OR682Math685CSI700

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OR682/Math685/CSI700

• Lecture 10
• Fall 2000

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Partial Differential Equations

• Partial differential equations (PDEs) involve
  partial derivatives with respect to more than one
  independent variable
• Typically 13 space dimensions (plus perhaps
  time)
• Can be initial-value problem, boundary-value
  problem, or combination
• May have an irregular spatial domain

Matlab functions pdetool
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PDEs more

• For simplicity, focus on a single PDE (not a
  system) with 2 independent variables either
• Two space variables x and y
• One space variable x and one time variable t
• Denote the solution by u with
• ux ?u/?x
• uxy ?2u/?x?y, etc.

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Classification of PDEs

• Classification based on the discriminant b2
  4ac of the general, linear, two-dimensional,
  second-order PDE
• Hyperbolic b2 4ac gt 0
• Parabolic b2 4ac 0
• Elliptic b2 4ac lt 0
• Classification more complex for nonlinear
  problems, problems with variable coefficients,
  and systems of equations

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Classification of PDEs (cont.)

• More intuitively
• Hyperbolic time-dependent processes (such as
  wave motion) that are not evolving to steady
  state
• Parabolic time-dependent processes (such as heat
  diffusion) that are evolving to steady state
• Elliptic processes that have already reached
  steady state, and are thus time independent

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Time-Dependent Problems

• Focus on simple model problems
• Heat equation ut c uxx (parabolic)
• u(0,x) f (x),
• u(t,0) 0, u(t,1) 0
• Mix of initial and boundary conditions
• Wave equation utt c uxx (hyperbolic)
• u(0,x) f (x), ut(0,x) g(x)
• u(t,0) 0, u(t,1) 0

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Semidiscrete Methods

• Discretize in space, but not in time
• Produces system of ODEs (initial value problem)
• Solve, for example, with ode15s

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Semidiscrete Methods (more)

• For example, use the approximation
• Then the heat equation becomes a system
• for i 1,,n where yi(t) u(t,i?x)
• From boundary conditions, y0 yn1 0 from
  initial conditions yi(0) f (xi)

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Stiffness, etc.

• This system has the form y? Ay for a
  tridiagonal matrix A with eigenvalues between
  4c/(?x)2 and 0. This system becomes stiff as ?x
  becomes small
• This semidiscrete approach is also called the
  method of lines
• A finite-element approach is also possible
  (stiffness also arises)

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Fully Discrete Methods

• Discretize in both space and time
• Continuous domain becomes a discrete mesh of
  points
• Replace derivatives by finite differences
• Numerical solution is a table of approximate
  values
• Accuracy depends on all step sizes
• Discretization produces an algebraic system of
  equations

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Consistency, Stability, Convergence

• For convergence as step sizes go to zero, two
  conditions must be met
• Consistency local truncation error goes to zero
  as step sizes go to zero (I.e., discrete problem
  approximates the continuous problem)
• Stability approximate solution remains bounded
• Consistency stability (together) are necessary
  and sufficient for convergence

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Hyperbolic vs. Parabolic

• Hyperbolic equations are conservative they
  propagate behavior undiminished (shocks, errors,
  etc.)
• Has implications for numerical methods (since
  algorithms are usually based on smooth
  approximations, such as polynomials or splines)
• Parabolic equations are dissipative solution
  tends toward steady state, damping out any lack
  of smoothness or consistency (more forgiving)

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Time-Independent Problems

• Consider the elliptic Helmholtz equation
• Important special cases
• Poissons equation (? 0)
• Laplaces equation (? 0 and f 0)
• Assume that the region is the unit square
• Types of boundary conditions
• Dirichlet specify u
• Neumann specify ux and/or uy

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Finite-Difference Methods

• Define a discrete mesh
• Replace derivatives by finite differences
• Results in a system of algebraic equations (no
  marching forward in time)
• Coefficient matrix is banded (for typical choices
  of finite difference approximations)
• Finite-element methods also possible

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For Next Class

• Homework see web site
• Reading
• Heath chapter 12