Two classes of implicit-explicit multistep methods for nonlinear stiff initial-value problems. (English) Zbl 1338.65179
Summary: The initial value problems of nonlinear ordinary differential equations which contain stiff and nonstiff terms often arise from many applications. In order to reduce the computation cost, implicit-explicit (IMEX) methods are often applied to these problems, i.e. the stiff and non-stiff terms are discretized by using implicit and explicit methods, respectively. In this paper, we mainly consider the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems, and present two classes of the IMEX multistep methods by combining implicit one-leg methods with explicit linear multistep methods and explicit one-leg methods, respectively. The order conditions and the convergence results of these methods are obtained. Some efficient methods are constructed. Some numerical examples are given to verify the validity of the obtained theoretical results.
MSC:
| 65L04 | Numerical methods for stiff equations |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
Keywords:
stiff problems; singular perturbation problems; implicit-explicit multistep methods; one-leg methods; linear multistep methods; convergenceSoftware:
IRKCReferences:
| [1] | Akrivis, G., Implicit-explicit multistep methods for nonlinear parabolic equations, Math. Comput., 82, 281, 45-68 (2013) · Zbl 1266.65152 |
| [2] | Akrivis, G.; Crouzeix, M.; Makridakis, C., Implicit-explicit multistep methods for quasilinear parabolic equations, Numer. Math., 82, 521-541 (1999) · Zbl 0936.65118 |
| [3] | Akrivis, G.; Smyrlis, Y. S., Implicit-explicit BDF methods for the Kuramoto-Sivashinsky equations, Appl. Numer. Math., 51, 151-169 (2004) · Zbl 1057.65069 |
| [4] | Ascher, U. M.; Ruuth, S. J.; Wetton, B. T.R., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32, 797-823 (1995) · Zbl 0841.65081 |
| [5] | Ascher, U. M.; Ruuth, S. J.; Spiteri, R. J., Implicit-explicit Runge-Kutta methods for time dependent partial differential equations, Appl. Numer. Math., 25, 151-167 (1997) · Zbl 0896.65061 |
| [6] | Auzinger, W.; Frank, R.; Kirlinger, G., Modern convergence theory for stiff initial value problems, J. Comput. Appl. Math., 45, 5-16 (1993) · Zbl 0782.65087 |
| [7] | Baiocchi, C.; Crouzeix, M., On the equivalence of A-stability and G-stability, Appl. Numer. Math., 5, 19-22 (1989) · Zbl 0671.65072 |
| [8] | Boscarino, S., On an accurate third order implicit-explicit Runge-Kutta methods for stiff problems, Appl. Numer. Math., 52, 1515-1528 (2009) · Zbl 1162.65367 |
| [9] | Boscarino, S., Error analysis of implicit-explicit Runge-Kutta methods derived from differential-algebraic systems, SIAM J. Numer. Anal., 45, 4, 1600-1621 (2007) · Zbl 1152.65088 |
| [10] | Boscarino, S.; Russo, G., On a class of uniformly accurate implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, SIAM J. Sci. Comput., 31, 3, 1926-1945 (2009) · Zbl 1193.65162 |
| [11] | Boscarino, S.; Pareschi, L.; Russo, G., Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 35, 1, A22-A51 (2013) · Zbl 1264.65150 |
| [12] | Dahlquist, G., G-stability is equivalent to A-stability, BIT, 18, 384-401 (1978) · Zbl 0413.65057 |
| [13] | Frank, J.; Hundsdorfer, W.; Verwer, J. G., On the stability of implicit-explicit linear multistep methods, Appl. Numer. Math., 25, 193-205 (1997) · Zbl 0887.65094 |
| [14] | Gjesdal, T., Implicit-explicit methods based on strong stability preserving multistep time discretization, Appl. Numer. Math., 57, 911-919 (2007) · Zbl 1122.65064 |
| [15] | Grooms, I.; Julien, K., Linearly implicit-explicit methods for nonlinear PDEs with linear dispersion and dissipation, J. Comput. Phys., 230, 3630-3650 (2011) · Zbl 1218.65099 |
| [16] | Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II (1996), Springer: Springer Stiff and Differential-algebraic Problems · Zbl 0859.65067 |
| [17] | Higueras, I., Strong stability for additive Runge-Kutta methods, SIAM J. Numer. Anal., 44, 4, 1735-1758 (2006) · Zbl 1126.65067 |
| [18] | in’t Hout, K. J., On the contractivity of implicit-explicit linear multistep methods, Appl. Numer. Math., 42, 201-212 (2002) · Zbl 1001.65090 |
| [19] | Hundsdorfer, W.; Steininger, B. L., Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems, BIT, 31, 124-143 (1991) · Zbl 0724.65071 |
| [21] | Hundsdorfer, W.; Ruuth, S. J., IMEX extensions of linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys., 225, 2016-2042 (2007) · Zbl 1123.65068 |
| [22] | Hundsdorfer, W.; Verwer, J. G., Numerical Solution of Time-dependent Advection-diffusion-reaction Equations (2003), Springer · Zbl 1030.65100 |
| [23] | Kanevsky, A.; Carpenter, M. H.; Gottlieb, D.; Hesthaven, J. S., Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes, J. Comput. Phys., 225, 1753-1781 (2007) · Zbl 1123.65097 |
| [24] | Kennedy, C. A.; Carpenter, M. H., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44, 139-181 (2003) · Zbl 1013.65103 |
| [25] | Knoth, O.; Wolke, R., Implicit-explicit Runge-Kutta methods for computing atmospheric reactive flows, Appl. Numer. Math., 28, 327-341 (1998) · Zbl 0934.76058 |
| [26] | Koto, T., IMEX Runge-Kutta schemes for reaction-diffusion equations, J. Comput. Appl. Math., 215, 182-195 (2008) · Zbl 1141.65072 |
| [27] | Koto, T., Stability of IMEX linear multistep methods for ordinary and delay differential equations, Front. Math. China, 4, 1, 113-129 (2009) · Zbl 1396.65114 |
| [28] | Kumar, H.; Mishra, S., Entropy stable numerical schemes for two-fluid plasma equations, J. Sci. Comput., 52, 401-425 (2012) · Zbl 1311.76150 |
| [29] | Kupka, F.; Happenhofer, N.; Higueras, I.; Koch, O., Total-variation-diminishing implicit-explicit Runge-Kutta methods for the simulation of double-diffusion convection in astrophysics, J. Comput. Phys., 231, 3561-3586 (2012) · Zbl 1241.85010 |
| [30] | Layton, W.; Trenchea, C., Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations, Appl. Numer. Math., 62, 112-120 (2012) · Zbl 1237.65101 |
| [31] | Li, Dongfang; Zhang, Chengjian; Wang, Wansheng; Zhang, Yangjing, Implicit-explicit predictor-corrector schemes for nonlinear parabolic differential equations, Appl. Math. Model., 35, 2711-2722 (2011) · Zbl 1219.65098 |
| [32] | Li, Shoufu, Theory of Computational Methods for Stiff Differential Equation (1997), Hunan Science and Technology Press, (in Chinesse) |
| [33] | Liu, Hongyu; Zou, Jun, Some new additive Runge-Kutta methods and their applications, J. Comput. Appl. Math., 190, 74-98 (2006) · Zbl 1089.65066 |
| [34] | Lubich, C., On the convergence of multistep methods for nonlinear stiff differential equations, Numer. Math., 58, 839-853 (1991) · Zbl 0729.65055 |
| [35] | Pareschi, L.; Russo, G., Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25, 1/2, 129-155 (2005) · Zbl 1203.65111 |
| [36] | Shampine, L. F.; Sommeijer, B. P.; Verwer, J. G., IRKC: an IMEX solver for stiff diffusion-reaction PDEs, J. Comput. Appl. Math., 196, 485-497 (2006) · Zbl 1100.65075 |
| [37] | Shen, Wensheng; Zhang, Changjiang; Zhang, Jun, Relaxation method for unsteady convection-diffusion equations, J. Comput. Math. Appl., 61, 908-920 (2011) · Zbl 1217.65176 |
| [38] | Verwer, J. G.; Blom, J. G.; Hundsdorfer, W., An implicit-explicit approach for atmospheric transport-chemistry problems, Appl. Numer. Math., 20, 191-209 (1996) · Zbl 0853.76092 |
| [39] | Verwer, J. G.; Sommeijer, B. P., An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations, SIAM J. Sci. Comput., 25, 5, 1824-1835 (2004) · Zbl 1061.65090 |
| [40] | Wang, Dong; Ruuth, S., Variable step-size implicit-explicit linear multistep methods for time-dependent partial differential equations, J. Comput. Math., 26, 6, 838-855 (2011) · Zbl 1174.65037 |
| [41] | Wolke, R.; Knoth, O., Implicit-explicit Runge-Kutta methods applied to atmospheric chemistry-transport modeling, Environ. Model. Software, 15, 711-719 (2000) |
| [42] | van Zuijler, A. H.; Bijl, H., Implicit and explicit higher order time integration schemes for structural dynamics and fluid-structure interaction computations, Comput. Struct., 83, 93-105 (2005) |
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