Bogacki–Shampine method
The Bogacki–Shampine method is a method for the numerical solution of ordinary differential equations, that was proposed by Przemysław Bogacki and Lawrence F. Shampine in 1989 . The Bogacki–Shampine method is a Runge–Kutta method of order three with four stages with the First Same As Last (FSAL) property, so that it uses approximately three function evaluations per step. It has an embedded second-order method which can be used to implement adaptive step size. The Bogacki–Shampine method is implemented in the codice_1 for fixed step solver and codice_2 for a variable step solver function  in MATLAB . Low-order methods are more suitable than higher-order methods like the Dormand–Prince method of order five, if only a crude approximation to the solution is required. Bogacki and Shampine argue that their method outperforms other third-order methods with an embedded method of order two. The Butcher tableau for the Bogacki–Shampine method is: Following the standard notation, the differential equation to be solved is formula_1. Furthermore, formula_2 denotes the numerical solution at time formula_3 and formula_4 is the step size, defined by formula_5. Then, one step of the Bogacki–Shampine method is given by: Here , formula_7 is a second-order approximation to the exact solution. The method for calculating formula_8 is due to . On the other hand, formula_8 is a third-order approximation, so the difference between formula_8 and formula_7 can be used to adapt the step size. The FSAL—first same as last—property is that the stage value formula_12 in one step equals formula_13 in the next step; thus, only three function evaluations are needed per step.