Download presentation
Presentation is loading. Please wait.
1
Chapter 4: Linear Multistep Methods Example: 3ed order Adams–Bashforth method Example: 2ed order Adams–Bashforth method Example: 2ed order backward difference formula (BDF2) Linear Multistep Methods:
2
Chapter 4: Linear Multistep Methods Two Polynomials Linear Multistep Methods: we write the method as [α, β], where Two Polynomials Other textbook(ρ, σ), Example: (BDF2) Example: 2ed order Adams–Bashforth method
![Chapter 4: Linear Multistep Methods Two Polynomials Linear Multistep Methods: we write the method as [α, β], where Two Polynomials Other textbook(ρ, σ), Example: (BDF2) Example: 2ed order Adams–Bashforth method](http://images.slideplayer.com./36/10563812/slides/slide_2.jpg "Chapter 4: Linear Multistep Methods Two Polynomials Linear Multistep Methods: we write the method as [α, β], where Two Polynomials Other textbook(ρ, σ), Example: (BDF2) Example: 2ed order Adams–Bashforth method")
3
Chapter 4: Linear Multistep Methods Starting methods Linear Multistep Methods: linear multistep methods require starting methods even to carry out a single step. Example: 2ed order Adams–Bashforth method One obvious approach to starting a k-step method is to carry out k − 1 steps with a Runge–Kutta method, preferably of the same order as the linear multistep method itself. given approximate
4
Sec 404: Consistency Linear Multistep Methods: Definition 404A A linear multistep method satisfying is said to be ‘preconsistent’ Example: (BDF2) Example: 2ed order Adams–Bashforth method
5
Sec 404: Consistency Linear Multistep Methods: Definition 404A A linear multistep method satisfying is said to be ‘consistent’ Example: (BDF2) Example: 2ed order Adams–Bashforth method
6
Sec 404: Consistency Linear Multistep Methods:
7
Sec 403: Stability Linear Multistep Methods: Definition 403A A linear multistep method [α, β] is ‘stable’ if the difference equation has only bounded solutions. all solutions are bounded if and only if the polynomial has all its zeros in the closed unit disc and all multiple zeros in the interior of this disc.
![Sec 403: Stability Linear Multistep Methods: Definition 403A A linear multistep method [α, β] is ‘stable’ if the difference equation has only bounded solutions.](http://images.slideplayer.com./36/10563812/slides/slide_7.jpg "all solutions are bounded if and only if the polynomial has all its zeros in the closed unit disc and all multiple zeros in the interior of this disc..")
8
Sec 403: Stability Linear Multistep Methods: Definition 403A A linear multistep method [α, β] is ‘stable’ if the difference equation has only bounded solutions. all solutions are bounded if and only if the polynomial has all its zeros in the closed unit disc and all multiple zeros in the interior of this disc. Example: (BDF2) Example: 2ed order Adams–Bashforth method
![Sec 403: Stability Linear Multistep Methods: Definition 403A A linear multistep method [α, β] is ‘stable’ if the difference equation has only bounded solutions.](http://images.slideplayer.com./36/10563812/slides/slide_8.jpg "all solutions are bounded if and only if the polynomial has all its zeros in the closed unit disc and all multiple zeros in the interior of this disc. Example: (BDF2) Example: 2ed order Adams–Bashforth method.")
9
Sec 403: Stability Linear Multistep Methods: Definition 402A
10
Sec 403: Stability Linear Multistep Methods: Theorem
11
Sec 403: Stability Linear Multistep Methods: Definition Example: 2ed order Adams–Bashforth method
12
Sec 403: Stability Linear Multistep Methods: Definition
Similar presentations
