MTH3007b Lecture 2

Me, in the lecture

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Lecture 1 established the explicit and implicit Euler methods and showed that both are first-order: global error $O(dt)$. This session introduces higher-order Runge-Kutta methods that achieve $O(d{t}^2)$ global error by evaluating the right-hand side $g(t,y)$ at intermediate points within each timestep.

General Form of Runge-Kutta Methods

A single-step method can always be written as

$$y_{i+1}=y_i+dt\cdot \varphi (t_i,y_i,dt)$$

where $\varphi$ is called the the increment function. For explicit Euler, $\varphi =g(t_i,y_i)$. Higher-order methods use more sophisticated choices of $\varphi$ that incorporate information about $g$ at multiple points.

Midpoint Method

Formula

The Midpoint method evaluates $g$ at the midpoint of the interval $[t_i,t_{i+1}]$:

$$\boxed{y_{i+1}=y_i+dt\cdot g\left(t_i+\frac{dt}{2},y_i+g(t_i,y_i)\frac{dt}{2}\right)}$$

The argument $y_i+g(t_i,y_i)dt/2$ is an Euler-style estimate of $y$ at the midpoint - this is then used to evaluate $g$ there.

Note

The midpoint method is a second-order method: global error $O(d{t}^2)$. This is a full order of magnitude improvement over explicit Euler.

How it achieves second order

The key idea is that by evaluating $g$ at the midpoint rather than the left endpoint, the method implicitly incorporates information about $g^{\prime }$ through the way $y$ changes across the interval. The Taylor expansion of the resulting scheme matches the exact solution to $O(d{t}^2)$ per step, giving $O(d{t}^2)$ global error (since the global error is one order lower in $dt$ than the LTE for consistent methods… wait, actually: LTE $=O(d{t}^3)$ per step, global $=O(d{t}^2)$).

Ralston’s Method

Formula

Ralston’s method is another second-order Runge-Kutta scheme. It uses two stages:

$$k_1=g(t_i,y_i)$$ $$k_2=g\left(t_i+\frac{2dt}{3},y_i+\frac{2dt{k}_1}{3}\right)$$ $$\boxed{y_{i+1}=y_i+dt\left(\frac{k_1}{4}+\frac{3k_2}{4}\right)}$$

The weights $1/4$ and $3/4$ are chosen to minimise the leading error coefficient among all second-order two-stage RK methods.

Important

Both the midpoint method and Ralston’s method are second-order: global error $\sim O(d{t}^2)$. Halving $dt$ reduces the error by a factor of four.

Comparison with Explicit Euler

For small enough $dt$, both second-order methods produce significantly smaller errors than explicit Euler. Ralston’s method is generally more accurate than the midpoint method at the same $dt$ due to its optimised coefficients.

Example

Applied to $\dot{y}=bt-ay$ (a stiff-ish equation for large $a$): at $dt=0.01$, a second-order method yields much smaller deviation from the exact solution than explicit Euler at the same step size.

The tradeoff is computational cost: each step of a two-stage RK method requires two evaluations of $g$ versus one for Euler. In practice, the improved accuracy per unit cost makes second-order methods preferable for most problems.

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Pre-Lecture Notes from University Notes

• General RK form: $y_{i+1}=y_i+dt\cdot \varphi (t_i,y_i,dt)$ where $\varphi$ is the increment function
• Midpoint method: $y_{i+1}=y_i+dt\cdot g(t_i+dt/2,y_i+g(t_i,y_i)dt/2)$ - second-order ($O(d{t}^2)$ global error)
• Ralston’s method: two-stage RK with $k_1=g(t_i,y_i)$, $k_2=g(t_i+2dt/3,y_i+2dt{k}_1/3)$, update $y_{i+1}=y_i+dt(k_1/4+3k_2/4)$
• Both methods are second-order; Ralston’s is generally more accurate than the midpoint method
• Cost: two $g$-evaluations per step (vs one for Euler), but $O(d{t}^2)$ error makes this worthwhile
• Next session: formal derivation of second-order RK coefficients via Taylor expansion, and the RK4 formula