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# 36 gives 2 1 z z2 1 q1z q2 z2 p0 equating

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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 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 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 the followin...
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