American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 185-194. doi: 10.3934/proc.2015.0185

Construction of highly stable implicit-explicit general linear methods

Angelamaria Cardone 1, , Zdzisław Jackiewicz 2, , Adrian Sandu 3, and  Hong Zhang 4,

1. 

Dipartimento di Matematica, Università di Salerno, I-84084 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

Received  September 2014 Revised  January 2015 Published  November 2015

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.
Keywords: construction of highly stable methods, general linear methods, stability analysis, IMEX methods, differential equations..
Mathematics Subject Classification: Primary: 65L05; Secondary: 65L2.
Citation: Angelamaria Cardone, Zdzisław Jackiewicz, Adrian Sandu, Hong Zhang. Construction of highly stable implicit-explicit general linear methods. Conference Publications, 2015, 2015 (special) : 185-194. doi: 10.3934/proc.2015.0185
References:
[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.   Google Scholar

[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.   Google Scholar

[3]

S. Boscarino, Error analysis of IMEX Runge-Kutta methods derived from differential-algebraic systems,, SIAM Journal on Numerical Analysis, 45 (2007), 1600.   Google Scholar

[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.   Google Scholar

[5]

M. Braś and A. Cardone, Construction of efficient general linear methods for non-stiff differential systems,, Math. Model. Anal., 17 (2012), 171.   Google Scholar

[6]

M. Braś, A. Cardone and R. D'Ambrosio, Implementation of explicit nordsieck methods with inherent quadratic stability,, Math. Model. Anal., 18 (2013), 289.   Google Scholar

[7]

J. C. Butcher, Diagonally-implicit multi-stage integration methods,, Appl. Numer. Math., 11 (1993), 347.   Google Scholar

[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.   Google Scholar

[9]

A. Cardone and Z. Jackiewicz, Explicit Nordsieck methods with quadratic stability,, Numer. Algorithms, 60 (2012), 1.   Google Scholar

[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.   Google Scholar

[11]

A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang, Extrapolated implicit-explicit Runge-Kutta methods,, Math. Model. Anal., 19 (2014), 18.   Google Scholar

[12]

A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang, Extrapolation-based implicit-explicit general linear methods,, Numer. Algorithms, 65 (2014), 377.   Google Scholar

[13]

J. Frank, W. Hundsdorfer and J. G. Verwer, On the stability of implicit-explicit linear multistep methods,, Appl. Numer. Math., 25 (1997), 193.   Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[16]

Z. Jackiewicz, General linear methods for ordinary differential equations,, John Wiley & Sons Inc., (2009).   Google Scholar

[17]

C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations,, Appl. Numer. Math., 44 (2003), 139.   Google Scholar

[18]

L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations,, in Recent trends in numerical analysis, (2001), 269.   Google Scholar

[19]

L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation,, J. Sci. Comput., 25 (2005), 129.   Google Scholar

[20]

W. M. Wright, The construction of order 4 DIMSIMs for ordinary differential equations,, Numer. Algorithms, 26 (2001), 123.   Google Scholar

[21]

H. Zhang and A. Sandu, A second-order diagonally-implicit-explicit multi-stage integration method,, Procedia CS, 9 (2012), 1039.   Google Scholar

[22]

H. Zhang, A. Sandu and S. Blaise, High order implicit-explicit general linear methods with optimized stability regions,, arXiv preprint, ().   Google Scholar

[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.   Google Scholar

[24]

E. Zharovski, A. Sandu and H. Zhang, A class of implicit-explicit two-step Runge-Kutta methods,, SIAM J. Numer. Anal., 53 (2015), 321.   Google Scholar

show all references

References:
[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.   Google Scholar

[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.   Google Scholar

[3]

S. Boscarino, Error analysis of IMEX Runge-Kutta methods derived from differential-algebraic systems,, SIAM Journal on Numerical Analysis, 45 (2007), 1600.   Google Scholar

[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.   Google Scholar

[5]

M. Braś and A. Cardone, Construction of efficient general linear methods for non-stiff differential systems,, Math. Model. Anal., 17 (2012), 171.   Google Scholar

[6]

M. Braś, A. Cardone and R. D'Ambrosio, Implementation of explicit nordsieck methods with inherent quadratic stability,, Math. Model. Anal., 18 (2013), 289.   Google Scholar

[7]

J. C. Butcher, Diagonally-implicit multi-stage integration methods,, Appl. Numer. Math., 11 (1993), 347.   Google Scholar

[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.   Google Scholar

[9]

A. Cardone and Z. Jackiewicz, Explicit Nordsieck methods with quadratic stability,, Numer. Algorithms, 60 (2012), 1.   Google Scholar

[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.   Google Scholar

[11]

A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang, Extrapolated implicit-explicit Runge-Kutta methods,, Math. Model. Anal., 19 (2014), 18.   Google Scholar

[12]

A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang, Extrapolation-based implicit-explicit general linear methods,, Numer. Algorithms, 65 (2014), 377.   Google Scholar

[13]

J. Frank, W. Hundsdorfer and J. G. Verwer, On the stability of implicit-explicit linear multistep methods,, Appl. Numer. Math., 25 (1997), 193.   Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[16]

Z. Jackiewicz, General linear methods for ordinary differential equations,, John Wiley & Sons Inc., (2009).   Google Scholar

[17]

C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations,, Appl. Numer. Math., 44 (2003), 139.   Google Scholar

[18]

L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations,, in Recent trends in numerical analysis, (2001), 269.   Google Scholar

[19]

L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation,, J. Sci. Comput., 25 (2005), 129.   Google Scholar

[20]

W. M. Wright, The construction of order 4 DIMSIMs for ordinary differential equations,, Numer. Algorithms, 26 (2001), 123.   Google Scholar

[21]

H. Zhang and A. Sandu, A second-order diagonally-implicit-explicit multi-stage integration method,, Procedia CS, 9 (2012), 1039.   Google Scholar

[22]

H. Zhang, A. Sandu and S. Blaise, High order implicit-explicit general linear methods with optimized stability regions,, arXiv preprint, ().   Google Scholar

[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.   Google Scholar

[24]

E. Zharovski, A. Sandu and H. Zhang, A class of implicit-explicit two-step Runge-Kutta methods,, SIAM J. Numer. Anal., 53 (2015), 321.   Google Scholar

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