Linear multi-step methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multi-step methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multi-step methods refer to several previous points and derivative values. In the case of linear multi-step methods, a linear combination of the previous points and derivative values is used.
Here, integration of the normalized two-body problem from t0 = 0 to t = 86400(s) for an eccentricity of e = 0.1 is implemented by Shampine-Gordon (variable-step, variable-order multi-step integrator) and compared with MATLAB’s ode113 (variable order Adams-Bashforth-Moulton PECE solver).
Meysam Mahooti (2021). Shampine-Gordon Integrator (https://www.mathworks.com/matlabcentral/fileexchange/74570-shampine-gordon-integrator), MATLAB Central File Exchange. Retrieved .
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