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...

In this paper, an improved trigonometrically fitted zero-dissipative explicit two-step hybrid method with fifth algebraic order is derived. The method is applied to several problems which solutions are oscillatory in nature. Numerical results obtained are compared with existing methods in the scientific literature. The comparison shows that the new method is more effective and efficient than th...

We introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces. The latter are functions f ◦ T : Rn × Q → Q with f ◦ T (x, ω) = f(T (x)ω) where Q is a compact (Hausdorff topological) space, f ∈ C(Q) and T (x) : Q → Q, x ∈ Rn, is an n-dimensional continuous dynamical system endowed with an invariant Radon probabilit...

Runge-Kutta methods are an important family of implicit and explicit iterative methods used for the approximation of solutions of ordinary differential equations. Explicit RungeKutta methods are unsuitable for the solution of stiff equations as their region of stability is small. Stiff equation is a differential equation for which certain numerical methods for solving the equation are numerical...

We present and analyze a new class of numerical methods for the solution of stiff stochastic differential equations (SDEs). These methods, called S-ROCK (for stochastic orthogonal Runge–Kutta Chebyshev), are explicit and of strong order 1 and possess large stability domains in the mean-square sense. For mean-square stable stiff SDEs, they are much more efficient than the standard explicit metho...

Continuous Explicit Runge-Kutta methods with the minimal number of stages are considered. These methods are continuously diierentiable if and only if one of the stages is the FSAL evaluation. A characterization of a subclass of these methods is developed for order 3,4 and 5. It is shown how the free parameters of these methods can be used either to minimize the continuous truncation error coeec...

Initial value solvers typically input a problem specification and an error tolerance, and output an approximate solution. Faced with this situation many users assume, or hope for, a linear relationship between the global error and the tolerance. In this paper we examine the potential for such 'tolerance proportionality' in existing explicit Runge-Kutta algorithms. We take account of recent deve...

Reaction-Advection-Diffusion problems occurring in Air Pollution Models have very stiff reaction terms and moderately stiff vertical mixing (diffusion and cloud transport) terms which both should be integrated implicitly in time. Standard implicit time stepping is by far too expensive for this type of problems whereas widely used splitting techniques may lead to unacceptably large errors for fa...

We prove optimal convergence estimates for eigenvalues and eigenvectors of a class of singular/stiff perturbed problems. Our profs are constructive in nature and use (elementary) techniques which are of current interest in computational Linear Algebra to obtain estimates even for eigenvalues which are in gaps of the essential spectrum. Further, we also identify a class of “regular” stiff pertur...

Partial difference equations are difference equations that involve functions with two or more independent integer variables. Such equations arise from considerations of random walk problems, molecular structure problems, and numerical difference approximation problems. Recently, there have been a large number of papers devoted to partial difference equations, and the problem of oscillatory of s...