5.6 Multistep Methods The methods discussed to this point in the chapter are called one-step methods because the approximation for the mesh point involves information from only one of the previous mesh points, . Although these methods might use functional evaluation information at points between and , they do not retain that information for direct use in future approximation. All the information used by these methods is obtained within the subinterval over which the solution is being approximated.

multistep method Method using the approximation at more than one previous mesh point to determine the approximation at the next point are called multistep method. The precise definition of these methods follow, together with the definition of the two types of multistep methods.

Definition 5.14 (5.22) (5.23)

multistep method When the method is called explicit, or open, since Eq. (5.23) then gives explicitly in terms of previously determined values. When the method is called implicit, or closed, since occurs on both sides of Eq. (5.23) and is specified only implicitly.

Example EXAMPLE The equations for each , define an explicit four-step method known as the fourth-order Adams-Bashforth technique . EXAMPLE The equations for each define an implicit three-step method known as the fourth-order Admas-Moulton technique.

local truncation error for multistep methods The local truncation error for multistep methods is defined analogously to that of one-step methods. As in the case of one-step methods, the local truncation error provides a measure of how the different equation fails to solve the difference equation.

Definition 5.15 (5.31)

Examples To determine the local truncation error for the three-step Adams-Bashforth method given as follows.

Examples Adams-Bashforth Two-Step Explicit Method: . where .The local truncation error is for some .

Examples Adams-Bashforth Four-step Explicit Method: . where .The local truncation error is for some .

Examples Adams-Moulton Two-step Implicit Method: . where .The local truncation error is for some .

Examples Adams-Moulton Three-step Implicit Method: . where .The local truncation error is for some .