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Construction of highly stable implicitexplicit general linear methods
1.  Dipartimento di Matematica, Università di Salerno, I84084 Fisciano (Sa), Italy 
2.  Department of Mathematics, Arizona State University, Tempe, Arizona 85287, and AGH University of Science and Technology, Kraków, Poland 
3.  Department of Computer Science, Virginia Polytechnic Institute & State University, Blacksburg, Virginia 24061, United States 
4.  Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, United States 
References:
[1] 
U. M. Ascher, S. J. Ruuth and R. J. Spiteri, Implicitexplicit RungeKutta methods for timedependent partial differential equations, Appl. Numer. Math, 25 (1997), 151167. 
[2] 
U. M. Ascher, S. J. Ruuth and B. T. R. Wetton, Implicitexplicit methods for timedependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797823. 
[3] 
S. Boscarino, Error analysis of IMEX RungeKutta methods derived from differentialalgebraic systems, SIAM Journal on Numerical Analysis, 45 (2007), 16001621. 
[4] 
S. Boscarino and G. Russo, On a class of uniformly accurate IMEX RungeKutta schemes and applications to hyperbolic systems with relaxation, SIAM J. Sci. Comput., 31 (2009), 19261945. 
[5] 
M. Braś and A. Cardone, Construction of efficient general linear methods for nonstiff differential systems, Math. Model. Anal., 17 (2012), 171189. 
[6] 
M. Braś, A. Cardone and R. D'Ambrosio, Implementation of explicit nordsieck methods with inherent quadratic stability, Math. Model. Anal., 18 (2013), 289307. 
[7] 
J. C. Butcher, Diagonallyimplicit multistage integration methods, Appl. Numer. Math., 11 (1993), 347363. 
[8] 
M. P. Calvo, J. de Frutos and J. Novo, Linearly implicit RungeKutta methods for advectionreactiondiffusion equations, Appl. Numer. Math., 37 (2001), 535549. 
[9] 
A. Cardone and Z. Jackiewicz, Explicit Nordsieck methods with quadratic stability, Numer. Algorithms, 60 (2012), 125. 
[10] 
A. Cardone, Z. Jackiewicz and H. Mittelmann, Optimizationbased search for Nordsieck methods of high order with quadratic stability, Math. Model. Anal., 17 (2012), 293308. 
[11] 
A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang, Extrapolated implicitexplicit RungeKutta methods, Math. Model. Anal., 19 (2014), 1843. 
[12] 
A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang, Extrapolationbased implicitexplicit general linear methods, Numer. Algorithms, 65 (2014), 377399. 
[13] 
J. Frank, W. Hundsdorfer and J. G. Verwer, On the stability of implicitexplicit linear multistep methods, Appl. Numer. Math., 25 (1997), 193205. 
[14] 
W. Hundsdorfer and S. J. Ruuth, IMEX extensions of linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys., 225 (2007), 20162042. 
[15] 
W. Hundsdorfer and J. Verwer, Numerical solution of timedependent advectiondiffusionreaction equations, vol. 33 of Springer Series in Comput. Mathematics, SpringerVerlag, 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 RungeKutta schemes for convectiondiffusionreaction equations, Appl. Numer. Math., 44 (2003), 139181. 
[18] 
L. Pareschi and G. Russo, Implicitexplicit RungeKutta 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, 269288. 
[19] 
L. Pareschi and G. Russo, ImplicitExplicit RungeKutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129155. 
[20] 
W. M. Wright, The construction of order 4 DIMSIMs for ordinary differential equations, Numer. Algorithms, 26 (2001), 123130. 
[21] 
H. Zhang and A. Sandu, A secondorder diagonallyimplicitexplicit multistage integration method, Procedia CS, 9 (2012), 10391046. 
[22] 
H. Zhang, A. Sandu and S. Blaise, High order implicitexplicit 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 ImplicitExplicit General Linear Methods for ordinary differential equations, J. Sci. Comput., 61 (2014), 119144. 
[24] 
E. Zharovski, A. Sandu and H. Zhang, A class of implicitexplicit twostep RungeKutta methods, SIAM J. Numer. Anal., 53 (2015), no. 1, 321341. 
show all references
References:
[1] 
U. M. Ascher, S. J. Ruuth and R. J. Spiteri, Implicitexplicit RungeKutta methods for timedependent partial differential equations, Appl. Numer. Math, 25 (1997), 151167. 
[2] 
U. M. Ascher, S. J. Ruuth and B. T. R. Wetton, Implicitexplicit methods for timedependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797823. 
[3] 
S. Boscarino, Error analysis of IMEX RungeKutta methods derived from differentialalgebraic systems, SIAM Journal on Numerical Analysis, 45 (2007), 16001621. 
[4] 
S. Boscarino and G. Russo, On a class of uniformly accurate IMEX RungeKutta schemes and applications to hyperbolic systems with relaxation, SIAM J. Sci. Comput., 31 (2009), 19261945. 
[5] 
M. Braś and A. Cardone, Construction of efficient general linear methods for nonstiff differential systems, Math. Model. Anal., 17 (2012), 171189. 
[6] 
M. Braś, A. Cardone and R. D'Ambrosio, Implementation of explicit nordsieck methods with inherent quadratic stability, Math. Model. Anal., 18 (2013), 289307. 
[7] 
J. C. Butcher, Diagonallyimplicit multistage integration methods, Appl. Numer. Math., 11 (1993), 347363. 
[8] 
M. P. Calvo, J. de Frutos and J. Novo, Linearly implicit RungeKutta methods for advectionreactiondiffusion equations, Appl. Numer. Math., 37 (2001), 535549. 
[9] 
A. Cardone and Z. Jackiewicz, Explicit Nordsieck methods with quadratic stability, Numer. Algorithms, 60 (2012), 125. 
[10] 
A. Cardone, Z. Jackiewicz and H. Mittelmann, Optimizationbased search for Nordsieck methods of high order with quadratic stability, Math. Model. Anal., 17 (2012), 293308. 
[11] 
A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang, Extrapolated implicitexplicit RungeKutta methods, Math. Model. Anal., 19 (2014), 1843. 
[12] 
A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang, Extrapolationbased implicitexplicit general linear methods, Numer. Algorithms, 65 (2014), 377399. 
[13] 
J. Frank, W. Hundsdorfer and J. G. Verwer, On the stability of implicitexplicit linear multistep methods, Appl. Numer. Math., 25 (1997), 193205. 
[14] 
W. Hundsdorfer and S. J. Ruuth, IMEX extensions of linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys., 225 (2007), 20162042. 
[15] 
W. Hundsdorfer and J. Verwer, Numerical solution of timedependent advectiondiffusionreaction equations, vol. 33 of Springer Series in Comput. Mathematics, SpringerVerlag, 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 RungeKutta schemes for convectiondiffusionreaction equations, Appl. Numer. Math., 44 (2003), 139181. 
[18] 
L. Pareschi and G. Russo, Implicitexplicit RungeKutta 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, 269288. 
[19] 
L. Pareschi and G. Russo, ImplicitExplicit RungeKutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129155. 
[20] 
W. M. Wright, The construction of order 4 DIMSIMs for ordinary differential equations, Numer. Algorithms, 26 (2001), 123130. 
[21] 
H. Zhang and A. Sandu, A secondorder diagonallyimplicitexplicit multistage integration method, Procedia CS, 9 (2012), 10391046. 
[22] 
H. Zhang, A. Sandu and S. Blaise, High order implicitexplicit 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 ImplicitExplicit General Linear Methods for ordinary differential equations, J. Sci. Comput., 61 (2014), 119144. 
[24] 
E. Zharovski, A. Sandu and H. Zhang, A class of implicitexplicit twostep RungeKutta methods, SIAM J. Numer. Anal., 53 (2015), no. 1, 321341. 
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