Chapter 2.8, Problem 9E

Interpretation Introduction

Interpretation:

Taylor series expansion of x at ${\text{t}}_{\text{1}}$, in time step $\text{Δt}$ is ${\text{x (t}}_{\text{1}}{\text{) = x (t}}_{\text{0}}\text{+Δt)}$. Show that the Runge-Kutta method produces a local error $\left|{\text{x (t}}_{\text{1}}{\text{) - x}}_1\right|$ of size $O{(\Delta t)}^5$.

Concept Introduction:

The Runge-Kutta method is a first-order numerical approach to solve the ordinary differential equations with some given initial value.

Iterative expression to calculate the function value from Runge-Kutta order $\text{4}$ is written as,

${\text{x}}_{\text{n+1}}{\text{= x}}_{\text{n}}\text{+ }\frac{\text{1}}{\text{6}}{\text{(k}}_{\text{1}}{\text{+2k}}_{\text{2}}{\text{+2k}}_{\text{3}}{\text{+k}}_{\text{4}}\text{)}$

${\text{t}}_{\text{n+1}}{\text{= t}}_{\text{n}}\text{+Δt}$

The expression for four $\text{k}$ terms in the above iteration is expressed as,

${\text{k}}_{\text{1}}{\text{= f(x}}_{\text{n}}\text{)(Δt)}$

${\text{k}}_{\text{2}}\text{= f}\left({\text{x}}_{\text{n}}\text{+}\frac{\text{1}}{\text{2}}{\text{k}}_{\text{1}}\right)\text{(Δt)}$

${\text{k}}_{\text{3}}\text{= f}\left({\text{x}}_{\text{n}}\text{+}\frac{\text{1}}{\text{2}}{\text{k}}_{\text{2}}\right)\text{(Δt)}$

${\text{k}}_{\text{4}}\text{= f}\left({\text{x}}_{\text{n}}\text{+}\frac{\text{1}}{\text{2}}{\text{k}}_{\text{3}}\right)\text{(Δt)}$