Runge-Kutta Integration RKI

Let us start by describing the simple RKI technique, as found in most textbooks on numerical analysis (e.g. Gerald 1978 or Jain 1984). Consider Fig. 5.1, showing a function u(t), to be solved for t+6t, point 

Starting from t, point A, given the first-order differential equation = /(u). (5.1) 

This form corresponds closely to our 3c/8t problem, where f(u) is a discrete approximation of the second derivative in the diffusion equation, using three c (or u) values. Let us say we have evaluated u and f(u) - the slope of u(t) at t - and now move along this slope to t+6t, point C, calling this the solution. This is called the first-order (Euler) method and can be expressed in the following notation  

This is what the standard explicit digital simulation method does. Then, /(u+A ) is the slope at the point D, and if we go only half-way along the slope /(u(t)), to point E and then the rest of the way with the new slope at D, we end up at point F, which is much closer to point B, the true solution. This loose description is mathematically expressed as (assuming Eq. 5.2) 

In numerical analysis texts, the symbol k is usually used we use A here to avoid a clash with rate constants. 

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