ChE306: Heat and Mass Transfer

1
Chapter 6
Numerical Solution of Partial Differential
Equations
2
Partial Differential Equations

• General form
• linear
• quasilinear
• nonlinear

only a function of independent variables
function of dependent variable, including
derivatives of lower order
coefficients are functions of derivatives of
same order as PDE
3
Classification of 2nd Order PDEs
homogeneous
elliptic
general canonical forms
parabolic
method of solution dictated by canonical form of
PDE
hyperbolic
4
Boundary Conditions

• Dirichlet (first kind)
• Neumann (second kind)
• Robbins (third kind)

values of dependent variable given at fixed
values of independent variable
Derivative of dependent variable given as a
constant or a function of independent variable
Derivative of dependent variable given as
function of dependent variable
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Representative PDEs
6
Representative PDEs
7
Representative PDEs
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Finite Difference Solution of PDEs

• Require 2D and 3D grids in 2 and 3 independent
  variables, respectively

analogous treatment of the terms
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Finite Difference Solution of PDEs
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Finite Difference Solution of PDEs

• For mixed partials,

Tables in text present summaries of results for
?, ?, and ? differencesfor 1st and 2nd order
PDEs for truncation of all but first order terms.
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Finite Difference Solution of PDEs
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Finite Difference Solution of PDEs
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Finite Difference Solution of PDEs
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Elliptic PDEs

• Found in description of steady-state conduction
  and diffusion problems

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Elliptic PDEs

• Consider the 2D elliptic PDE (Laplace's eqn)
• Replace 2nd order derivatives with finite
  differences
• Rearrange to obtain linear algebraic eqn in terms
  of dependent variable at five adjacent grid points

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Elliptic PDEs
y

• Total grid points
• Internal grid points
• Simultaneous linearalgebraic equations
• unknowns

q segments
x
p segments
corner points don't appear in equations
Boundary conditions provide missing information
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Elliptic PDEs
Examples of boundary condition treatment Use
FFD to avoid problem (eqn with same order of
accuracy)
Neumann
ficticious point outside of solution bounds
Robbins
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Elliptic PDEs
The difference equation for the PDE at each
internal node with boundary conditions are a set
of linear algebraic equations. Therefore, Gauss
methods may be used to solve. Converge using
Gauss-Seidelrelaxation method
Solve explicitly for uij
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Elliptic PDEs
Poisson Equation (nonhomogenous Laplace)
substitute CFD eqns
solve explicitly for uij
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Elliptic PDEs
Poisson Equation Neumann or Robbins BCs
forward
lower x
upper x
backward
lower y
forward
upper y
backward
N node on x or y 0 or L
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Elliptic PDEs
Example 6.1 elliptic.m
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Parabolic PDEs

• Unsteady state heat conduction diffusion
• Explicit methods (central differences)
• Combine, rearrange
• Application is straightforward given IC and 2 BCs
• Explicit formula unstable (particularly if
  negative terms on RHS)

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Parabolic PDEs

• Positivity Rule
• If A, B, and C are gt 0, AND ABC?1 ? stable
• Replace 1st order derivative with forward
  difference
• For a stable solution,

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Parabolic PDEs

• 1D
• 2D

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Parabolic PDEs

• Positivity rule dictates stability condition

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Parabolic PDEs

• Implicit Methods
• Second order term at ½ point expressed as
  weighted average of central differences at
  (i,n1) and (i,n)

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Parabolic PDEs

• Variable-weighted implicit parabolic PDE
  approximation
• ? 0 classic explicit formula
• ? 1 backward implicit approximation
• Can also be obtained by approximating as 1st
  order partial with backward FD at (i,n1) or 2nd
  order partial with central FD at (i,n1)

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Parabolic PDEs

• ? ½ Crank-Nicolson implicit formula

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Parabolic PDEs

• nonhomogeneous 1D parabolic PDE,
• Also need to evaluate f, assumed to be an
  average
• Crank-Nicolson implicit for nonhomogenous
  parabolic PDEs

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Parabolic PDEs

• Writing Crank-Nicolson for entire grid
• Simultaneous linear algebraic equations
• Matrix of coefficients usually tridiagonal
• Solved by a Gauss elimination method
• Implicit formula are unconditionally stable
• Explicit formula are conditionally stable, but
  computationally much easier to solve
• Solution Methodology
• Method of lines converts PDE into a set of ODEs
  by discretizing only the spatial derivatives with
  finite differences, with i 0 and i N
  equations modified to fit boundary conditions.

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Hyperbolic PDEs

• 2D, nonhomogeneous
• For 1D homogeneous case, expand derivatives with
  CFDs
• Rearrange (explicit solution) positivity
  rule

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Hyperbolic PDEs

• 2D, homogeneous
• Stable for

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Hyperbolic PDEs

• Implicit methods for hyperbolic equations by
  variable-weight

explicit method Crank-Nicolson-type algorithm
Method leads to a tri-diagonal matrix of
coefficients forwhich a Gauss Elimination method
can be used to solve
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Irregular Boundaries

• In the x-direction
• Eliminate 2nd order term by substitution
• Eliminate 1st order term

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Irregular Boundaries

• Repeat process for y-direction

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Irregular Boundaries

• Specification of Neumann orRobbins boundary
  conditions
• where n is the unit vector normalto the
  boundary, and ? is the angle between n and the
  x-axis
• Expand partials on RHS of equal sign in a Taylor
  series of i,j derivatives, and use difference
  equations on i,j derivatives (backward in this
  example). Substitute into expression above.

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Polar Coordinates

• Laplacian operator in Polar Coordinates