An explicit runge kutta formula results when aij 0

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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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