Consider the following 2D linear PDE: $$ a_1 \frac{\partial^2 \phi}{\partial x^2} + 2 a_2 \frac{\partial^2 \phi}{\partial x \partial y} + a_3 \frac{\partial^2 \phi}{\partial y^2} + a_4 \frac{\partial \phi}{\partial x} + a_5 \frac{\partial \phi}{\partial y} + a_6 \phi = b $$ where the coefficients $a_1,\dots,a_6,b$ are all functions of $x,y$ (i.e. this is the most general second-order 2D linear PDE). A lot of books on PDEs seem to make a big deal about what type the equation is, i.e. elliptic, hyperbolic, or parabolic. For example an elliptic PDE is defined as one in which $$ a_2^2 - a_1 a_3 < 0 $$ From the point of view of actually wanting to solve the equation numerically (given appropriate boundary conditions), I don't see why knowing the type of equation it is has any relevance.

I would use a 2nd order Taylor series expansion to discretize all the derivatives into a finite difference approximation $$ A \boldsymbol{\phi} = \mathbf{b} $$ and then just solve the resulting sparse matrix system. It seems to me this would work just fine no matter what the value of $a_2^2 - a_1 a_3$ is. I suspect a similar statement could be made if using Finite Elements or Finite Volume methods. So my question is why do people care so much about the type of PDE when a numerical solution is desired?

A related question is related to Multigrid methods. Multigrid methods seem to be applied primarily to elliptic PDEs. But for the example above, if I choose appropriate mesh sizes, why can't I apply the same multigrid method regardless of the value of $a_2^2 - a_1 a_3$? Do multigrid methods work only when this discriminent is negative for some reason?

  • $\begingroup$ “and then just solve the resulting sparse matrix system” The problem here lies in the use of the word ’just’. If it was 'just' to solve such a linear system then everything would be easy. But it’s not always that easy (in terms of speed and accuracy / stability). Thats why we have so many different methods to choose from and the ‘best’ method to use for a particular equation is often related to it's type. $\endgroup$ – Winther Aug 6 '18 at 2:50
  • $\begingroup$ Since you are dealing with numerics, a PDE can be discretized in many ways. Different ways of discretizing a PDE lead to favoring some properties of the PDE over others. As well, there are issues of conditioning, error propagation, stability etc. that exist for all numerical methods. Hence, studying types of PDE often helps in understanding what properties to expect and construct methods specifically customized for conservation of those. Finite Volume Method for example is favorable in advection problems, Finite Element Method are favorable in energy-conservative PDEs etc... $\endgroup$ – Snifkes Aug 9 '18 at 13:37

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