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

Construction of highly stable implicit-explicit general linear methods

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.
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.

[1]

Andrew J. Steyer, Erik S. Van Vleck. Underlying one-step methods and nonautonomous stability of general linear methods. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2859-2877. doi: 10.3934/dcdsb.2018108

[2]

Hui Liang, Hermann Brunner. Collocation methods for differential equations with piecewise linear delays. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1839-1857. doi: 10.3934/cpaa.2012.11.1839

[3]

Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289

[4]

Yuhong Dai, Ya-xiang Yuan. Analysis of monotone gradient methods. Journal of Industrial and Management Optimization, 2005, 1 (2) : 181-192. doi: 10.3934/jimo.2005.1.181

[5]

William Guo. Unification of the common methods for solving the first-order linear ordinary differential equations. STEM Education, 2021, 1 (2) : 127-140. doi: 10.3934/steme.2021010

[6]

Wenxiong Chen, Shijie Qi. Direct methods on fractional equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1269-1310. doi: 10.3934/dcds.2019055

[7]

Konstantinos Drakakis. A review of the available construction methods for Golomb rulers. Advances in Mathematics of Communications, 2009, 3 (3) : 235-250. doi: 10.3934/amc.2009.3.235

[8]

Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control and Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012

[9]

Yoonsang Lee, Bjorn Engquist. Variable step size multiscale methods for stiff and highly oscillatory dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1079-1097. doi: 10.3934/dcds.2014.34.1079

[10]

Yixuan Wu, Yanzhi Zhang. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 851-876. doi: 10.3934/dcdss.2022016

[11]

Angelamaria Cardone, Dajana Conte, Beatrice Paternoster. Two-step collocation methods for fractional differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2709-2725. doi: 10.3934/dcdsb.2018088

[12]

Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283

[13]

Xiaobing Feng, Shu Ma. Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 687-711. doi: 10.3934/dcdss.2021071

[14]

B. S. Goh, W. J. Leong, Z. Siri. Partial Newton methods for a system of equations. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 463-469. doi: 10.3934/naco.2013.3.463

[15]

Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems and Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019

[16]

Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems and Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163

[17]

Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim, Dae-Woon Lim. New construction methods of quaternary periodic complementary sequence sets. Advances in Mathematics of Communications, 2010, 4 (1) : 61-68. doi: 10.3934/amc.2010.4.61

[18]

Antonella Zanna. Symplectic P-stable additive Runge—Kutta methods. Journal of Computational Dynamics, 2022, 9 (2) : 299-328. doi: 10.3934/jcd.2021030

[19]

Jiangshan Wang, Lingxiong Meng, Hongen Jia. Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model. Electronic Research Archive, 2020, 28 (3) : 1191-1205. doi: 10.3934/era.2020065

[20]

M. Sumon Hossain, M. Monir Uddin. Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 173-186. doi: 10.3934/naco.2019013

 Impact Factor: 

Metrics

  • PDF downloads (152)
  • HTML views (0)
  • Cited by (0)

[Back to Top]