Content deleted Content added
Jitse Niesen (talk | contribs) →Multistep Method Families: add very short section on BDF |
Jitse Niesen (talk | contribs) m section headings |
||
Line 27:
The exact solution is <math> y(t) = \mathrm{e}^t </math>.
===One-
A simple numerical method is Euler's method:
: <math> y_{n+1} = y_n + hf(t_n, y_n). \, </math>
Line 40:
\end{align} </math>
===Two-
Euler's method is a one-step method. A simple multistep method is the two-step Adams–Bashforth method
: <math> y_{n+2} = y_{n+1} + \tfrac32 hf(t_{n+1},y_{n+1}) - \tfrac12 hf(t_n,y_n). </math>
Line 55:
=== Adams–Bashforth methods ===
The Adams–Bashforth methods are explicit methods. The coefficients are <math>a_{s-1}=-1</math> and <math>a_{s-2} = \cdots = a_0 = 0</math>, while the <math>b_j</math> are chosen such that the methods has order ''s'' (this determines the methods uniquely).
Line 101 ⟶ 100:
The central concepts in the analysis of linear multistep methods, and indeed any numerical method for differential equations, are [[Numerical ordinary differential equations#Analysis|convergence, order, and stability]].
=== Consistency and order ===
The first question is whether the method is consistent: is the difference equation
:<math> \begin{align}
Line 120:
In particular, the method is consistent if it has order one, which is the case if <math>\rho(1)=0</math> and <math>\rho'(1)=\sigma(1)</math>.
=== Stability and convergence ===
If the roots of the characteristic polynomial <math>\rho</math> all have modulus less than or equal to 1 and the roots of modulus 1 are of multiplicity 1, we say that the [[root condition]] is satisfied.
The method is convergent [[if and only if]] it is consistent and the root condition is satisfied. Consequently, a consistent method is stable if and only if this condition is satisfied, and thus the method is convergent if and only if it is stable.
| |||