Related papers: Multi-rate Runge-Kutta methods: stability analysis…
In this technical note a general procedure is described to construct internally consistent splitting methods for the numerical solution of differential equations, starting from matching pairs of explicit and diagonally implicit Runge-Kutta…
Many HPC applications that solve differential equations rely on the Runge-Kutta family of methods for time integration. Among these methods, the fourth-order accurate RK4 scheme is especially popular. This time integration scheme requires…
We propose a new method that extends conservative explicit multirate methods to implicit explicit-multirate methods. We develop extensions of order one and two with different stability properties on the implicit side. The method is suitable…
This paper is devoted to examining the stability of Runge-Kutta methods for solving nonlinear Volterra delay-integro-differential-algebraic equations (DIDAEs) with constant delay. Hybrid numerical schemes combining Runge-Kutta methods and…
Meta-learning has emerged as an important framework for learning new tasks from just a few examples. The success of any meta-learning model depends on (i) its fast adaptation to new tasks, as well as (ii) having a shared representation…
A space-time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes,…
In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et. al. (2019, 2022) to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no…
A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive…
In this work we consider a mixed precision approach to accelerate the implemetation of multi-stage methods. We show that Runge-Kutta methods can be designed so that certain costly intermediate computations can be performed as a…
Earth system models are complex integrated models of atmosphere, ocean, sea ice, and land surface. Coupling the components can be a significant challenge due to the difference in physics, temporal, and spatial scales. This study explores…
In this paper we consider the development of Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such systems the scaling depends on an additional parameter which modifies the nature of the…
This survey provides an overview of state-of-the art multirate schemes, which exploit the different time scales in the dynamics of a differential equation model by adapting the computational costs to different activity levels of the system.…
Multirate time integration methods apply different step sizes to resolve different components of the system based on the local activity levels. This local selection of step sizes allows increased computational efficiency while achieving the…
Applied to the master equation, the usual numerical integration methods, such as Runge-Kutta, become inefficient when the rates associated with various transitions differ by several orders of magnitude. We introduce an integration scheme…
A wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation of entropy or various energies, which arise implicitly under the exact solution of their governing PDEs. However, standard temporal schemes,…
Many important differential equations model quantities whose value must remain positive or stay in some bounded interval. These bounds may not be preserved when the model is solved numerically. We propose to ensure positivity or other…
This work constructs a new class of multirate schemes based on the recently developed generalized additive Runge-Kutta (GARK) methods (Sandu and Guenther, 2013). Multirate schemes use different step sizes for different components and for…
Simulation of complex dynamical systems arising in many applications is computationally challenging due to their size and complexity. Model order reduction, machine learning, and other types of surrogate modeling techniques offer cheaper…
In this article, a family of two- and three-stage explicit multiquadric (MQ) and inverse multiquadric (IMQ) radial basis functions (RBFs) Runge-Kutta methods are introduced for solving ordinary differential equations. These methods are…
This work focuses on the construction of a new class of fourth-order accurate methods for multirate time evolution of systems of ordinary differential equations. We base our work on the Recursive Flux Splitting Multirate (RFSMR) version of…