Some $A$-stable methods for stiff ordinary differential equations

Additive Runge-Kutta Methods for Stiff Ordinary Differential Equations

Certain pairs of Runge-Kutta methods may be used additively to solve a system of n differential equations x' = J(t)x + g(t, x). Pairs of methods, of order p < 4, where one method is semiexplicit and /(-stable and the other method is explicit, are obtained. These methods require the LU factorization of one n X n matrix, and p evaluations of g, in each step. It is shown that such methods have a s...

Heterogeneous multiscale methods for stiff ordinary differential equations

The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are ba...

Comparison of some recent numerical methods for initial-value problems for stiff ordinary differential equations

We consider the combustion equation as one of the candidates from the class of stiff ordinary differential equations. A solution over a length of time that is inversely proportional to δ > 0 (where δ > 0 is a small disturbance of the pre-ignition state) is sought. This problem has a transient at the midpoint of the integration interval. The solution changes from being non-stiff to stiff, and af...

A quasi-interpolation method for solving stiff ordinary differential equations

A Parallel Algorithm for Stiff Ordinary Differential Equations

The problem associated with the stiff ordinary differential equation (ODE) systems in parallel processing is that the calculus can not be started simultaneously on many processors with an explicit formula. The proposed algorithm is constructed for a special classes of stiff ODE, those of the form y'(t)=A(t)y(t)+g(t). It has a high efficiency in the implementation on a distributed memory multipr...