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.

Next-generation exascale machines with extreme levels of parallelism will provide massive computing resources for large scale numerical simulations complex physical systems at unprecedented parameter ranges. However, novel methods, scalable algorithms and re-design current state-of-the art solvers are required scaling to these minimal overheads. One such approach partial differential equations ...

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...

This paper is devoted to a phenomenon in hyperbolic balance laws, first described by Fiedler and Liebscher [2], which is similar in spirit to the Turing instability. The combination of two individually stabilising effects can lead to quite rich dynamical behaviour, like instabilities, oscillations, or pattern formation. Our problem is composed of two ingredients. First, we have a strictly hyper...

We propose and analyze an implicit mass-matrix penalization (IMMP) technique which enables efficient and exact sampling of the (Boltzmann/Gibbs) canonical distribution associated with highly oscillatory systems. The penalization is based on an extended Hamiltonian whith artificial constraints associated with each expected fast degree of freedom (fDOF). The penalty parameters enable arbitrary tu...

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...