In the context of multiscale computations, techniques have recently been developed that enable microscopic simulators to perform macroscopic level tasks (equation-free multiscale computation). The main tool is the so-called coarse-grained time-stepper, which implements an approximation of the unavailable macroscopic time-stepper using only the microscopic simulator. Several schemes were develop...

In this paper we deal with the Landau-LifshitzGilbert equation which describes dynamics of ferromagnetism. Based on the strong nonlinearities a stabilised discretisation method is necessary to skip the time-step restriction. In this paper we propose a implicit full discrete scheme with an embedded operatorsplitting method.

We apply Patankar Runge–Kutta methods to y′ = M(y)y and focus on the case where M(y) is a graph Laplacian as resulting scheme will preserve positivity total mass. The second order Heun method tested using four test problems (stiff non-stiff) cast into this form. local error estimated step size chosen adaptively. Concerning accuracy efficiency, results are comparable those obtained with traditio...

This paper constructs extrapolated implicit-explicit time stepping methods that allow one to efficiently solve problems with both stiff and nonstiff components. The proposed methods are based on Euler steps and can provide very high order discretizations of ODEs, index-1 DAEs, and PDEs in the method of lines framework. Implicit-explicit schemes based on extrapolation are simple to construct, ea...

In the operator splitting solution of atmospheric transport-chemistry problems modeling air pollution, a major task is the numerical integration of the stii ODE systems describing the chemical transformations. In this note a numerical comparison is presented between two special purpose solvers developed for this task. Note: This report is one of a series on the development of algorithms for lon...

Abstract This work introduces a new one-step method with three intermediate points for solving stiff differential systems. These types of problems appear in different disciplines and, particular, derived from chemical reactions. In fact, the term “stiff”’ was coined by Curtiss and Hirschfelder an article on kinetics (Hirschfelder, Proc Natl Acad Sci USA 38:235–243, 1952). The techniques interpo...

We give an algorithm for efficient step size control in numerical integration of non– stiff initial value problems, based on a formula tailormade to methods where the numerical solution is compared with a solution of lower order.

A partitioned implicit-explicit orthogonal Runge-Kutta method (PIROCK) is proposed for the time integration of diffusion-advection-reaction problems with possibly severely stiff reaction terms and stiff stochastic terms. The diffusion terms are solved by the explicit second order orthogonal Chebyshev method (ROCK2), while the stiff reaction terms (solved implicitly) and the advection and noise ...

A partitioned implicit-explicit orthogonal Runge-Kutta method (PIROCK) is proposed for the time integration of diffusion-advection-reaction problems with possibly severely stiff reaction terms and stiff stochastic terms. The diffusion terms are solved by the explicit second order orthogonal Chebyshev method (ROCK2), while the stiff reaction terms (solved implicitly) and the advection and noise ...

For many systems of differential equations modeling problems in science and engineering, there are often natural splittings of the right hand side into two parts, one of which is non-stiff or mildly stiff, and the other part is stiff. Such systems can be efficiently treated by a class of implicit-explicit (IMEX) diagonally implicit multistage integration methods (DIMSIMs), where the stiff part ...