Exercise · ZW-RUNGE-KUTTA-METHODS-010 · ★★★ Advanced
For the midpoint method with \(y'=y\), \(y(0)=1\), \(h=0.2\), what is \(y_1\)? Use \(k_1=f(t_0,y_0)\), \(k_2=f(t_0+h/2,\ y_0+hk_1/2)\), \(y_1=y_0+hk_2\).
Choose one option.
Show answer
- B. \(1.22\)
Explanation
Here \(k_1=1\), \(k_2=1.1\) and \(y_1=1+0.2\cdot 1.1=1.22\); the value \(1.2\) corresponds to using Euler instead of the midpoint method.
Review the concept: Runge-Kutta methods
Runge-Kutta methods are families of one-step numerical schemes that approximate the solution of an initial value problem by evaluating the function \(f\) at several intermediate points within each step.
\[y_{n+1}=y_n+h\sum_{i=1}^{s} b_i k_i\]
More practice problems: runge-kutta methods
- ZW-RUNGE-KUTTA-METHODS-001 How many stages does the classical RK4 method have?
- ZW-RUNGE-KUTTA-METHODS-002 Which Runge-Kutta method coincides with the explicit Euler method?
- ZW-RUNGE-KUTTA-METHODS-003 What global error order does the RK4 method have?
- ZW-RUNGE-KUTTA-METHODS-004 In an explicit Runge-Kutta method, what shape does the coefficient matrix a_ij have?
- ZW-RUNGE-KUTTA-METHODS-005 What is the value of c_1 in any explicit Runge-Kutta method?
- ZW-RUNGE-KUTTA-METHODS-006 For y'=2y, y(0)=1 and h=0.5, what is y_1 using Euler's method?
- ZW-RUNGE-KUTTA-METHODS-007 A Runge-Kutta method has c_2=1/2, a_21=1/2, b_1=0, b_2=1. What is b_1+b_2?
- ZW-RUNGE-KUTTA-METHODS-008 With y'=y, y(0)=1 and h=0.2, what is the slope k_1 in the first step of any explicit Runge-Kutta me…