Study on Banded Implicit Runge–Kutta Methods for Solving Stiff Differential Equations

Effectiveness of Implicit Methods for Stiff Stochastic Differential Equations

In this paper we study the behavior of a family of implicit numerical methods applied to stochastic differential equations with multiple time scales. We show by a combination of analytical arguments and numerical examples that implicit methods in general fail to capture the effective dynamics at the slow time scale. This is due to the fact that such implicit methods cannot correctly capture non...

A Class of Linearly Implicit Numerical Methods for Solving Stiff Ordinary Differential Equations

We introduce ABC-schemes, a new class of linearly implicit one-step methods for numerical integration of stiff ordinary differential equation systems. Formulas of ABC-schemes invoke the Jacobian of differential system similary to the methods of Rosenbrock type, but unlike the latter they include also the square of the Jacobian matrix.

Explicit methods for stiff stochastic differential equations

Multiscale differential equations arise in the modeling of many important problems in the science and engineering. Numerical solvers for such problems have been extensively studied in the deterministic case. Here, we discuss numerical methods for (mean-square stable) stiff stochastic differential equations. Standard explicit methods, as for example the EulerMaruyama method, face severe stepsize...

A quasi-interpolation method for solving stiff ordinary differential equations

Haar wavelet method for solving stiff differential equations

Application of the Haar wavelet approach for solving stiff differential equations is discussed. Solution of singular perturbation problems is also considered. Efficiency of the recommended method is demonstrated by means of four numerical examples, mostly taken from well-known textbooks.