A self adjusting multirate algorithm for robust time discretization of partial differential equations. (English) Zbl 1454.65045
Summary: We show the distinctive potential advantages of a self adjusting multirate method based on diagonally implicit solvers for the robust time discretization of partial differential equations. The properties of the specific ODE methods considered are reviewed, with special focus on the TR-BDF2 solver. A general expression for the stability function of a generic one stage multirate method is derived, which allows to study numerically the stability properties of the proposed algorithm in a number of examples relevant for applications. Several numerical experiments, aimed at the time discretization of hyperbolic partial differential equations, demonstrate the efficiency and accuracy of the resulting approach.
MSC:
| 65L04 | Numerical methods for stiff equations |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
Keywords:
partial differential equations; time discretization; stiff problems; diagonally implicit solversSoftware:
TR-BDF2References:
| [1] | Andrus, J., Numerical solution of systems of ordinary differential equations separated into subsystems, SIAM J. Numer. Anal., 16, 605-611 (1979) · Zbl 0421.65044 |
| [2] | Gear, C.; Wells, D., Multirate linear multistep methods, BIT Numer. Math., 24, 484-502 (1984) · Zbl 0555.65046 |
| [3] | Klemp, J.; Wilhelmson, R., The simulation of three-dimensional convective storm dynamics, J. Atmos. Sci., 35, 1070-1096 (1978) |
| [4] | Andrus, J., Stability of a multi-rate method for numerical integration of ODEs, Comput. Math. Appl., 25, 3-14 (1993) · Zbl 0771.65037 |
| [5] | Günther, M.; Kvaernø, A.; Rentrop, P., Multirate partitioned Runge-Kutta methods, BIT Numer. Math., 41, 1504-1514 (2001) · Zbl 0990.65081 |
| [6] | Baldauf, M., Linear stability analysis of Runge-Kutta-based partial time-splitting schemes for the Euler equations, Mon. Weather Rev., 138, 4475-4496 (2010) |
| [7] | Skamarock, W.; Klemp, J., The stability of time-split numerical methods for the hydrostatic and the nonhydrostatic elastic equations, Mon. Weather Rev., 120, 2109-2127 (1992) |
| [8] | Savcenco, V.; Hundsdorfer, W.; Verwer, J., A multirate time stepping strategy for stiff ordinary differential equations, BIT Numer. Math., 47, 137-155 (2007) · Zbl 1113.65071 |
| [9] | Savcenco, V., Comparison of the asymptotic stability properties for two multirate strategies, J. Comput. Appl. Math., 220, 508-524 (2008) · Zbl 1146.65060 |
| [10] | Hundsdorfer, W.; Savcenco, V., Analysis of a multirate theta-method for stiff ODEs, Appl. Numer. Math., 59, 693-706 (2009) · Zbl 1163.65049 |
| [11] | Kuhn, K.; Lang, J., Comparison of the asymptotic stability for multirate Rosenbrock methods, J. Comput. Appl. Math., 262, 139-149 (2014) · Zbl 1301.65072 |
| [12] | Ranade, A., Multirate Algorithms Based on DIRK Methods For Large Scale System Simulation (2016), Politecnico di Milano, (Ph.D. thesis) |
| [13] | Bank, R.; Coughran, W.; Fichtner, W.; Grosse, E.; Rose, D.; Smith, R., Transient simulation of silicon devices and circuits, IEEE Trans. Electron Devices, 32, 1992-2007 (1985) |
| [14] | Hosea, M.; Shampine, L., Analysis and implementation of TR-BDF2, Appl. Numer. Math., 20, 21-37 (1996) · Zbl 0859.65076 |
| [15] | Giraldo, F.; Kelly, J.; Constantinescu, E., Implicit-explicit formulations of a three-dimensional nonhydrostatic unified model of the atmosphere (NUMA), SIAM J. Sci. Comput., 35, 5, 1162-1194 (2013) · Zbl 1280.86008 |
| [16] | Delpopolo, L., Application of the Multirate TR-BDF2 Method to the Time Discretization of Nonlinear Conservation Laws (2015), Degree in Mathematical Engineering, Politecnico di Milano, (Master’s thesis) |
| [17] | Tumolo, G.; Bonaventura, L., A semi-implicit, semi-Lagrangian, DG framework for adaptive numerical weather prediction, Q. J. R. Meteorol. Soc., 141, 2582-2601 (2015) |
| [18] | Bonaventura, L.; Nieto, E. F.; Diaz, J. G.; Reina, G. N., Multilayer shallow water models with locally variable number of layers and semi-implicit time discretization, J. Comput. Phys., 309, 209-234 (2018) · Zbl 1392.76012 |
| [19] | Fok, P., A linearly fourth order multirate Runge-Kutta method with error control, J. Sci. Comput., 1-19 (2015) |
| [20] | Söderlind, G.; Wang, L., Evaluating numerical ODE/DAE methods, algorithms and software, J. Comput. Appl. Math., 185, 244-260 (2006) · Zbl 1081.65533 |
| [21] | Lambert, J., Numerical Methods for Ordinary Differential Systems: The Initial Value Problem (1991), Wiley · Zbl 0745.65049 |
| [22] | Bonaventura, L.; Rocca, A. D., Unconditionally strong stability preserving extensions of the TR-BDF2 method, J. Sci. Comput., 70, 859-895 (2017) · Zbl 1361.65046 |
| [23] | Delpopolo, L.; Bonaventura, L.; Scotti, A.; Formaggia, L., A conservative implicit multirate method for hyperbolic problems, Comput. Geosci., 23, 647-664 (2019) · Zbl 1420.65092 |
| [24] | Bonaventura, L.; Casella, F.; Delpopolo, L.; Ranade, A., A Self Adjusting Multirate Algorithm Based on the TR-BDF2 MethodTech. Rep. 8/2018 (2018), MOX - Politecnico di Milano |
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.
