Runge-Kutta Methods - Engineering Mathematics, Lecture notes of Engineering Mathematics

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Module 3

Lesson 3

Runge-Kutta Methods

Edgar M. Adina

Instructor

CE50P- 2

Numerical Solutions to Engineering Problems

Recall: Euler’s Method Φ Step size, h x y x 0 ,y 0 True value y 1 , Predicted value Graphical interpretation of the first step of Euler’s method Consider the first order differential equation in the form of f ( x y ) dx dy = , With initial condition: 𝑦^0 =^ 𝑦(𝑥^0 ) Then, 𝑦𝑖+ 1 = 𝑦𝑖 + 𝑓 𝑥𝑖, 𝑦𝑖 ℎ Where ℎ = 𝑥𝑖 − 𝑥𝑖− 1 𝑓𝑜𝑟 𝑖 = 1 , 2 , 3 , … , 𝑛

3 rd Order Runge-Kutta Consider the first order differential equation in the form of f ( x y ) dx dy = , With initial condition: 𝑦 0 =^ 𝑦(𝑥 0 ) Then, 𝑦𝑖+ 1 = 𝑦𝑖 +

Where ℎ = 𝑥𝑖 − 𝑥𝑖− 1 𝑓𝑜𝑟 𝑖 = 1 , 2 , 3 , … , 𝑛

4 th Order Runge-Kutta Consider the first order differential equation in the form of f ( x y ) dx dy = , With initial condition: 𝑦 0 =^ 𝑦(𝑥 0 ) Then, (^) 𝑦 𝑖+ 1 =^ 𝑦𝑖 +^

Where ℎ = 𝑥𝑖 − 𝑥𝑖− 1 𝑓𝑜𝑟 𝑖 = 1 , 2 , 3 , … , 𝑛