Multistep methods and predictor-corrector pairs

In contrast to the one-step metiiods, e.g. the Runge-Kutta method, where the solution at Xn+ is calculated from the solution at x , multistep methods calculate the solution from previous steps, e g. x , x -, x -2- In other words, multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Therefore, these methods are more efficient than the single-step Runge Kutta metiiods. 

Among the explicit multistep methods, the Adams-Bashforth methods are the most widely used. The second-order (global error) Adams-Bashforth two-step method is 

The fourth-order, four-step Adams-Bashforth method, 

Neither the Runge-Kutta nor tiie Adams Bashforth methods can handle stiff differential equations well. The Adams-Moulton method is an implicit multistep method that can handle stiff problems better (stiff problems are dicussed later in this chapter). The two-step Adams-Moulton method (third-order accurate) is 

The implicit multistep methods add stability but require more computation to evaluate the implicit part. In addition, the error coefficient of the Adams-Moulton method of order k is smaller than that of the Adams Bashforth method of the same order. As a consequence, the implicit methods should give improved accuracy. In fact, the error coefficient for the imphcit fourth-order Adams Moulton method is 19/720, and for the explicit fourth-order Adams Bashforth method it is 251/720. The difference is thus about an order of magnitude. Pairs of exphcit and implicit multistep methods of the same order are therefore often used as predictor-corrector pairs. In this case, the explicit method is used to calculate the solution,, at v +i. Furthermore, the imphcit method (corrector) uses y + to calculate /(x +i,y +i), which replaces /(x +i,y +i). This allows the solution, y +i, to be improved using the implicit method. The combination of the Adams Bashforth and the Adams Moulton methods as predictor orrector pairs is implemented in some ODE solvers. The Matlab odel 13 solver is an example of a variable-order Adams Bashforth Moulton multistep solver. 

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