Article Contents
Article Contents

# Construction of highly stable implicit-explicit general linear methods

• This paper deals with the numerical solution of systems of differential equations with a stiff part and a non-stiff one, typically arising from the semi-discretization of certain partial differential equations models. It is illustrated the construction and analysis of highly stable and high-stage order implicit-explicit (IMEX) methods based on diagonally implicit multistage integration methods (DIMSIMs), a subclass of general linear methods (GLMs). Some examples of methods with optimal stability properties are given. Finally numerical experiments confirm the theoretical expectations.
Mathematics Subject Classification: Primary: 65L05; Secondary: 65L20.

 Citation:

•  [1] U. M. Ascher, S. J. Ruuth and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math, 25 (1997), 151-167. [2] U. M. Ascher, S. J. Ruuth and B. T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823. [3] S. Boscarino, Error analysis of IMEX Runge-Kutta methods derived from differential-algebraic systems, SIAM Journal on Numerical Analysis, 45 (2007), 1600-1621. [4] S. Boscarino and G. Russo, On a class of uniformly accurate IMEX Runge-Kutta schemes and applications to hyperbolic systems with relaxation, SIAM J. Sci. Comput., 31 (2009), 1926-1945. [5] M. Braś and A. Cardone, Construction of efficient general linear methods for non-stiff differential systems, Math. Model. Anal., 17 (2012), 171-189. [6] M. Braś, A. Cardone and R. D'Ambrosio, Implementation of explicit nordsieck methods with inherent quadratic stability, Math. Model. Anal., 18 (2013), 289-307. [7] J. C. Butcher, Diagonally-implicit multi-stage integration methods, Appl. Numer. Math., 11 (1993), 347-363. [8] M. P. Calvo, J. de Frutos and J. Novo, Linearly implicit Runge-Kutta methods for advection-reaction-diffusion equations, Appl. Numer. Math., 37 (2001), 535-549. [9] A. Cardone and Z. Jackiewicz, Explicit Nordsieck methods with quadratic stability, Numer. Algorithms, 60 (2012), 1-25. [10] A. Cardone, Z. Jackiewicz and H. Mittelmann, Optimization-based search for Nordsieck methods of high order with quadratic stability, Math. Model. Anal., 17 (2012), 293-308. [11] A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang, Extrapolated implicit-explicit Runge-Kutta methods, Math. Model. Anal., 19 (2014), 18-43. [12] A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang, Extrapolation-based implicit-explicit general linear methods, Numer. Algorithms, 65 (2014), 377-399. [13] J. Frank, W. Hundsdorfer and J. G. Verwer, On the stability of implicit-explicit linear multistep methods, Appl. Numer. Math., 25 (1997), 193-205. [14] W. Hundsdorfer and S. J. Ruuth, IMEX extensions of linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys., 225 (2007), 2016-2042. [15] W. Hundsdorfer and J. Verwer, Numerical solution of time-dependent advection-diffusion-reaction equations, vol. 33 of Springer Series in Comput. Mathematics, Springer-Verlag, 2003. [16] Z. Jackiewicz, General linear methods for ordinary differential equations, John Wiley & Sons Inc., Hoboken, NJ, 2009. [17] C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44 (2003), 139-181. [18] L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations, in Recent trends in numerical analysis, vol. 3 of Adv. Theory Comput. Math., Nova Sci. Publ., Huntington, NY, 2001, 269-288. [19] L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129-155. [20] W. M. Wright, The construction of order 4 DIMSIMs for ordinary differential equations, Numer. Algorithms, 26 (2001), 123-130. [21] H. Zhang and A. Sandu, A second-order diagonally-implicit-explicit multi-stage integration method, Procedia CS, 9 (2012), 1039-1046. [22] H. Zhang, A. Sandu and S. Blaise, High order implicit-explicit general linear methods with optimized stability regions, arXiv preprint, URL http://arxiv.org/abs/1407.2337. [23] H. Zhang, A. Sandu and S. Blaise, Partitioned and Implicit-Explicit General Linear Methods for ordinary differential equations, J. Sci. Comput., 61 (2014), 119-144. [24] E. Zharovski, A. Sandu and H. Zhang, A class of implicit-explicit two-step Runge-Kutta methods, SIAM J. Numer. Anal., 53 (2015), no. 1, 321-341.
Open Access Under a Creative Commons license