pde by changcheng2


									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
• 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.
  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

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.

                  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

            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.
            Example: Poisson Equation in 2D

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

             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.
                    Finite Differences
• Assume we have a uniform mesh with a point x in the
   – 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).

           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.

      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).

                  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.

              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

                  Parallel Computation

• Serious calculations today are mostly done on a parallel
• 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.

Example: Original Mesh

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

Example: Mesh on Two Processors

                • Dividing into two connected
                  subsets and renumber within the
                • 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.

                     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
   • …

                   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

                    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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