Runge–Kutta Methods for Ordinary Differential Equations

Since their first discovery by Runge (Math Ann 46:167–178, 1895), Heun (Z Math Phys 45:23–38, 1900) and Kutta (Z Math Phys 46:435–453, 1901), Runge–Kutta methods have been one of the most important procedures for the numerical solution of ordinary differential equation systems. This survey paper ranges over many aspects of Runge–Kutta methods, including order conditions, order barriers, the efficient implementation of implicit methods, effective order methods and strong stability-preserving methods. Finally, applications to the analysis and implementation of G-symplectic methods will be discussed.

[1]  J. M. Sanz-Serna,et al.  Runge-kutta schemes for Hamiltonian systems , 1988 .

[2]  S. Gill,et al.  A process for the step-by-step integration of differential equations in an automatic digital computing machine , 1951, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  John C. Butcher,et al.  On the attainable order of Runge-Kutta methods , 1965 .

[4]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[5]  John C. Butcher,et al.  An algebraic theory of integration methods , 1972 .

[6]  John C. Butcher.  The cohesiveness of G-symplectic methods , 2015, Numerical Algorithms.

[7]  C. Runge.  Ueber die numerische Auflösung von Differentialgleichungen , 1895 .

[8]  John C. Butcher,et al.  A generalization of singly-implicit Runge-Kutta methods , 1997 .

[9]  John C. Butcher,et al.  Towards efficient Runge-Kutta methods for stiff systems , 1990 .

[10]  W. Kutta.  Beitrag zur Naherungsweisen Integration Totaler Differentialgleichungen , 1901 .

[11]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[12]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[13]  J. C. Butcher,et al.  Dealing with Parasitic Behaviour in G-Symplectic Integrators , 2013 .

[14]  J. Butcher.  Coefficients for the study of Runge-Kutta integration processes , 1963, Journal of the Australian Mathematical Society.

[15]  The Control of Parasitism in G-symplectic Methods , 2014, SIAM J. Numer. Anal..

[16]  R. Alexander.  Diagonally implicit runge-kutta methods for stiff odes , 1977 .

[17]  David I. Ketcheson,et al.  Strong stability preserving runge-kutta and multistep time discretizations , 2011 .