Heun's method

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In mathematics and computational science, Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule), or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Both variants can be seen as extensions of the Euler method into two-stage second-order Runge–Kutta methods. The procedure for calculating the numerical solution to the initial value problem: y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , {\displaystyle y'(t)=f(t,y(t)),\qquad \qquad y(t_{0})=y_{0},} by way of Heun's method, is to first calculate the intermediate value y ~ i + 1 {\displaystyle {\tilde {y}}_{i+1}} and then the final approximation y i + 1 {\displaystyle y_{i+1}} at the next integration point. y ~ i + 1 = y i + h f ( t i , y i ) {\displaystyle {\tilde {y}}_{i+1}=y_{i}+hf(t_{i},y_{i})} y i + 1 = y i + h 2 [ f ( t i , y i ) + f ( t i + 1 , y ~ i + 1 ) ] , {\displaystyle y_{i+1}=y_{i}+{\frac {h}{2}}[f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1})],} where h {\displaystyle h} is the step size and t i + 1 = t i + h {\displaystyle t_{i+1}=t_{i}+h} .