Type of pdes
 

I read the classifications of PDES form Tannehill CFD book. But I couldn’t understand the physical concept of elliptic, parabolic and Hyperbolic equations; From mathematical point of view, it is simple to classify the PDES but I want to know how the type of PDES(parabolic, Hyperbolic or Elliptic) would affect the fluid behavior.

I would be too pleased if anyone can help me.

I would be most grateful if I can find the answer of my question as soon as possible.

Dear Ashkan,

The general rule with regard to the physical behaviour of problems governed by the various classification of PDEs can be summarised below.

1. Elliptic PDEs: Steady state diffusion Eg: Laplace and Poisson equations are elliptic, a standard problem is steady state heat conduction. A more common example is steady incompressible flows.

2. Parabolic PDEs: These represent time dependent diffusive phenomena. Unsteady heat conduction is parabolic and is a balance between the unsteady term and the diffusive term.

3. Hyperbolic PDEs: Of most interest possibly to most CFD people, these represent transoport phenomena where there is a definite speed and direction of propogation of information. A typical example is the wave equation or the linear convective equation, which simply represents the convection of wave(s) over time at some finite speeds.

These are reflected through the matehmatical description of these equations via. the characteristics, domain of dependence and domain of influence. While elliptic PDEs do not have any real characteristics (domain of dependence and influence are the entire domain itself), parabolic PDEs have a time-like characteristic. Hyperbolic PDEs have real and distinct characrteristics and well-defined domains of dependence and influence, which is a effectively exploited in the design of numerical algorithms for fluid flows.

Hope this helps.

Regards,

Ganesh