Construction of Additive Semi-Implicit Runge–Kutta Methods with Low-Storage Requirements

@article{Higueras2016ConstructionOA,
title={Construction of Additive Semi-Implicit Runge–Kutta Methods with Low-Storage Requirements},
journal={Journal of Scientific Computing},
year={2016},
volume={67},
pages={1019-1042}
}
• Published 1 October 2015
• Mathematics, Computer Science
• Journal of Scientific Computing
Space discretization of some time-dependent partial differential equations gives rise to systems of ordinary differential equations in additive form whose terms have different stiffness properties. In these cases, implicit methods should be used to integrate the stiff terms while efficient explicit methods can be used for the non-stiff part of the problem. However, for systems with a large number of equations, memory storage requirement is also an important issue. When the high dimension of the…
5 Citations
High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations
• Computer Science, Mathematics
J. Sci. Comput.
• 2016
The main purpose of the paper is to show how to use implicit–explicit Runge–Kutta methods in a much more general context than usually found in the literature, obtaining very effective schemes for a
On the necessity of negative coefficients for high order DIRK Low-Storage methods
containing terms with different stiffness properties. This kind of problems may arise from the semidiscretization of some partial differential equations. Depending on the spatial discretization, a
Improving the IMEX method with a residual balanced decomposition
In numerical time-integration with implicit-explicit (IMEX) methods, a within-step adaptable decomposition called residual balanced decomposition is introduced. With this decomposition, the
A shortcut for IMEX methods: integrate the residual explicitly
The efficiency of SIMEX-RK methods in overcoming parabolic stiffness is explored by exploring the trade-off between the computational effort placed in solving the implicit equation and the size of the numerically stable time-step.
Order Barrier for Low-Storage DIRK Methods with Positive Weights
• Mathematics, Computer Science
J. Sci. Comput.
• 2018
It is proved that third order low-storage DIRK methods must contain negative weights, obtaining the order barrier $$p\le 2$$p≤2 for these schemes.

References

SHOWING 1-10 OF 31 REFERENCES
Additive Semi-Implicit Runge-Kutta Methods for Computing High-Speed Nonequilibrium Reactive Flows
This paper is concerned with time-stepping numerical methods for computing stiff semi-discrete systems of ordinary differential equations for transient hypersonic flows with thermo-chemical
Design and Implementation of Predictors for Additive Semi-Implicit Runge--Kutta Methods
• Computer Science, Mathematics
SIAM J. Sci. Comput.
• 2009
The aim of the present paper is to develop efficient initializers for additive semi-implicit Runge-Kutta (ASIRK) methods, which are used for solving nonlinear systems of ordinary differential equations.
On some new low storage implementations of time advancing Runge-Kutta methods
• Computer Science, Mathematics
J. Comput. Appl. Math.
• 2012
In this paper, explicit Runge-Kutta schemes with minimum storage requirements for systems with very large dimension that arise in the spatial discretization of some partial differential equations are considered and the results of some numerical experiments are presented to show the behavior of fourth-order optimal schemes for some nonlinear problems.
Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations
• Mathematics
• 2003
Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and
Low-storage, Explicit Runge-Kutta Schemes for the Compressible Navier-Stokes Equations
• Mathematics
• 2000
The derivation of low-storage, explicit Runge-Kutta (ERK) schemes has been performed in the context of integrating the compressible Navier-Stokes equations via direct numerical simulation.
Highly Efficient Strong Stability-Preserving Runge-Kutta Methods with Low-Storage Implementations
• D. Ketcheson
• Mathematics, Computer Science
SIAM J. Sci. Comput.
• 2008
This work considers the problem of finding explicit Runge-Kutta methods with optimal SSP time-step restrictions, first for linear autonomous ordinary differential equations and then for nonlinear or nonautonomous equations, and finds new optimal low- storage methods and new low-storage implementations of known optimal methods.
On an accurate third order implicit-explicit Runge--Kutta method for stiff problems
Most of the popular implicit-explicit (IMEX) Runge-Kutta (R-K) methods existing in the literature suffer from the phenomenon of order reduction in the stiff regime when applied to stiff problems
Error Analysis of IMEX Runge-Kutta Methods Derived from Differential-Algebraic Systems
• S. Boscarino
• Computer Science, Mathematics
SIAM J. Numer. Anal.
• 2007
This analysis expands the global error in powers of $\varepsilon$ and shows that the coefficients of the error are the global errors of the IMEX Runge-Kutta method applied to a differential-algebraic system.
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
• Mathematics
• 1997
Abstract Implicit-explicit (IMEX) linear multistep time-discretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable
Runge-Kutta methods with minimum storage implementations
• D. Ketcheson
• Mathematics, Computer Science
J. Comput. Phys.
• 2010
This work investigates Runge-Kutta methods that require only two storage locations per ODE, and presents a new, more general class of methods that provide error estimates and/or the ability to restart a step while still employing the minimum possible number of memory registers.