Chapter 4: Linear Multistep Methods Example: 3ed order Adams–Bashforth method Example: 2ed order Adams–Bashforth method Example: 2ed order backward difference.

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Presentation transcript:

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:

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 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

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

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

Sec 404: Consistency Linear Multistep Methods:

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. 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 402A

Sec 403: Stability Linear Multistep Methods: Theorem

Sec 403: Stability Linear Multistep Methods: Definition Example: 2ed order Adams–Bashforth method

Sec 403: Stability Linear Multistep Methods: Definition