Partial Differential Equation.ppt

7/27/2019 Partial Differential Equation.ppt

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Definition of a PDE and Notation

A PDE is an equation with derivatives of at least twovariables in it.

Let u be a function of x and y. There are several ways towrite a PDE, e.g.,

The equations above are linear and first order. The order isdetermined 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. Mostinteresting problems are nonlinear and time dependent.

yuxuuu yx

//


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Characterization of Simple Second Order PDEs

Let

Then the type of PDE is determined by the discriminant

< 0 elliptic

= 0parabolic

> 0 hyperbolic

gfueuducubuau yxyyxyxx 2

acb 2


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Characterization of n Variable Second Order PDEs

A general linear PDE of order 2:

Assume symmetry in coefficients so that A = [ aij ] issymmetric. 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.

.11, dcuubua iji xin

ixxij

n

ji


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PDE Model Problems

Laplaces Equation (elliptic):

Heat Equation (parabolic):

Wave Equations (hyperbolic):

All problems can be mapped to one of these! in theory

0

0

0

0

yyxxtt

yxt

yyxxt

yyxx

uuu

uuu

uuu

uu


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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.


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Example: Poisson Equation in 2D

uxxuyy = 1 in (0,1)2 ; u = 0 on (0,1)2 .


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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.


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Finite Differences

Assume we have a uniform mesh with a point x in theinterior..

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).


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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 samematrix problem. Choosing the elements differently can

lead to a 9 point operatorinstead.


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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.

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).

)(

).(

5.)(

),(),( 1

ii

jiij

iiijiijji

jj

n

j

fIntb

deldelaIntA

VbVVAVI

yxVyxV


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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.


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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


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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.


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Example: Original Mesh

Consider solving a problem on the

given grid. Assume that only half

of the nodes fit on a processor.


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Dividing into two connected

subsets and renumber within the

subdomains.

Communication occurs betweenneighbors that cross the

processor boundary.

Ghost points (or overlap) can

reduce communication

sometimes at the expense ofextra computation.

Computation is o(1/1000)

communication per word.

Example: Mesh on Two Processors


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Mesh Decomposition

Goals are to maximize interior while minimizing

connections between subdomains. Critical parameter:

minimize communication.

Such decomposition problems have been studied in loadbalancing for parallel computation.

Lots of choices:

METIS package from the University of Minnesota.

PARTI package from the University of Maryland


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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 spectralexpansion.


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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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Partial Differential Equation.ppt