The problem with the one-step method is that it only considers one previous point when calculating the next step. The process continues to reach the solution but a multistep method takes into account multiple previous steps when reaching the solution thus gaining efficiency.
Some of the popular multi-step methods are Adam-bashforth, Adams-Moulton methods, backward differentiation formula, etc.
One thing to note is that in the case of the Runga-Kutta 4 method we were focusing on dividing the interval between two points into multiple intervals and thus attaining higher order equations. What if there was a way to solve for ODE considering the previous points as well.
Consider the Picards method form. If you were to assume the function under integration to be a constant we reach Euler’s method and if you are to consider a linear polynomial of previous points you surprisingly reach to Adams-Bashforth method.
Similarly, if you increase the order of the polynomial that you consider to can develop different methods. When generating this method you will be stuck where you do not no the value of the function at x1 and an initial guess of y1.
These can easily be solved using the RK-2 method or any other method. Since we are using polynomials to get the value of the integral we get the same issue as with approximating polynomials.
Since extrapolation is more error-prone we use an implicit method that is Adams-Moulton method. This method takes into considering the use of the Adams-Bashforth and RK-4 method to calculate the initial unknown points.
Thus, we are left with a three-step method. This gives a more efficient and accurate answer. Since we were getting an efficient answer using the AB-2 or AB-4 method but it was not that accurate comparatively.
For Discussion Comment below or Go to the Next Chapter System of linear equations