Skip to main content Accessibility help
×
Home

Application of gPCRK Methods to Nonlinear Random Differential Equations with Piecewise Constant Argument

  • Chengjian Zhang (a1) and Wenjie Shi (a1) (a2)

Abstract

We propose a class of numerical methods for solving nonlinear random differential equations with piecewise constant argument, called gPCRK methods as they combine generalised polynomial chaos with Runge-Kutta methods. An error analysis is presented involving the error arising from a finite-dimensional noise assumption, the projection error, the aliasing error and the discretisation error. A numerical example is given to illustrate the effectiveness of this approach.

Copyright

Corresponding author

*Corresponding author. Email addresses: cjzhang@mail.hust.edu.cn (C. Zhang), wenjieshihust@gmail.com (W. Shi)

References

Hide All
[1] Babuska, I., Nobile, F. and Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal. 45, 10051034 (2007).
[2] Beck, J., Nobile, F., Tamellini, L. and Tempone, R., Implementation of optimal Galerkin and collocation approximations of PDEs with random coefficients, ESIAM Proc. 33, 1021 (2011).
[3] Ernst, O.G., Mugler, A., Starkloff, H.J. and Ullmann, E., On the convergence of generalized polynomial chaos expansions, ESIAM Math. Model. Numer. 46, 317339 (2012).
[4] Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York (1991).
[5] Hairer, E., Nørsett, S. P. and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin (1993).
[6] Liu, M., Ma, S. and Yang, Z., Stability analysis of Runge-Kutta methods for unbounded retarded differential equations with piecewise continuous arguments, Appl. Math. Comput. 191, 5766 (2007).
[7] Liu, X. and Liu, M., Asymptotic stability of Runge-Kutta methods for nonlinear differential equations with piecewise continuous arguments, J. Comput. Appl. Math. 280, 265274 (2015).
[8] Narayan, A. and Zhou, T., Stochastic collocation on unstructured multivariate meshes, Commun. Comput. Phys. 18, 136 (2015).
[9] Shi, W. and Zhang, C., Error analysis of generalized polynomial chaos for nonlinear random ordinary differential equations, Appl. Numer. Math. 62, 19541964 (2012).
[10] Shi, W. and Zhang, C., Generalized polynomial chaos for nonlinear random Pantograph equations, Acta Math. Appl. Sin. English Ser. 32, 685700 (2016).
[11] Tang, T. and Zhou, T., Recent developments in high order numerical methods for uncertainty quantification, Sci. Sin. Math. 45, 891928 (2015).
[12] Wang, W. and Li, S., Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay, Appl. Math. Lett. 21, 983991 (2008).
[13] Wang, W., Stability of solutions of nonlinear neutral differential equations with piecewise constant delay and their discretisations, Appl. Math. Comput. 219, 45904600 (2013).
[14] Wiener, J., Generalized Solutions of Functional Differential Equations, World Scientific, Singapore (1993).
[15] Xiu, D., Fast numerical methods for stochastic computations: A review, Commun. Comput. Phys. 5, 242272 (2009).
[16] Xiu, D., Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, New Jersey (2010).
[17] Xiu, D. and Karniadakis, G.E., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput. 24, 619644 (2002).
[18] Zhou, T., A stochastic collocation method for delay differential equations with random input, Adv. Appl. Math. Mech. 6, 403418 (2014).

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed