RK-stable second derivative multistage methods with strong stability preserving based on Taylor series conditions. (English) Zbl 1524.65271
Summary: Time stepping methods are often required for solving system of ordinary differential equations arising from spatial discretization of partial differential equations. In our prior work, we derived sufficient conditions for a subclass of second derivative general linear methods (SGLMs), so-called second derivative diagonally implicit multistage integration methods, to preserve the strong stability properties of spatial discretizations when coupled with forward Euler and a Taylor series conditions which allows for more flexibility in the choice of spatial discretizations. In this work, we extend this theory to the class of the high number of external stages SGLMs with Runge-Kutta stability. Proposed methods up to order five are examined on some one spatial dimension linear and nonlinear problems in PDEs confirming the verification of the theoretical order and potential of such schemes in maintaining some nonlinear stability properties such as and total variation.
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
strong stability preserving; monotonicity; total variation; second derivative methods; Taylor seriesSoftware:
SSPTSmethodsReferences:
| [1] | Abdi, A., Construction of high-order quadratically stable second-derivative general linear methods for the numerical integration of stiff ODEs, J Comput Appl Math, 303, 218-228 (2016) · Zbl 1382.65190 · doi:10.1016/j.cam.2016.02.054 |
| [2] | Abdi, A.; Behzad, B., Efficient Nordsieck second derivative general linear methods: construction and implementation, Calcolo, 55, 28, 1-16 (2018) · Zbl 1416.65199 |
| [3] | Abdi, A.; Conte, D., Implementation of second derivative general linear methods, Calcolo, 57, 20, 1-29 (2020) · Zbl 1453.65155 |
| [4] | Abdi, A.; Hojjati, G., An extension of general linear methods, Numer Algorithms, 57, 149-167 (2011) · Zbl 1228.65111 · doi:10.1007/s11075-010-9420-y |
| [5] | Abdi, A.; Hojjati, G., Maximal order for second derivative general linear methods with Runge-Kutta stability, Appl Numer Math, 61, 1046-1058 (2011) · Zbl 1227.65063 · doi:10.1016/j.apnum.2011.06.004 |
| [6] | Abdi, A.; Hojjati, G., Implementation of Nordsieck second derivative methods for stiff ODEs, Appl Numer Math, 94, 241-253 (2015) · Zbl 1325.65098 · doi:10.1016/j.apnum.2015.04.002 |
| [7] | Abdi, A.; Braś, M.; Hojjati, G., On the construction of second derivative diagonally implicit multistage integration methods, Appl Numer Math, 76, 1-18 (2018) · Zbl 1288.65104 · doi:10.1016/j.apnum.2013.08.006 |
| [8] | Barghi Oskouie, N.; Hojjati, G.; Abdi, A., Efficient second derivative methods with extended stability regions for non-stiff IVPs, Comput Appl Math, 37, 2001-2016 (2018) · Zbl 1404.65062 · doi:10.1007/s40314-017-0436-y |
| [9] | Braś, M.; Cardone, A., Construction of efficient general linear methods for non-stiff differential systems, Math Model Anal, 17, 171-189 (2012) · Zbl 1245.65084 · doi:10.3846/13926292.2012.655789 |
| [10] | Braś, M.; Cardone, A.; D’Ambrosio, R., Implementation of explicit Nordsieck methods with inherent quadratic stability, Math Model Anal, 18, 289-397 (2013) · Zbl 1266.65122 · doi:10.3846/13926292.2013.785039 |
| [11] | Butcher, JC, Numerical methods for ordinary differential equations (2016), New York: Wiley, New York · Zbl 1354.65004 · doi:10.1002/9781119121534 |
| [12] | Butcher, JC; Hojjati, G., Second derivative methods with RK stability, Numer Algorithms, 40, 415-429 (2005) · Zbl 1084.65069 · doi:10.1007/s11075-005-0413-1 |
| [13] | Califano, G.; Izzo, G.; Jackiewicz, Z., Strong stability preserving general linear methods with Runge-Kutta stability, J Sci Comput, 76, 943-968 (2018) · Zbl 1402.65065 · doi:10.1007/s10915-018-0646-5 |
| [14] | Cardone, A.; Jackiewicz, Z., Explicit Nordsieck methods with quadratic stability, Numer Algorithms, 60, 1-25 (2012) · Zbl 1247.65104 · doi:10.1007/s11075-011-9509-y |
| [15] | Cardone, A.; Jackiewicz, Z.; Mittelmann, HD, Optimization-based search for Nordsieck methods of high order with quadratic stability, Math Model Anal, 17, 293-308 (2012) · Zbl 1260.65063 · doi:10.3846/13926292.2012.685497 |
| [16] | Christlieb, AJ; Gottlieb, S.; Grant, Z.; Seal, DC, Explicit strong stability preserving multistage two-derivative time-stepping schemes, J Sci Comput, 68, 914-942 (2016) · Zbl 1352.65289 · doi:10.1007/s10915-016-0164-2 |
| [17] | Constantinescu, EM; Sandu, A., Optimal explicit strong-stability-preserving general linear methods, SIAM J Sci Comput, 32, 3130-3150 (2010) · Zbl 1217.65179 · doi:10.1137/090766206 |
| [18] | Dahlquist, G., A special stability problem for linear multistep methods, BIT Numer Math, 3, 27-43 (1963) · Zbl 0123.11703 · doi:10.1007/BF01963532 |
| [19] | Ditkowski, A.; Gottlieb, S.; Grant, ZJ, Two-derivative error inhibiting schemes and enhanced error inhibiting schemes, SIAM J Numer Anal, 58, 3197-3225 (2020) · Zbl 1453.65164 · doi:10.1137/19M1306129 |
| [20] | Ezzeddine, AK; Hojjati, G.; Abdi, A., Sequential second derivative general linear methods for stiff systems, Bull Iran Math Soc, 40, 83-100 (2014) · Zbl 1302.65163 |
| [21] | Ferracina, L.; Spijker, MN, An extension and analysis of the Shu-Osher representation of Runge-Kutta methods, Math Comput, 74, 201-219 (2004) · Zbl 1058.65098 · doi:10.1090/S0025-5718-04-01664-3 |
| [22] | Ferracina, L.; Spijker, MN, Stepsize restrictions for total-variation-boundedness in general Runge-Kutta procedures, Appl Numer Math, 53, 265-279 (2005) · Zbl 1073.65076 · doi:10.1016/j.apnum.2004.08.024 |
| [23] | Ferracina, L.; Spijker, MN, Strong stability of singly-diagonally-implicit Runge-Kutta methods, Appl Numer Math, 58, 1675-1686 (2008) · Zbl 1153.65080 · doi:10.1016/j.apnum.2007.10.004 |
| [24] | Gottlieb, S., On high order strong stability preserving Runge-Kutta and multi step time discretizations, J Sci Comput, 25, 105-128 (2005) · Zbl 1203.65166 |
| [25] | Gottlieb, S.; Ruuth, SJ, Optimal strong-stability-preserving time-stepping schemes with fast downwind spatial discretizations, J Sci Comput, 27, 289-303 (2006) · Zbl 1115.65092 · doi:10.1007/s10915-005-9054-8 |
| [26] | Gottlieb, S.; Shu, C-W; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev, 43, 89-112 (2001) · Zbl 0967.65098 · doi:10.1137/S003614450036757X |
| [27] | Gottlieb, S.; Ketcheson, DI; Shu, C-W, High order strong stability preserving time discretizations, J Sci Comput, 38, 251-289 (2009) · Zbl 1203.65135 · doi:10.1007/s10915-008-9239-z |
| [28] | Gottlieb, S.; Ketcheson, DI; Shu, C-W, Strong stability preserving Runge-Kutta and multistep time discretizations (2011), Hackensack: World Scientific, Hackensack · Zbl 1241.65064 · doi:10.1142/7498 |
| [29] | Grant, Z.; Gottlieb, S.; Seal, DC, A strong stability preserving analysis for explicit multistage two-derivative time-stepping schemes based on Taylor series conditions, Commun Appl Math Comput, 1, 21-59 (2019) · Zbl 1449.65223 |
| [30] | Higueras, I., On strong stability preserving time discretization methods, J Sci Comput, 21, 193-223 (2004) · Zbl 1074.65095 · doi:10.1023/B:JOMP.0000030075.59237.61 |
| [31] | Higueras, I., Monotonicity for Runge-Kutta methods: inner product norms, J Sci Comput, 24, 97-117 (2005) · Zbl 1078.65554 · doi:10.1007/s10915-004-4789-1 |
| [32] | Higueras, I., Representations of Runge-Kutta methods and strong stability preserving methods, SIAM J Numer Anal, 43, 924-948 (2005) · Zbl 1097.65078 · doi:10.1137/S0036142903427068 |
| [33] | Hundsdorfer, W.; Ruuth, SJ, On monotonicity and boundedness properties of linear multistep methods, Math Comput, 75, 655-672 (2005) · Zbl 1092.65059 · doi:10.1090/S0025-5718-05-01794-1 |
| [34] | Hundsdorfer, W.; Ruuth, SJ; Spiteri, RJ, Monotonicity-preserving linear multistep methods, SIAM J Numer Anal, 41, 605-623 (2003) · Zbl 1050.65070 · doi:10.1137/S0036142902406326 |
| [35] | Izzo, G.; Jackiewicz, Z., Strong stability preserving general linear methods, J Sci Comput, 65, 271-298 (2015) · Zbl 1329.65164 · doi:10.1007/s10915-014-9961-7 |
| [36] | Izzo, G.; Jackiewicz, Z., Strong stability preserving multistage integration methods, Math Model Anal, 20, 552-577 (2015) · Zbl 1488.65175 · doi:10.3846/13926292.2015.1085921 |
| [37] | Izzo, G.; Jackiewicz, Z., Strong stability preserving transformed DIMSIMs, J Comput Appl Math, 343, 174-188 (2019) · Zbl 1524.65268 · doi:10.1016/j.cam.2018.03.018 |
| [38] | Jackiewicz, Z., General linear methods for ordinary differential equations (2009), New Jersey: Wiley, New Jersey · Zbl 1211.65095 · doi:10.1002/9780470522165 |
| [39] | Jiang, G-S; Shu, C-W, Efficient implementation of weighted ENO schemes, J Comput Phys, 126, 202-228 (1996) · Zbl 0877.65065 · doi:10.1006/jcph.1996.0130 |
| [40] | Ketcheson, DI; Gottlieb, S.; Macdonald, CB, Strong stability preserving two-step Runge-Kutta methods, SIAM J Numer Anal, 49, 2618-2639 (2011) · Zbl 1240.65278 · doi:10.1137/10080960X |
| [41] | Moradi, A.; Farzi, J.; Abdi, A., Strong stability preserving second derivative general linear methods, J Sci Comput, 81, 392-435 (2019) · Zbl 1427.65110 · doi:10.1007/s10915-019-01021-1 |
| [42] | Moradi, A.; Abdi, A.; Farzi, J., Strong stability preserving second derivative diagonally implicit multistage integration methods, Appl Numer Math, 150, 536-558 (2020) · Zbl 1439.65077 · doi:10.1016/j.apnum.2019.11.001 |
| [43] | Moradi, A.; Abdi, A.; Farzi, J., Strong stability preserving second derivative general linear methods with Runge-Kutta stability, J Sci Comput, 85, 1, 1-39 (2020) · Zbl 1450.65064 · doi:10.1007/s10915-020-01306-w |
| [44] | Moradi, A.; Sharifi, M.; Abdi, A., Transformed implicit-explicit second derivative diagonally implicit multistage integration methods with strong stability preserving explicit part, Appl Numer Math, 156, 14-31 (2020) · Zbl 1442.65118 · doi:10.1016/j.apnum.2020.04.007 |
| [45] | Moradi, A.; Abdi, A.; Hojjati, G., Strong stability preserving implicit and implicit-explicit second derivative general linear methods with RK stability, Comput Appl Math, 41, 135, 1-23 (2022) · Zbl 1499.65286 |
| [46] | Moradi, A.; Abdi, A.; Hojjati, G., High order explicit second derivative methods with strong stability properties based on Taylor series conditions, ANZIAM J, 64, 264-291 (2022) · Zbl 1512.65139 · doi:10.1017/S1446181122000128 |
| [47] | Movahedinejad, A.; Hojjati, G.; Abdi, A., Second derivative general linear methods with inherent Runge-Kutta stability, Numer Algorithms, 73, 371-389 (2016) · Zbl 1351.65051 · doi:10.1007/s11075-016-0099-6 |
| [48] | Ramazani, P.; Abdi, A.; Hojjati, G.; Moradi, A., Explicit Nordsieck second derivative general linear methods, ANZIAM J, 64, 69-88 (2022) · Zbl 1492.65195 |
| [49] | Ruuth, SJ; Hundsdorfer, W., High-order linear multistep methods with general monotonicity and boundedness properties, J Comput Phys, 209, 226-248 (2005) · Zbl 1074.65085 · doi:10.1016/j.jcp.2005.02.029 |
| [50] | Schütz, J.; Seal, DC; Jaust, A., Implicit multiderivative collocation solvers for linear partial differential equations with discontinuous Galerkin spatial discretizations, J Sci Comput, 73, 1145-1163 (2017) · Zbl 1381.65078 · doi:10.1007/s10915-017-0485-9 |
| [51] | Shu, C-W, Total-variation diminishing time discretizations, J Sci Comput, 9, 1073-1084 (1988) · Zbl 0662.65081 |
| [52] | Spijker, MN, Stepsize conditions for general monotonicity in numerical initial value problems, SIAM J Numer Anal, 45, 1226-1245 (2007) · Zbl 1144.65055 · doi:10.1137/060661739 |
| [53] | Spiteri, RJ; Ruuth, SJ, A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J Numer Anal, 40, 469-491 (2002) · Zbl 1020.65064 · doi:10.1137/S0036142901389025 |
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