2 research outputs found
Efficient implementation of partitioned stiff exponential Runge-Kutta methods
Multiphysics systems are driven by multiple processes acting simultaneously,
and their simulation leads to partitioned systems of differential equations.
This paper studies the solution of partitioned systems of differential
equations using exponential Runge-Kutta methods. We propose specific
multiphysics implementations of exponential Runge-Kutta methods satisfying
stiff order conditions that were developed in [Hochbruck et al., SISC, 1998]
and [Luan and Osterman, JCAM, 2014]. We reformulate stiffly--accurate
exponential Runge--Kutta methods in a way that naturally allows of the
structure of multiphysics systems, and discuss their application to both
component and additively partitioned systems. The resulting partitioned
exponential methods only compute matrix functions of the Jacobians of
individual components, rather than the Jacobian of the full, coupled system. We
derive modified formulations of particular methods of order two, three and
four, and apply them to solve a partitioned reaction-diffusion problem. The
proposed methods retain full order for several partitionings of the discretized
problem, including by components and by physical processes
Partitioned Exponential Methods for Coupled Multiphysics Systems
Multiphysics problems involving two or more coupled physical phenomena are
ubiquitous in science and engineering. This work develops a new partitioned
exponential approach for the time integration of multiphysics problems. After a
possible semi-discretization in space, the class of problems under
consideration is modeled by a system of ordinary differential equations where
the right-hand side is a summation of two component functions, each
corresponding to a given set of physical processes.
The partitioned-exponential methods proposed herein evolve each component of
the system via an exponential integrator, and information between partitions is
exchanged via coupling terms. The traditional approach to constructing
exponential methods, based on the variation-of-constants formula, is not
directly applicable to partitioned systems. Rather, our approach to developing
new partitioned-exponential families is based on a general-structure additive
formulation of the schemes. Two method formulations are considered, one based
on a linear-nonlinear splitting of the right hand component functions, and
another based on approximate Jacobians. The paper develops classical
(non-stiff) order conditions theory for partitioned exponential schemes based
on particular families of T-trees and B-series theory. Several practical
methods of third order are constructed that extend the Rosenbrock-type and
EPIRK families of exponential integrators. Several implementation optimizations
specific to the application of these methods to reaction-diffusion systems are
also discussed. Numerical experiments reveal that the new
partitioned-exponential methods can perform better than traditional
unpartitioned exponential methods on some problems.Comment: Fixed a definition and other minor typos. Results remain unchange