Abstract
In this paper a singularly perturbed Riccati equation is considered. The problem is solved by using a hybrid finite difference method on a Shishkin mesh. The method is composed of the midpoint upwind scheme and the backward Euler scheme based on the relation between the local mesh width and the perturbation parameter. It is proved that the scheme is almost second-order convergent, in the global maximum norm, independently of the singular perturbation parameter. Numerical experiments support the theoretical results.
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de Boor, C.: Good approximation by splines with variable knots. In: Meir, A., Sharma, A. (eds.) Spline Functions and Approximation Theory, Proceedings of Symposium held at the University of Alberta, Edmonton, May 29–June 1, 1972. Basel, Birkhäuser (1973)
Carroll, J.: Exponentially fitted one-step methods for the numerical solution of the scalar Riccati equation. J. Comput. Appl. Math. 16, 9–25 (1986)
Cen, Z., Erdogan, F., Xu, A.: An almost second order uniformly convergent scheme for a singularly perturbed initial value problem. Numer. Algor. 67, 457–476 (2014)
Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman & Hall Press, London (2000)
González-Pinto, S., Casasús, L., González-Vera, P.: Two uniform schemes for the singularly perturbed Riccati equation. Comput. Math. Appl. 23, 75–85 (1992)
Grodt, T., Gajic, Z.: The recursive reduced-order solution of the singularly perturbed matrix differential Riccati equation. IEEE Trans. Autom. Control 33, 751–754 (1988)
Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comp. 32, 1025–1039 (1978)
Kenney, C.S., Leipnik, R.B.: Numerical integration of the differential matrix Riccati equation. IEEE Trans. Autom. Control 30, 962–970 (1985)
Kumar, M., Singh, P., Mishra, H.K.: A recent survey on computational techniques for solving singularly perturbed boundary value problems. Int. J. Computer Math. 84, 1439–1463 (2007)
Linß, T.: Sufficient conditions for uniform convergence on layer-adapted grids. Appl. Numer. Math. 37, 241–255 (2001)
O Reilly, M.J.: A uniform scheme for the singularly perturbed Riccati equation. Numer. Math. 50, 483–501 (1987)
O Reilly, M.J., O’Riordan, E.: A Shishkin mesh for a singularly perturbed Riccati equation. J. Comput. Appl. Math. 182, 372–387 (2005)
O’Riordan, E., Quinn, J.: Parameter-uniform numerical methods for some singularly perturbed nonlinear initial value problems. Numer. Algor. 61, 579–611 (2012)
Roos, H.-G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems, 2nd edn., vol. 24. Springer, Berlin (2008)
Stynes, M., Roos, H.-G.: The midpoint upwind scheme. Appl. Numer. Math. 23, 361–374 (1997)
Vulanović, R.: Boundary Shcok Problems and Singularly Perturbed Riccati Equations. In: Hegarty, A.F., Kopteva, N., O’Riordan, E., Stynes, M. (eds.): BALL 2008 - Boundary and Interior Layers (Lecture Notes in Computational Science and Engineering, Vol. 69), pp 277–285. Springer, Berlin/Heidelberg (2009)
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Cen, Z., Xu, A. & Le, A. On the hybrid finite difference scheme for a singularly perturbed Riccati equation. Numer Algor 71, 417–436 (2016). https://doi.org/10.1007/s11075-015-0001-y
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DOI: https://doi.org/10.1007/s11075-015-0001-y