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 RungeKutta methods (Theorem 3.3.1). j=0 k=0
1 1
1+z 2
1 + z + z2 =2 1 1
1;z 1+z=2
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 2sorder sstage implicit RungeKutta formula and it corresponds to the (s s) Pade approximation.
Proof. cf. Butcher 7]. We'll be able to construct several implicit RungeKutta methods having unbounded
absolutestability 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 Astable if its region of absolute stability in cludes the entire lefthalf plane Re(h ) 0. 23 The relationship between Astability and the Pade approximations is established by
the followin...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, yn, Tn, Numerical ordinary differential equations

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