The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
Numerical methods for ordinary differential systems: the initial value problem
Numerical methods for ordinary differential systems: the initial value problem
LAPACK's user's guide
Parallel and sequential methods for ordinary differential equations
Parallel and sequential methods for ordinary differential equations
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Consider the system of ordinary differential equation (ODEs) dy/dt=f(t,y) where (a) t∈[a,b] with ba , (b) y is a vector containing s components and (c) y(a) is given The θ-method is applied to solve approximately the system of ODEs on a set of prescribed grid-points If N is the number of time-steps that are to be carried out, then this numerical method can be defined by using the following set of relationships yn=yn−1+h (1−θ) f(tn−1, yn−1) , θ∈[0.5, 1.0] , n=1, 2, ..., N , h=(b−a) / N , tn=tn−1+h=t0+nh , t0=a , tN=b As a rule, the accuracy of the approximations { yn | n=1, 2, ..., N } , can be improved by applying the Richardson Extrapolation under the assumption that the stability of the computational process is preserved Therefore, it is natural to require that the combined numerical method (Richardson Extrapolation plus the θ-method) is in some sense stable It can be proved that the combined method is strongly A-stable when θ∈[2/3, 1.0] This is the main result in the paper The usefulness of this result in the solution of many problems arising in different scientific and engineering areas is demonstrated by performing a series of experiments with an extremely badly-scaled and very stiff atmospheric chemistry scheme which is actually used in several well-known large-scale air pollution models.