MATH 685/ CSI 700/ OR 682
Lecture Notes
Lecture 12.
Partial differential equations.
Partial differential equations
Partial differential equations
Advection equation
Example
Typical solution of advection equation, with initial function “advected”
(shifted) over time
Characteristics
Classification of PDEs
Classification of PDEs
Classification of PDEs, cont.
Time-dependent problems
Semidiscrete methods
Semidiscrete finite difference
Methods of lines
Stiffness
Semidiscrete collocation
method
Semidiscrete collocation
Semidiscrete collocation
Semidiscrete collocation
Fully discrete methods
Fully discrete methods
Fully discrete methods
Example: heat equation
Heat equation, cont.
Heat equation, cont.
Example: wave equation
Example: wave equation
Wave equation
Stability
Severe restriction on time step can make explicit methods
relatively inefficient
Implicit finite difference
methods
Implicit finite difference
Method is unconditionally
stable and second-order accurate
in time
Implicit finite difference
Convergence
Stability
CFL condition
Wave equation
Wave equation
Time-dependent problems
Finite difference methods
Example: Laplace equation
Laplace equation, cont.
Laplace equation, cont.
Laplace equation, cont.
Laplace equation, cont.
Finite element methods
Finite element methods