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Nested implicit Runge-Kutta pairs of Gauss and lobatto types with local and global error controls for stiff ordinary differential equations. (English. Russian original) Zbl 07264985
Comput. Math. Math. Phys. 60, No. 7, 1134-1154 (2020); translation from Zh. Vychisl. Mat. Mat. Fiz. 60, No. 7, 1170-1192 (2020).
Summary: The problem of efficient global error estimation and control is studied in embedded nested implicit Runge-Kutta pairs of Gauss and Lobatto types as applied to stiff ordinary differential equations (ODEs). Stiff problems may arise in many areas of engineering, and their accurate numerical solution is an important issue of computational and applied mathematics. A cheap global error estimation technique designed recently for the mentioned Runge-Kutta pairs can severely overestimate the global error when applied to stiff ODEs and, hence, this reduces the efficiency of those solvers. In the present paper, we explain the cause of that error overestimation and show how to improve the mentioned computation techniques for stiff problems. Such modifications not only boost the efficiency of numerical integration of stiff ODEs, but also make the embedded nested implicit Runge-Kutta pairs with scaled modified local and global error controls superior to stiff built-in MATLAB ODE solvers with only local error control when applied to important test examples.
MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L04 Numerical methods for stiff equations
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[1] Kulikov, G. Yu.; Kulikova, M. V., Accurate numerical implementation of the continuous-discrete extended Kalman filter, IEEE Trans. Autom. Contr., 59, 273-279 (2014) · Zbl 1360.93700
[2] Kulikov, G. Yu.; Kulikova, M. V., High-order accurate continuous-discrete extended Kalman filter for chemical engineering, Eur. J. Contr., 21, 14-26 (2015) · Zbl 1403.93184
[3] Kulikov, G. Yu.; Kulikova, M. V., The accurate continuous-discrete extended Kalman filter for radar tracking, IEEE Trans. Signal Process., 64, 948-958 (2016) · Zbl 1412.94052
[4] Kulikov, G. Yu.; Kulikova, M. V., Estimating the state in stiff continuous-time stochastic systems within extended Kalman filtering, SIAM J. Sci. Comput., 38, A3565-A3588 (2016) · Zbl 1353.65009
[5] Kulikov, G. Yu.; Lima, P. M.; Morgado, M. L., Analysis and numerical approximation of singular boundary value problems with the p-Laplacian in fluid mechanics, J. Comput. Appl. Math., 262, 87-104 (2014) · Zbl 1301.65079
[6] Kulikov, G. Yu.; Kulikova, M. V., Accurate cubature and extended Kalman filtering methods for estimating continuous-time nonlinear stochastic systems with discrete measurements, Appl. Numer. Math., 111, 260-275 (2017) · Zbl 1353.65008
[7] Kulikov, G. Yu.; Kulikova, M. V., Accurate continuous-discrete unscented Kalman filtering for estimation of nonlinear continuous-time stochastic models in radar tracking, Signal Process., 139, 25-35 (2017)
[8] Kulikov, G. Yu.; Kulikova, M. V., Accurate state estimation of stiff continuous-time stochastic models in chemical and other engineering, Math. Comput. Simul., 142, 62-81 (2017)
[9] Kulikov, G. Yu.; Kulikova, M. V., Square-root Kalman-like filters for estimation of stiff continuous-time stochastic systems with ill-conditioned measurements, IET Control Theory Appl., 11, 1420-1425 (2017)
[10] Kulikov, G. Yu.; Kulikova, M. V., Moore-Penrose-pseudo-inverse-based Kalman-like filtering methods for estimation of stiff continuous-discrete stochastic systems with ill-conditioned measurements, IET Control Theory Appl., 12, 2205-2212 (2018)
[11] Kulikov, G. Yu.; Kulikova, M. V., Estimation of maneuvering target in the presence of non-Gaussian noise: A coordinated turn case study, Signal Process., 145, 241-257 (2018)
[12] Kulikov, G. Yu.; Kulikova, M. V., Practical implementation of extended Kalman filtering in chemical systems with sparse measurements, Russ. J. Numer. Anal. Math. Model., 33, 41-53 (2018) · Zbl 06858144
[13] Kulikov, G. Yu.; Kulikova, M. V., Numerical robustness of extended Kalman filtering based state estimation in ill-conditioned continuous-discrete nonlinear stochastic chemical systems, Int. J. Robust Nonlinear Control, 29, 1377-1395 (2019) · Zbl 1410.93123
[14] Kulikov, G. Yu.; Kulikova, M. V., Square-root accurate continuous-discrete extended-unscented Kalman filtering methods with embedded orthogonal and J-orthogonal QR decompositions for estimation of nonlinear continuous-time stochastic models in radar tracking, Signal Process., 166, 107253 (2020)
[15] Kulikov, G. Yu.; Kulikova, M. V., NIRK-based Cholesky-factorized square-root accurate continuous-discrete unscented Kalman filters for state estimation in nonlinear continuous-time stochastic models with discrete measurements, Appl. Numer. Math., 147, 196-221 (2020) · Zbl 07137370
[16] Aggoun, L.; Elliot, R. J., Measure Theory and Filtering: Introduction and Applications (2005), Cambridge, UK: Cambridge Univ. Press, Cambridge, UK
[17] Aström, K. J., Introduction to Stochastic Control Theory (1970), New York: Academic, New York · Zbl 0226.93027
[18] Bar-Shalom, Y.; Li, X.-R.; Kirubarajan, T., Estimation with Applications to Tracking and Navigation (2001), New York: Wiley, New York
[19] Crassidis, J. L.; Junkins, J. L., Optimal Estimation of Dynamic Systems (2004), New York: CRC, New York · Zbl 1072.93001
[20] Grewal, M. S.; Andrews, A. P., Kalman Filtering: Theory and Practice (2001), New Jersey: Prentice Hall, New Jersey
[21] Grewal, M. S.; Weill, L. R.; Andrews, A. P., Global Positioning Systems, Inertial Navigation, and Integration (2001), New York: Wiley, New York
[22] Jazwinski, A. H., Stochastic Processes and Filtering Theory (1970), New York: Academic, New York · Zbl 0203.50101
[23] Øksendal, B., Stochastic Differential Equations: An Introduction with Applications (2003), New York: Springer, New York · Zbl 1025.60026
[24] Rawlings, J. B.; Mayne, D. Q., Model Predictive Control: Theory and Design (2013), Madison, Wisconsin: Bob Hill, LLC, Madison, Wisconsin
[25] Butcher, J. C., Numerical Methods for Ordinary Differential Equations (2008), Chichester: Wiley, Chichester · Zbl 1167.65041
[26] Dekker, K.; Verwer, M. P., Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations (1984), Amsterdam: North-Holland, Amsterdam · Zbl 0571.65057
[27] Hairer, E.; Lubich, C.; Wanner, G., Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations (2002), Berlin: Springer-Verlag, Berlin · Zbl 0994.65135
[28] Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems (1993), Berlin: Springer-Verlag, Berlin · Zbl 0789.65048
[29] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (1996), Berlin: Springer-Verlag, Berlin · Zbl 0859.65067
[30] Jackiewicz, Z., General Linear Methods for Ordinary Differential Equations (2009), Hoboken: Wiley, Hoboken · Zbl 1211.65095
[31] Sanz-Serna, J. M.; Calvo, M. P., Numerical Hamilton Problems (1994), London: Chapman and Hall, London
[32] Shampine, L. F., Numerical Solution of Ordinary Differential Equations (1994), New York: Chapman and Hall, New York · Zbl 0832.65063
[33] G. Yu. Kulikov and S. K. Shindin, “On a family of cheap symmetric one-step methods of order four,” in Computational Science—ICCS 2006, Proceedings, Part I, 6th International Conference, Reading, UK, May 28-31,2006, Ed. by V. N. Alexandrov, et al., Lecture Notes in Computer Science (Springer-Verlag, Berlin, 2006), Vol. 3991, pp. 781-785.
[34] G. Yu. Kulikov and S. K. Shindin, “Numerical tests with Gauss-type nested implicit Runge-Kutta formulas,” in Computational Science—ICCS 2007, Proceedings, Part I, 7th International Conference, Beijing, China, May 27-30,2007, Ed. by Y. Shi, et al., Lecture Notes in Computer Science (Springer-Verlag, Berlin, 2007), Vol. 4487, pp. 136-143.
[35] Kulikov, G. Yu.; Shindin, S. K., Adaptive nested implicit Runge-Kutta formulas of Gauss type, Appl. Numer. Math., 59, 707-722 (2009) · Zbl 1161.65055
[36] Kulikov, G. Yu., Automatic error control in the Gauss-type nested implicit Runge-Kutta formula of order 6, Russ. J. Numer. Anal. Math. Model., 24, 123-144 (2009) · Zbl 1168.65368
[37] Kulikov, G. Yu., Embedded symmetric nested implicit Runge-Kutta methods of Gauss and Lobatto types for solving stiff ordinary differential equations and Hamiltonian systems, Comput. Math. Math. Phys., 55, 983-1003 (2015) · Zbl 1325.65105
[38] Cash, J. R., A class of implicit Runge-Kutta methods for the numerical solution of stiff ordinary differential equations, J. ACM, 22, 504-511 (1975) · Zbl 0366.65029
[39] Cash, J. R., On a class of implicit Runge-Kutta procedures, IMA J. Numer. Anal., 19, 455-470 (1977) · Zbl 0364.65051
[40] Cash, J. R., On a note of the computational aspects of a class of implicit Runge-Kutta procedures, IMA J. Numer. Anal., 20, 425-441 (1977) · Zbl 0386.65033
[41] Cash, J. R.; Singhal, A., Mono-implicit Runge-Kutta formulas for the numerical integration of stiff differential systems, IMA J. Numer. Anal., 2, 211-227 (1982) · Zbl 0488.65031
[42] Skvortsov, L. M., How to avoid accuracy and order reduction in Runge-Kutta methods as applied to stiff problems, Comput. Math. Math. Phys., 57, 1124-1139 (2017) · Zbl 1379.65051
[43] Skvortsov, L. M., Implicit Runge-Kutta methods with explicit internal stages, Comput. Math. Math. Phys., 58, 307-321 (2018) · Zbl 06909570
[44] Shampine, L. F.; Reichelt, M. W., The MATLAB ODE suite, SIAM J. Sci. Comput., 18, 1-22 (1997) · Zbl 0868.65040
[45] Higham, D.; Higham, N., MATLAB Guide (2005), Philadelphia: SIAM, Philadelphia
[46] Kulikov, G. Yu.; Weiner, R., Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation, J. Comput. Appl. Math., 233, 2351-2364 (2010) · Zbl 1206.65179
[47] Baker, T. S.; Dormand, J. R.; Gilmore, J. P.; Prince, P. J., Continuous approximation with embedded Runge-Kutta methods, Appl. Numer. Math., 22, 51-62 (1996) · Zbl 0871.65077
[48] Dormand, J. R.; Prince, P. J., Runge-Kutta triples, Comput. Math. Appl. A, 12, 1007-1017 (1986) · Zbl 0618.65059
[49] Dormand, J. R.; Prince, P. J., Practical Runge-Kutta processes, SIAM J. Sci. Stat. Comput., 10, 977-989 (1989) · Zbl 0704.65053
[50] Enright, W. H., Analysis of error control strategies for continuous Runge-Kutta methods, SIAM J. Numer. Anal., 26, 588-599 (1989) · Zbl 0676.65073
[51] Enright, W. H.; Jackson, K. R.; Nørsett, S. P.; Thomsen, P. G., Interpolants for Runge-Kutta formulas, ACM Trans. Math. Software, 12, 193-218 (1986) · Zbl 0617.65068
[52] Higham, D. J., Highly continuous Runge-Kutta interpolants, ACM Trans. Math. Software, 17, 368-386 (1991) · Zbl 0900.65235
[53] Sharp, P. W.; Verner, J. H., Generation of high-order interpolants for explicit Runge-Kutta pairs, ACM Trans. Math. Software, 24, 13-29 (1998) · Zbl 0928.65086
[54] Kulikov, G. Yu., Cheap global error estimation in some Runge-Kutta pairs, IMA J. Numer. Anal., 33, 136-163 (2013) · Zbl 1271.65119
[55] Weiner, R.; Kulikov, G. Yu., Local and global error estimation and control within explicit two-step peer triples, J. Comput. Appl. Math., 262, 261-270 (2014) · Zbl 1301.65067
[56] Kulikov, G. Yu.; Weiner, R., A singly diagonally implicit two-step peer triple with global error control for stiff ordinary differential equations, SIAM J. Sci. Comput., 37, A1593-A1613 (2015) · Zbl 1433.65122
[57] Weiner, R.; Kulikov, G. Yu.; Beck, S.; Bruder, J., New third- and fourth-order singly diagonally implicit two-step peer triples with local and global error controls for solving stiff ordinary differential equations, J. Comput. Appl. Math., 316, 380-391 (2017) · Zbl 1372.65199
[58] Kulikov, G. Yu.; Weiner, R., Global error estimation and control in linearly-implicit parallel two-step peer W-methods, J. Comput. Appl. Math., 236, 1226-1239 (2011) · Zbl 1269.65075
[59] Aïd, R.; Levacher, L., Numerical investigations on global error estimation for ordinary differential equations, J. Comput. Appl. Math., 82, 21-39 (1997) · Zbl 0887.65096
[60] Shampine, L. F., Global error estimation for stiff ODEs, Lect. Notes Math., 1066, 159-168 (1984)
[61] Shampine, L. F., Error estimation and control for ODEs, J. Sci. Comput., 25, 3-16 (2005) · Zbl 1203.65122
[62] Shampine, L. F.; Watts, H. A., Global error estimation for ordinary differential equations, ACM Trans. Math. Software, 2, 172-186 (1976) · Zbl 0328.65041
[63] Skeel, R. D., Thirteen ways to estimate global error, Numer. Math., 48, 1-20 (1986) · Zbl 0562.65050
[64] Skeel, R. D., Global error estimation and the backward differentiation formulas, Appl. Math. Comput., 31, 197-208 (1989) · Zbl 0675.65084
[65] H. J. Stetter, “Global error estimation in ODE-solvers,” in Numerical Integration of Differential Equations and Large Linear Systems: Proceedings, Ed. by J. Hinze (Bielefeld, 1980), Lecture Notes in Mathematics, Vol. 968 (Springer-Verlag, Berlin, 1982), pp. 269-279.
[66] Dormand, J. R.; Duckers, R. R.; Prince, P. J., Global error estimation with Runge-Kutta methods, IMA J. Numer. Anal., 4, 169-184 (1984) · Zbl 0577.65054
[67] Dormand, J. R.; Gilmore, J. P.; Prince, P. J., Globally embedded Runge-Kutta schemes, Ann. Numer. Math., 1, 97-106 (1994) · Zbl 0824.65054
[68] Dormand, J. R.; Lockyer, M. A.; McGorrigan, N. E.; Prince, P. J., Global error estimation with Runge-Kutta triples, Comput. Math. Appl., 18, 835-846 (1989) · Zbl 0683.65054
[69] Lang, J.; Verwer, J. G., On global error estimation and control for initial value problems, SIAM J. Sci. Comput., 29, 1460-1475 (2007) · Zbl 1145.65047
[70] Macdougall, T.; Verner, J. H., Global error estimators for order 7, 8 Runge-Kutta pairs, Numer. Algorithms, 31, 215-231 (2002) · Zbl 1014.65066
[71] Makazaga, J.; Murua, A., New Runge-Kutta based schemes for ODEs with cheap global error estimation, BIT, 43, 595-610 (2003) · Zbl 1046.65055
[72] Shampine, L. F.; Baca, L. S., Global error estimates for ODEs based on extrapolation methods, SIAM J. Sci. Stat. Comput., 6, 1-14 (1985) · Zbl 0578.65077
[73] Tirani, R., A parallel algorithm for the estimation of the global error in Runge-Kutta methods, Numer. Algorithms, 31, 311-318 (2002) · Zbl 1012.65082
[74] Calvo, M.; Higham, D. J.; Montijano, J. I.; Rández, L., Stepsize selection for tolerance proportionality in explicit Runge-Kutta codes, Adv. Comput. Math., 7, 361-382 (1997) · Zbl 0891.65097
[75] Calvo, M.; González-Pinto, S.; Montijano, J. I., Global error estimation based on the tolerance proportionality for some adaptive Runge-Kutta codes, J. Comput. Appl. Math., 218, 329-341 (2008) · Zbl 1149.65052
[76] Higham, D. J., Global error versus tolerance for explicit Runge-Kutta methods, IMA J. Numer. Anal., 11, 457-480 (1991) · Zbl 0738.65073
[77] Higham, D. J., The tolerance proportionality of adaptive ODE solvers, J. Comput. Appl. Math., 45, 227-236 (1993) · Zbl 0780.65050
[78] Kulikov, G. Yu.; Merkulov, A. I.; Shindin, S. K., Asymptotic error estimate for general Newton-type methods and its application to differential equations, Russ. J. Numer. Anal. Math. Model., 22, 567-590 (2007) · Zbl 1145.65033
[79] Kulikov, G. Yu.; Shindin, S. K., One-leg variable-coefficient formulas for ordinary differential equations and local-global step size control, Numer. Algorithms, 43, 99-121 (2006) · Zbl 1109.65061
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