MATH 685/ CSI 700/ OR 682 Lecture Notes

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MATH 685/ CSI 700/ OR 682 Lecture Notes

• Lecture 12.
• Partial differential equations.

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Partial differential equations
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Partial differential equations
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Advection equation
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Example
Typical solution of advection equation, with
initial function advected (shifted) over time
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Characteristics
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Classification of PDEs
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Classification of PDEs
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Classification of PDEs, cont.
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Time-dependent problems
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Semidiscrete methods
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Semidiscrete finite difference
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Methods of lines
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Stiffness
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Semidiscrete collocation method
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Semidiscrete collocation
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Semidiscrete collocation
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Semidiscrete collocation
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Fully discrete methods
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Fully discrete methods
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Fully discrete methods
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Example heat equation
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Heat equation, cont.
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Heat equation, cont.
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Example wave equation
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Example wave equation
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Wave equation
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Stability
Severe restriction on time step can make explicit
methods relatively inefficient
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Implicit finite difference methods
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Implicit finite difference
Method is unconditionally stable and second-order
accurate in time
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Implicit finite difference
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Convergence
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Stability
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CFL condition
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Wave equation
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Wave equation
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Time-dependent problems
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Finite difference methods
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Example Laplace equation
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Laplace equation, cont.
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Laplace equation, cont.
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Laplace equation, cont.
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Laplace equation, cont.
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Finite element methods
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Finite element methods