53a or 353b may be added to yn or yn 2 respectively to

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Unformatted text preview: Using the triangular inequality p p p jdp j jyn ; yn+1j + jyn+1 ; y(tn)j: n The last term on the right is the local error of the order p + 1 method (3.5.4b) and is O(hp+2) thus, p p jdp j jyn ; yn+1j + jdp+1j: n n The higher-order error term on the right may be neglected to get an error estimate of the form p p jdp j jyn ; yn+1j: n (3.5.5) Embedding, like Richardson's extrapolation, is also an expensive way of estimating errors. If the number of Runge-Kutta stages s p, then embedding requires approximately m(p + 1) additional function evaluations per step for a system of m ODEs. The number of function evaluations can be substantially reduced by embedding the p th-order method within an (s +1)-stage method of order p +1. For explicit Runge-Kutta methods, the tableau of the (s + 1)-stage method would have the form 0 c2 c3 ... a21 a31 ... a32 ... ... cs+1 as+1 1 as+1 2 as+1 s ^1 ^2 ^s ^s+1 b b b b (Zero's on an above the diagonal in A are not shown.) Assuming that the p th-order Runge-Kutta method has s stages, it would be required to have the form 0 c2 a21 c3 a31 a32 ... ... ... . . . cs as1 as2 as s 1 b1 b2 bs+1 bs With this form, only one additio...
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