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

4.1.4.2. What common flow conditions are hyperbolic, elliptic and parabolic?

  • 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

4.1.4.3. What are some examples of hyperbolic, elliptic and parabolic PDEs?

  • 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 \(\longrightarrow\) \(\rho \left( {u {\partial u \over \partial x} + v {\partial u \over \partial y}} \right) = \mu {\partial^2 u \over \partial y^2}\)