Stage value predictors for additive and partitioned Runge-Kutta methods

Additive and partitioned Runge-Kutta methods are widely used for the numerical integration of some special ODEs. They usually involve the numerical solution of nonlinear systems that require starting values as accurate as possible. In this paper we consider stage value predictors for these kind of methods. First, we deal with partitioned Runge-Kutta methods. The results obtained are transferred to additive Runge-Kutta methods. The theory developed is used to construct starting values for the Lobatto IIIA-IIIB methods and some IMEX methods from the literature. Some numerical results show that the use of the stage value predictors considered in this paper reduces the number of iterations per step and hence the computational cost.

[1]  Steven J. Ruuth,et al.  Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations , 1997 .

[2]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[3]  Donato Trigiante,et al.  Recent trends in numerical analysis , 2000 .

[4]  L. Jay.  Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems , 1996 .

[5]  J. I. Montijano,et al.  Stabilized starting algorithms for collocation Runge-Kutta methods☆ , 2003 .

[6]  M. Calvo,et al.  Are high order variable step equistage initializers better than standard starting algorithms , 2004 .

[7]  Lorenzo Pareschi,et al.  Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation , 2010, 1009.2757.

[9]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[10]  G. Russo,et al.  Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations , 2000 .

[11]  M. Calvo.  High order starting iterates for implicit Runge–Kutta methods: an improvement for variable‐step symplectic integrators , 2002 .

[12]  J. Butcher.  The Numerical Analysis of Ordinary Di erential Equa-tions , 1986 .

[13]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[14]  Juan I. Montijano,et al.  Variable-order starting algorithms for implicit Runge-Kutta methods on stiff problems , 2002 .

[15]  I. Higueras,et al.  Starting algorithms for low stage order RKN methods , 2002 .

[16]  Manuel Calvo,et al.  Two-Step High Order Starting Values for Implicit Runge–Kutta Methods , 2003, Adv. Comput. Math..

[17]  M. Carpenter,et al.  Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations , 2003 .

[18]  J. I. Montijano,et al.  On the Starting Algorithms for Fully Implicit Runge-Kutta Methods , 2000 .

[19]  M. P. Laburta.  Starting algorithms for IRK methods , 1997 .

[20]  Inmaculada Higueras,et al.  IRK methods for DAE: starting algorithms , 1999 .

[21]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[22]  Xiaolin Zhong,et al.  Additive Semi-Implicit Runge-Kutta Methods for Computing High-Speed Nonequilibrium Reactive Flows , 1996 .

[23]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[24]  Hans Olsson,et al.  Stage Value Predictors and Efficient Newton Iterations in Implicit Runge-Kutta Methods , 1998, SIAM J. Sci. Comput..

[25]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[26]  Inmaculada Higueras,et al.  Starting algorithms for some DIRK methods , 2004, Numerical Algorithms.

[27]  M. P. Laburta,et al.  Starting algorithms for Gauss Runge-Kutta methods for Hamiltonian systems☆ , 2003 .

[28]  C. Murray,et al.  Solar System Dynamics: Expansion of the Disturbing Function , 1999 .