Unformatted text preview: cillate rapidly. We'll study explicit methods in
this section and take up implicit methods in the next.
Runge-Kutta formulas are derived in the same manner as the second-order methods
of Section 3.1. Thus, we
8 1. expand the exact solution of the ODE in a Taylor's series about, e.g., tn 1 ; 2. substitute the exact solution of the ODE into the Runge-Kutta formula and expanded the result in a Taylor's series about, e.g., tn 1 and
; 3. match the two Taylor's series expansions to as high an order as possible. The
coe cients are usually not uniquely determined by this process thus, there are
families of methods having a given order.
A Runge-Kutta method that is consistent to order k (or simply of order k) will match
the terms of order hk in both series. Clearly the algebra involved in obtaining these formulas increases combinatorically with increasing order. A symbolic manipulation system,
such as MAPLE or MATHEMATICA, can be used to reduce complexity. Fortunately,
the derivation is adequately demonstrated by the second-order methods presented in
Section 3.1 and,...
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- Spring '14
- Numerical Analysis, yn, Tn, Numerical ordinary differential equations