Study on banded implicit Runge-Kutta methods for solving stiff differential equations. (English) Zbl 1435.65103
Summary: The implicit Runge-Kutta method with A-stability is suitable for solving stiff differential equations. However, the fully implicit Runge-Kutta method is very expensive in solving large system problems. Although some implicit Runge-Kutta methods can reduce the cost of computation, their accuracy and stability are also adversely affected. Therefore, an effective banded implicit Runge-Kutta method with high accuracy and high stability is proposed, which reduces the computation cost by changing the Jacobian matrix from a full coefficient matrix to a banded matrix. Numerical solutions and results of stiff equations obtained by the methods involved are compared, and the results show that the banded implicit Runge-Kutta method is advantageous to solve large stiff problems and conducive to the development of simulation.
MSC:
| 65L04 | Numerical methods for stiff equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
Software:
RODASReferences:
| [1] | Hairer, E.; Wanner, G.; Nørsett, S. P., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2-11 (1996), Berlin, China: Springer-Verlag, Berlin, China · Zbl 0859.65067 |
| [2] | Nirmala, V.; Parimala, V.; Rajarajeswari, P., Application of Runge-Kutta method for finding multiple numerical solutions to intuitionistic fuzzy differential equations, Journal of Physics: Conference Series, 1139, 1 (2018) · doi:10.1088/1742-6596/1139/1/012012 |
| [3] | Su, B.; Tuo, X.; Xu, L., Two modified symplectic partitioned Runge-Kutta methods for solving the elastic wave equation, Journal of Geophysics and Engineering, 14, 4, 811-821 (2017) · doi:10.1088/1742-2140/aa6e7a |
| [4] | Athanasakis, I. E.; Papadopoulou, E. P.; Saridakis, Y. G., Runge-Kutta and Hermite Collocation for a biological invasion problem modeled by a generalized Fisher equation, Journal of Physics: Conference Series, 490, 1 (2014) · doi:10.1088/1742-6596/490/1/012133 |
| [5] | Tian, J. H.; Zou, J.; Wang, Y. S.; Liu, J.; Yuan, J.; Zhou, Y., “Simulation of bipolar charge transport with trapping and recombination in polymeric insulators using Runge-Kutta discontinuous Galerkin method, Journal of Physics D: Applied Physics, 41, 19 (2008) · doi:10.1088/0022-3727/41/19/195416 |
| [6] | Chowdhury, S.; Das, P. K. G., Numerical Solutions of Initial Value Problems Using Mathematica, 1-11 (2018), Saint-Raphaël, France: Morgan & Claypool Publishers, Saint-Raphaël, France |
| [7] | Samsudin, A.; Rosli, N.; Ariffin, A. N., Stability analysis of explicit and implicit stochastic Runge-Kutta methods for stochastic differential equations, Journal of Physics: Conference Series, 890, 1 (2017) · doi:10.1088/1742-6596/890/1/012084 |
| [8] | Kazemi-Kamyab, V.; Van Zuijlen, A. H.; Bijl, H., Analysis and application of high order implicit Runge-Kutta schemes to collocated finite volume discretization of the incompressible Navier-Stokes equations, Computers & Fluids, 108, 15, 107-115 (2015) · Zbl 1390.76455 · doi:10.1016/j.compfluid.2014.11.025 |
| [9] | Jameson, A., Evaluation of fully implicit Runge Kutta schemes for unsteady flow calculations, Journal of Scientific Computing, 73, 2-3, 819-852 (2017) · Zbl 1397.65104 · doi:10.1007/s10915-017-0476-x |
| [10] | Liu, K.; Liao, X.; Li, Y., Parallel simulation of power systems transient stability based on implicit Runge-Kutta methods and W-transformation, Electric Power Components and Systems, 45, 20, 2246-2256 (2017) · doi:10.1080/15325008.2017.1400606 |
| [11] | Antohe, V.; Gladwell, I., Performance of Gauss implicit Runge-Kutta methods on separable Hamiltonian systems, Computers & Mathematics with Applications, 45, 1-3, 481-501 (2003) · Zbl 1035.65153 · doi:10.1016/s0898-1221(03)80032-9 |
| [12] | Franco, J. M.; Gómez, I.; Rández, L., SDIRK methods for stiff ODEs with oscillating solutions, Journal of Computational and Applied Mathematics, 81, 2, 197-209 (1997) · Zbl 0887.65078 · doi:10.1016/s0377-0427(97)00056-3 |
| [13] | Franco, J. M.; Gomez, I., Two three-parallel and three-processor SDIRK methods for stiff initial-value problems, Journal of Computational and Applied Mathematics, 87, 1, 119-134 (1997) · Zbl 0898.65046 · doi:10.1016/s0377-0427(97)00182-9 |
| [14] | Liao, W.; Yan, Y., Singly diagonally implicit Runge-Kutta method for time-dependent reaction-diffusion equation, Numerical Methods for Partial Differential Equations, 27, 6, 1423-1441 (2011) · Zbl 1243.65112 · doi:10.1002/num.20589 |
| [15] | Wing, M. K.; Senu, N.; Suleiman, M., A five-stage singly diagonally implicit Runge-Kutta-Nyström method with reduced phase-lag, AIP Conference Proceedings, 1482, 1, 315-320 (2012) · doi:10.1063/1.4757486 |
| [16] | Ababneh, O. Y.; Ahmad, R., “Construction of third-order diagonal implicit Runge-Kutta methods for stiff problems, Chinese Physics Letters, 26, 8 (2009) · Zbl 1191.65090 · doi:10.1088/0256-307x/26/8/080503 |
| [17] | Butcher, J. C.; Chartier, P., A generalization of singly-implicit Runge-Kutta methods, Applied Numerical Mathematics, 24, 2-3, 343-350 (1997) · Zbl 0906.65076 · doi:10.1016/s0168-9274(97)00031-7 |
| [18] | Burrage, K.; Butcher, J. C.; Chipman, F. H., An implementation of singly-implicit Runge-Kutta methods, BIT Numerical Mathematics, 20, 3, 326-340 (1980) · Zbl 0456.65040 · doi:10.1007/bf01932774 |
| [19] | Burrage, K., A special family of Runge-Kutta methods for solving stiff differential equations, BIT Numerical Mathematics, 18, 1, 22-41 (1978) · Zbl 0384.65034 · doi:10.1007/bf01947741 |
| [20] | Alexande, R., Diagonally implicit Runge-Kutta methods for stiff O. D. E.’s, SIAM Journal on Numerical Analysis, 14, 6, 1006-1021 (1977) · Zbl 0374.65038 · doi:10.1137/0714068 |
| [21] | Chen, D. J. L., The efficiency of Singly-implicit Runge-Kutta methods for stiff differential equations, Numerical Algorithms, 65, 3, 533-554 (2014) · Zbl 1291.65199 · doi:10.1007/s11075-013-9797-5 |
| [22] | Butcher, J. C., Implicit Runge-Kutta processes, Mathematics of Computation, 18, 85, 50-64 (1964) · Zbl 0123.11701 · doi:10.1090/s0025-5718-1964-0159424-9 |
| [23] | Golub, G. H.; Van Loan, C. F., Matrix Computations, 3, 152-156 (2012), London, UK: John Hopkins University Press, London, UK |
| [24] | Demmel, J. W., Applied numerical linear algebra, 56, 79-82 (1997), Philadelphia, PA, USA: SIAM, Philadelphia, PA, USA · Zbl 0879.65017 |
| [25] | Ehle, B. L., On padé approximations to the exponential function and a-stable methods for the numerical solution of initial value problems, Doctoral dissertation, Waterloo, Canada: University of Waterloo, Waterloo, Canada |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
