By Ernst Hairer, Gerhard Wanner
The topic of this e-book is the answer of stiff differential equations and of differential-algebraic structures. This moment variation comprises new fabric together with new numerical checks, fresh growth in numerical differential-algebraic equations, and better FORTRAN codes.
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"A really good book...Throughout, illuminating photographs, sketches and costs from papers of researchers within the box upload a component of straightforward informality and inspire the text." --MATHEMATICS TODAY
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Additional info for Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems
6. 20) with n=8. Explain the result with the help of Fig. 1. 7. 14). Why? 8. 32)) and denote its analytic continuation as eigenvalue of C by A(ß). Prove that a) If ReA;fO, then for some y E R A(ß) = A' (1-~ (l-ReA) + ißy + O(ß2)) This shows that IA(ß)I < lAI for small ß> 0 if Re A < 1. 2. 13. 14. Explicit Euler on the beam problem (every 50th step drawn) submatrix, then ,x(ß) = ,x. L Hint. Write the characteristic polynomial of 8 in the form det(,\J -8) = ,x(,xp(,x) + ßq(,x)) , where p(,x) = det(,\J - C) is the characteristic polynomial of C, and differentiate with respect to ß.
14). In both eases the stability function satisfies R( 00) = O. 5). 3. , if one of the c's, usually c6 , equals 1. 52 IV. Stiff Problems - One Step Methods 4. (Pade (1899), see also Lagrange (1776)). ".. 1 a: 2 7·94 1+--1+ ... leads to the diagonal Pade approximations for e"'. Hint. Compute the first partial fractions. 5. 5. The trapezoidal rule 0 0 1 1 o. 8?
20 IV. 3. Stability domains for GBS extrapolation methods Analysis of the Examples of IV.! 04 o 10 4 Y3 -104Y3 - 6 . 6. The third one produces stiffnes8. 0015. This again confirms the numerical observations. 2. 6') (Brusselator reaction with diffusion) is a large 2N x 2N matrix. 16) are 4a (. 1f'k)2 (N)2 ( . 19) and are loeated between -4a( N +1)2 and O. 16) with much smaller eoefficients ean be regarded as a small perturbation. 16) will remain dose to those of the unperturbed matrix and He in a strip neighbouring the interval [-4a( N + 1)2,0].