Numerical Solutions to Partial

1
Ch 12.
Numerical Solutions to Partial Differential
Equations
Applied mathematics.
Korea University.
2
Index
Ch12.1
Elliptic Partial Differential Equations
Ch 12.2
Parabolic Partial Differential Equations
Ch 12.3
Hyperbolic Partial Differential Equations
Ch 12.4
An Introduction to the Finite-Element Mothod
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• Elliptic Partial Differential Equations
  poisson equation

• Parabolic Partial Differential Equations Heat,
  diffusion equation

• Hyperbolic Partial Differential Equations wave
  equation

4
Elliptic Partial Differential Eqeations
Ch 12.1

• Poisson equation

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Poisson equation at the points
Boundary condition
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Finite Difference method with truncation error
of order
Boundary condition
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Parabolic Partial Differential Eqeations
Ch 12.2

• Parabolic partial differential equation

boundary condition
initial condition
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Using Taylor series in
Using Taylor series in
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local truncation error
initial condition
boundary condition
12
let
initial condition
Forward Difference method
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If
is made in representing the initial data
At n-th time step the error in is
.
The method is stable
The Forward Difference method is therefore stable
only if
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eigenvalues of A
or
The Forward Difference method is conditionally
stable with rate of convergence
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To obtain a method that is unconditionally stable
Backward-Difference method
where
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The matrix representation
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eigenvalues of A
At n-th time step the error in is
.
The Backward-Difference method is unconditionally
stable method.
The local truncation error for method is of order
.
Richardsons method
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Crank-Nicolson method
Forward-Difference method at j-th step in
local truncation error
Backward-Difference method at (j1)th step in
local truncation error
Assume that
average difference method
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The matrix representation
where
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Hyperbolic Partial Differential Eqeations
Ch 12.3

• Hyperbolic partial differential equation

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Using centered-difference quotient
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An Introduction to Finite-Element Method
Ch 12.4
boundary condition
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Polynomials of linear type in and
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linear system
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