temperature in a two dimensional thermodynamic model based on Fick's Law. Of interest to the computational scientist is the fact that solutions of Laplace's equation, and elliptic equations in general, can support large gradients only in response to external stresses manifested as a source/sink term ( in equation (7)) or as an abrupt change in type of or value of a boundary condition. Almost invariably the computational analysis of an elliptic equation reduces to a linear algebra problem of solving a system of diagonally dominant linear equations. Armed with this knowledge, the computational scientist has apriori knowledge of the types of algorithms and architectures that may provide an efficient numerical solution of an elliptic equation of the form (7).

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 . 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.

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.

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