4.4. Implicit Multistep Methods#
Implicit (closed) methods are derived by using the additional point \(\,(x_{j+1},y_{j+1})\,\) as an interpolation node in the approximation of the integral \(\displaystyle \int_{x_j}^{x_{j+1}} f(x,y) \dx\). The simplest of the implicit methods is the Backward Euler method:
with the local truncation error: \(\displaystyle \tau_{j+1} = -\frac{1}{2}y^{(2)}(z_j)h^2, ~\ x_j < z_j < x_{j+1}\).
Some of the more common implicit methods are listed below:
Adams–Moulton One–Step Method, \(k=1\)
\(\displaystyle y_{j+1} = y_j + \frac{h}{2}\bigl[\,f_{j+1} + f_j\,\bigr]\) (this is the Trapezoidal method)
with the local truncation error: \(\displaystyle \tau_{j+1} = -\frac{1}{12}y^{(3)}(z_j)h^3, ~\ x_j < z_j < x_{j+1}\).Adams–Moulton Two–Step Method, \(k=2\)
\(\displaystyle y_{j+1} = y_j + \frac{h}{12}\bigl[\,5f_{j+1} + 8f_j - f_{j-1}\,\bigr]\)
with the local truncation error: \(\displaystyle \tau_{j+1} = -\frac{1}{24}y^{(4)}(z_j)h^4, ~\ x_j < z_j < x_{j+1}\).Adams–Moulton Three–Step Method, \(k=3\)
\(\displaystyle y_{j+1} = y_j + \frac{h}{24}\bigl[\,9f_{j+1} + 19f_j - 5f_{j-1} + f_{j-2}\,\bigr]\)
with the local truncation error: \(\displaystyle \tau_{j+1} = -\frac{19}{720}y^{(5)}(z_j)h^5, ~\ x_j < z_j < x_{j+1}\).Adams–Moulton Four–Step Method, \(k=4\)
\(\displaystyle y_{j+1} = y_j + \frac{h}{720}\bigl[\,251f_{j+1} + 646f_j - 264f_{j-1} + 106f_{j-2} - 19f_{j-3}\,\bigr]\)
with the local truncation error: \(\displaystyle \tau_{j+1} = -\frac{3}{160}y^{(6)}(z_j)h^6, ~\ x_j < z_j < x_{j+1}\).
It is important to compare an \(m\)–step Adams–Bashforth open method to an \((m\!-\!1)\)–step Adams–Moulton closed method. Both require \(m\) evaluations of \(f\) per step, and both have the terms \(\,y^{m\!+\!1}(z)\,h^{m\!+\!1}\,\) in their local truncation errors. In general, the coefficients of the terms involving \(f\) and those in the local truncation error are smaller for the Adams–Moulton methods. This means a greater stability for the closed methods and smaller rounding errors.