4.1.4. ODEs and PDEs¶

4.1.4.1. How to decide if a PDE is elliptic, hyperbolic or parabolic?¶

\[a {{\partial^2 u} \over {\partial x^2}} + b {{\partial^2 u} \over {\partial x \partial y}} + c {{\partial^2 u} \over {\partial y^2}} = g(x, y)\]

or for a \(n \times n\) system:

\[{\partial \vec{U} \over \partial t} + \vec{A} {{\partial \vec{U}} \over {\partial x}}=0\]

The type depends on the determinant:

\(b^2-4ac>0 \longrightarrow\) all n eigenvalues are real \(\longrightarrow\) Hyperbolic
\(b^2-4ac=0 \longrightarrow\) all n eigenvalues are real and identical \(\longrightarrow\) Parabolic
\(b^2-4ac<0 \longrightarrow\) all n eigenvalues are complex \(\longrightarrow\) Elliptic

Hyperbolic \(\longrightarrow\) (Steady \(\lor\) Unsteady) \(\land\) Compressible \(\land\) Inviscid \(\land\) Supersonic
Parabolic \(\longrightarrow\) (Compressible \(\lor\) Incompressible) \(\land\) Viscous Boundary Layer
Elliptic \(\longrightarrow\) (Compressible \(\lor\) Incompressible) \(\land\) Inviscid \(\land\) Steady \(\land\) Subsonic
Mixed (most cases are mixed) \(\longrightarrow\) There is a parabolic boundary layer, but parts of the domain away from the boundary are hyperbolic or elliptic

Wave Equation \(\longrightarrow\) Hyperbolic \(\longrightarrow\) \({\partial u \over \partial t} + a {{\partial u} \over {\partial x}}=0\)
Laplace Equation \(\longrightarrow\) Elliptic \(\longrightarrow\) \({\partial^2 u \over \partial x^2} + {{\partial^2 u} \over {\partial y^2}}=0\)
Poisson Equation \(\longrightarrow\) Elliptic \(\longrightarrow\) \({\partial^2 p \over \partial x^2} + {{\partial^2 p} \over {\partial y^2}}=f(u)\)
Stokes Flow \(\longrightarrow\) Parabolic \(\longrightarrow\) \({\partial p \over \partial x} = \mu {\partial^2 u \over \partial x^2}\)
Heat Conduction \(\longrightarrow\) Parabolic \(\longrightarrow\) \({\partial T \over \partial t} = \alpha {\partial^2 T \over \partial x^2}\)
Boundary Layer \(\longrightarrow\) Parabolic \(\longrightarrow\) \(\rho \left( {u {\partial u \over \partial x} + v {\partial u \over \partial y}} \right) = \mu {\partial^2 u \over \partial y^2}\)