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Avaji, M.; Jodayree Akbarfam, A.; Haghighi, A.

Stability analysis of high order Runge-Kutta methods for index 1 stochastic differential-algebraic equations with scalar noise. (English) Zbl 1433.65010

Appl. Math. Comput. 362, Article ID 124544, 14 p. (2019).
Summary: In this paper, a new class of implicit stochastic Runge-Kutta (SRK) methods is constructed for numerically solving systems of index 1 stochastic differential-algebraic equations (SDAEs) with scalar multiplicative noise. By applying rooted tree theory analysis, the family of coefficients of the proposed methods of order 1.5 are calculated in the mean-square sense. In particular, we derive some four-stage stiffly accurate semi-implicit SRK methods for approximating index 1 SDAEs with scalar noise. For these methods, first, MS-stability functions, applied to a scalar linear test equation with multiplicative noise, are calculated. Then, their regions of MS-stability are compared with the corresponding MS-stability region of the original SDE. Accordingly, we illustrate that the proposed schemes seems to have good stability properties. Numerical results will be presented to check the convergence order and computational efficiency of the new methods.

Cited in 3 Documents

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L80 Numerical methods for differential-algebraic equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness

Keywords:

stochastic differential-algebraic equations; stochastic Runge-Kutta methods; stiffly accurate; MS-stability; A-stability

Software:

RODAS
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Full Text: DOI

References:

[1] Schein, O.; Denk, G., Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits, J. Comput. Appl. Math., 100, 1, 77-92 (1998) · Zbl 0928.65014
[2] Shaker Verlag, Aachen · Zbl 0952.60067
[3] Römisch, W.; Winkler, R., Stochastic DAEs in circuit simulation, (Antreich, K.; Bulirsch, R.; Gilg, A.; Rentrop, P., Modeling, Simulation, and Optimization of Integrated Circuits (2003), Birkhäuser Basel: Birkhäuser Basel Basel), 303-318 · Zbl 1045.65008
[4] Winkler, R., Stochastic differential algebraic equations of index 1 and applications in circuit simulation, J. Comput. Appl. Math., 157, 2, 447-505 (2003) · Zbl 1043.65010
[5] Brenan, K.; Campbell, S.; Petzold, L., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 14 (1996), SIAM · Zbl 0844.65058
[6] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems (2010), Springer, London New York · Zbl 1192.65097
[7] Lamour, R.; März, R.; Tischendorf, C., Differential-algebraic equations: a projector based analysis (2013), Springer Science & Business Media · Zbl 1276.65045
[8] Kloeden, P.; Platen, E., Numerical Solution of Stochastic Differential Equations (1999), Springer, Berlin
[9] Mao, X., Stochastic differential equations and applications (2007), Elsevier · Zbl 1138.60005
[10] Khasminskii, R., Stochastic stability of differential equations, 66 (2011), Springer Science and Business Media
[11] Alabert, A.; Ferrante, M., Linear stochastic differential-algebraic equations with constant coefficients, Electron. Commun. Probability, 11, 316-335 (2006) · Zbl 1135.60033
[12] Sickenberger, T.; Weinmüller, E.; Winkler, R., Local error estimates for moderately smooth problems: Part II-SDEs and SDAEs with small noise, BIT Numer. Math., 49, 217-245 (2009) · Zbl 1170.65005
[13] Wang, K.; Crow, M. L., Numerical simulation of stochastic differential algebraic equations for power system transient stability with random loads, 2011 IEEE Power and Energy Society General Meeting, 1-8 (2011), IEEE
[14] Küpper, D.; Kaænø, A.; Rößler, A., A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scaler noise, BIT Numer. Math., 53, 2, 437-455 (2012) · Zbl 1247.65006
[15] Küpper, D.; Kaænø, A.; Rößler, A., Stability analysis and classification of Runge-Kutta methods for index 1 stochastic differential-algebraic equations with scalar noise, Appl. Numer. Math., 96, 24-44 (2015) · Zbl 1321.65013
[16] Saito, Y.; Mitsui, T., Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33, 6, 2254-2267 (1996) · Zbl 0869.60052
[17] Liesen, J.; Mehrmann, V., Linear Algebra (2015), Springer · Zbl 1334.15001
[18] Rößler, A., Runge-kutta methods for the strong approximation of solutions of stochastic differential equations, SIAM J. Numer. Anal., 48, 3, 922-952 (2010) · Zbl 1231.65015
[19] Rößler, A., Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations, Stochastic Anal. Appl., 24, 1, 97-134 (2006) · Zbl 1094.65008
[20] Rößler, A., Strong and weak approximation methods for stochastic differential equations some recent developments, Recent developments in applied probability and statistics, 127-153 (2010), Springer · Zbl 1210.60073
[21] Shaker Verlag, Aachen · Zbl 1017.65002
[22] Rößler, A., Second order Runge-Kutta methods for itô stochastic differential equations, SIAM J. Numer. Anal., 47, 3, 1713-1738 (2009) · Zbl 1193.65006
[23] Amiri, S.; Hosseini, S. M., A class of balanced stochastic Runge-Kutta methods for stiff sde systems, Numer. Algorithms, 69, 3, 531-552 (2015) · Zbl 1323.65004
[24] Rathinasamy, A.; Nair, P., Asymptotic mean-square stability of weak second-order balanced stochastic rungekutta methods for multi-dimensional itô stochastic differential systems, Appl. Math. Comput., 332, 276-303 (2018) · Zbl 1427.65008
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