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					Fun and Games with Partial
  Differential Equations


        Craig Douglas
           CS 521
        April 11, 2000
           Definition of a PDE and Notation
• A PDE is an equation with derivatives of at least two
  variables in it.
• Let u be a function of x and y. There are several ways to
  write a PDE, e.g.,
        ux  uy
         u / x  u / y
• The equations above are linear and first order. The order is
  determined by the maximum number of derivatives of any
  term.
• A nonlinear PDE has the solution times a partial derivative
  or a partial derivative raised to some power in it. Most
  interesting problems are nonlinear and time dependent.
                                                              2
  Characterization of Simple Second Order PDE’s

• Let
    auxx  2buxy  cu yy  dux  eu y  fu  g
• Then the type of PDE is determined by the discriminant
     b 2  ac
   – < 0 elliptic
   – = 0 parabolic
   – > 0 hyperbolic



                                                           3
Characterization of n Variable Second Order PDE’s

• A general linear PDE of order 2:
       in, j 1 aij uxi x j  in1 biuxi  cu  d .

• Assume symmetry in coefficients so that A = [ aij ] is
  symmetric. Eig(A) are real. Let P and Z denote the
  number of positive and zero eigenvalues of A.
   – Elliptic: Z = 0 and P = n or Z = 0 and P = 0..
   – Parabolic: Z > 0 (det(A) = 0).
   – Hyperbolic: Z=0 and P = 1 or Z = 0 and P = n-1.
   – Ultra hyperbolic: Z = 0 and 1 < P < n-1.

                                                           4
                  PDE Model Problems

• Laplace’s Equation (elliptic):    u xx  u yy  0


• Heat Equation (parabolic):       ut  u xx  u yy  0


• Wave Equations (hyperbolic):     ut  u x  u y  0
                                   utt  u xx  u yy  0
• All problems can be mapped to one of these!… in theory

                                                           5
            Boundary and Initial Conditions,
                Well and Ill Posedness

• Boundary conditions on G  GD U GN U GR.
   – Dirichlet: u = g on GD.
   – Neumann: un = g on GN.
   – Robin: au + b un = g on GR.
• Initial conditions at t=0.
   – U(t=0,x,y) = u0(x,y).
• Well posed PDE if and only if
   – A solution to the problem exists.
   – The solution is unique.
   – The solution depends continuously on the problem data.
• Ill posed if not well posed.
                                                              6
            Example: Poisson Equation in 2D


• – uxx – uyy = 1 in (0,1)2 ; u = 0 on (0,1)2 .




                                                   7
             Finite Whosiwhatsit Methods

• There are three common methods of producing a finite
  dimensional problem whose solution can be computed,
  which approximates the solution of the original, infinite
  dimensional problem:
   – Finite elements
   – Finite differences
   – Finite volumes
• Each has its place, supporters, and detractors.
• There are also other methods, e.g., collocation, spectral
  methods, pseudo-blah-blah-blah methods, etc.
                                                              8
                    Finite Differences
• Assume we have a uniform mesh with a point x in the
  interior..
   – Forward difference: D+h u(x) = u(x+h) – u(x).
   – Backward difference: D-h u(x) = u(x) – u(x-h).
   – Central difference: x u(x) = u(x+h/2) – u(x-h/2) or
       x2 u(x) = u(x+h) – 2u(x) + u(x-h).
• Taylor Series and Truncation Error
   – Look at the difference between the approximation and
      the Taylor series. When they do not match, there is a
      remainder, which is known as the truncation error. It is
      usually specified as O(hp).

                                                             9
           Poisson Equation Example, Again

• The Poisson equation example used central differences to
  solve a block matrix problem of the form
     A = [-I, T, -I ],
  where I is the nxn identity matrix and T is a nxn
  tridiagonal matrix [ -1, 4, -1 ]. There are n rows of blocks
  in A (i.e., A is n2xn2). This is known as a 5 point operator.
• Choosing the right finite element method on a square (right
  triangles with piecewise linear elements) leads to the same
  matrix problem. Choosing the elements differently can
  lead to a 9 point operator instead.

                                                             10
      Finite Elements (Variational Formulations)

• Find u in test space H such that a(u,v) = f(v) for all v in H,
  where a is a bilinear form and f is a linear functional.
        V ( x, y )   nj 1 V j j ( x, y )
         I (V )  .5 i  j AijViV j   i biVi
         Aij  Int (a deli . del j )
         bi  Int ( fi )
• The coefficients Vj are computed and the function V(x,y)
  is evaluated anyplace that a value is needed.
• The basis functions should have local support (i.e., have a
  limited area where they are nonzero).

                                                                11
                  Matrix Free Methods

• Many problems have simple matrices associated with the
  linear algebra (e.g., the Poisson equation example).
• By using methods (e.g., Krylov space or relaxation
  methods) that only multiply the matrix A times a vector x,
  code to calculate y=Ax can be written instead of storing
  the matrix A.
• This reduces the cost of the computer (which is mostly
  memory chips) and allows for vastly larger simulations.



                                                           12
              Time Stepping Methods

• Standard methods are common:
   – Forward Euler (explicit)
   – Backward Euler (implicit)
   – Crank-Nicolson (implicit)
• Variable length time stepping
   – Most common in Method of Lines (MOL) codes or
     Differential Algebraic Equation (DAE) solvers




                                                     13
                  Parallel Computation

• Serious calculations today are mostly done on a parallel
  computer.
• The domain is partitioned into subdomains that may or
  may not overlap slightly.
• Goal is to calculate as many things in parallel as possible
  even if some things have to be calculated on several
  processors in order to avoid communication.
• Communication is the Darth Vader of parallel computing.



                                                                14
Example: Original Mesh


           Consider solving a problem on the
           given grid. Assume that only half
           of the nodes fit on a processor.




                                               15
Example: Mesh on Two Processors


                • Dividing into two connected
                  subsets and renumber within the
                  subdomains.
                • Communication occurs between
                  neighbors that cross the
                  processor boundary.
                • Ghost points (or overlap) can
                  reduce communication
                  sometimes at the expense of
                  extra computation.
                • Computation is o(1/1000)
                  communication per word.

                                               16
                     Mesh Decomposition


• Goals are to maximize interior while minimizing
  connections between subdomains. Critical parameter:
  minimize communication.
• Such decomposition problems have been studied in load
  balancing for parallel computation.
• Lots of choices:
   • METIS package from the University of Minnesota.
   • PARTI package from the University of Maryland
   • …

                                                          17
                   Benchmarking: Speedup
• Speedup for 5 layer SEOM. Dashed lines for large Pacific simulation
  (3552 elements) and the solid lines are for the small Atlantic Basin
  simulation (792 elements). Both simulations use 7th order spectral
  expansion.




                                                                     18
                    Benchmarking: Timing
• Timings versus processors for 5 layer SEOM. Dashed lines for large
  Pacific simulation (3552 elements) and the solid lines are for the small
  Atlantic Basin simulation (792 elements). Both simulations use 7th
  order spectral expansion.




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posted:1/16/2012
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