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Unformatted text preview: d the (1,1) approximation corresponds to the midpoint rule. (The (1,1) approximation also corresponds to the trapezoidal rule.) Methods
corresponding to the (s s) diagonal Pade approximations are Butcher's maximum order
implicit Runge-Kutta methods (Theorem 3.3.1). j=0 k=0
1 + z + z2 =2 1 1
1;z=2 1+2z=3+z 2 =6
1;z=3 2 1
1;z +z 2 =2 1+z=3
1;2z=3+z 2 =6 1+z=2+z 2 =12
1;z=2+z 2 =12 Table 3.3.1: Some Pade approximations of ez . Theorem 3.3.2. There is one and only one 2s-order s-stage implicit Runge-Kutta formula and it corresponds to the (s s) Pade approximation.
Proof. cf. Butcher 7]. We'll be able to construct several implicit Runge-Kutta methods having unbounded
absolute-stability regions. We'll want to characterize these methods according to their
behavior as jzj ! 1 and this requires some additional notions of stability. De nition 3.3.2. A numerical method is A-stable if its region of absolute stability in- cludes the entire left-half plane Re(h ) 0. 23 The relationship between A-stability and the Pade approximations is established by
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14