Solving ordinary differential equations: Stiff and differential-algebraic problems
The subject of this book is the solution of stiff differential equations and of differential-algebraic systems (differential equations with constraints). There is a chapter on one-step and extrapolation methods for stiff problems, another one on multistep methods and general linear methods for stiff problems, and a third one on the treatment of singular perturbation problems and differential-algebraic systems. The beginning of each chapter is of an introductory nature, followed by practical applications, the discusssion of numerical results, theoretical investigations on the order of accuracy, linear and non-linear stability convergence and asymptotic expansions. Stiff and differential-algebraic problems arise everywhere in scientific computations ( e.g., in physics, chemistry, biology, control engineering, electrical network analysis, mechanical sytems). Many applications as well as computer programmes are presented.
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Stiff Problems OneStep Methods
Diagonally Implicit RK Methods
Multistep Methods for Stiff Problems
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A-stable algebraically stable applied assumption BDF methods becomes bounded Burrage coefficients collocation methods compute consider convergence results corresponding Dahlquist defined denote derivatives diagonal differential equation differential-algebraic eigenvalues equivalent error constant Euler method exact solution Exercise explicit extrapolation follows given gives global error Hairer hence implicit Euler method implicit Runge-Kutta methods implies initial values Inserting integration Jacobian Jeltsch Lemma linear methods linear multistep methods linear system linearly implicit Lobatto IIIC Lubich matrix method of order N0rsett Nevanlinna numerical solution obtain one-leg method order conditions order star ordinary differential equations Pade approximations parameters poles proof of Theorem Prove quadrature formula rational function recursion replace RK-method root locus root locus curve Rosenbrock method Runge-Kutta methods Section shows singular perturbation solved stability domain stability function stiff differential equations stiff equations stiff problems SUBROUTINE sufficiently small Suppose Table term transformation trapezoidal rule trees vector y-component yields zeros