Numerical solution of PDEs

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Consider the following 2D linear PDE:
$$
a_1 fracpartial^2 phipartial x^2 + 2 a_2 fracpartial^2 phipartial x partial y + a_3 fracpartial^2 phipartial y^2 + a_4 fracpartial phipartial x + a_5 fracpartial phipartial y + a_6 phi = b
$$
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
$$
A boldsymbolphi = mathbfb
$$
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?




pde numerical-methods




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    Consider the following 2D linear PDE:
    $$
    a_1 fracpartial^2 phipartial x^2 + 2 a_2 fracpartial^2 phipartial x partial y + a_3 fracpartial^2 phipartial y^2 + a_4 fracpartial phipartial x + a_5 fracpartial phipartial y + a_6 phi = b
    $$
    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
    $$
    A boldsymbolphi = mathbfb
    $$
    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?




    pde numerical-methods




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      Consider the following 2D linear PDE:
      $$
      a_1 fracpartial^2 phipartial x^2 + 2 a_2 fracpartial^2 phipartial x partial y + a_3 fracpartial^2 phipartial y^2 + a_4 fracpartial phipartial x + a_5 fracpartial phipartial y + a_6 phi = b
      $$
      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
      $$
      A boldsymbolphi = mathbfb
      $$
      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?




      pde numerical-methods




      share|cite|improve this question













      Consider the following 2D linear PDE:
      $$
      a_1 fracpartial^2 phipartial x^2 + 2 a_2 fracpartial^2 phipartial x partial y + a_3 fracpartial^2 phipartial y^2 + a_4 fracpartial phipartial x + a_5 fracpartial phipartial y + a_6 phi = b
      $$
      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
      $$
      A boldsymbolphi = mathbfb
      $$
      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?





      pde numerical-methods





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