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), 151-167.

[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, (). 

[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.

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), 151-167.

[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, (). 

[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.

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