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Discretization of singular systems and error estimation

100%

Karampetakis N. P. , Karamichalis R.

International Journal of Applied Mathematics and Computer Science

2014

tom Vol. 24, no. 1

65--73

This paper proposes a discretization technique for a descriptor differential system. The methodology used is both triangular first order hold discretization and zero order hold for the input function. Upper bounds for the error between the continuous and the discrete time solution are produced for both discretization methods and are shown to be better than any other existing method in the literature.

Local Recovery of Sub-Crustal Stress Due to Mantle Convection from Satellite-to-Satellite Tracking Data

80%

Šprlák M. , Eshagh M.

Acta Geophysica

2016

tom Vol. 64, no. 4

904--929

Two integral transformations between the stress function, differentiation of which gives the meridian and prime vertical components of the sub-crustal stress due to mantle convection, and the satellite-to-satellite tracking (SST) data are presented in this article. In the first one, the SST data are the disturbing potential differences between twin-satellites and in the second one the line-of-sight (LOS) gravity disturbances. It is shown that the corresponding integral kernels are well-behaving and therefore suitable for inversion and recovery of the stress function from the SST data. Recovery of the stress function and the stress components is also tested in numerical experiments using simulated SST data. Numerical studies over the Himalayas show that inverting the disturbing potential differences leads to a smoother stress function than from inverting LOS gravity disturbances. Application of the presented integral formulae allows for recovery of the stress from the satellite mission GRACE and its planned successor.

Local accuracy and error bounds of the improved Runge-Kutta numerical methods

80%

Qureshi S. , Memon Z. , Shaikh A. A.

Journal of Applied Mathematics and Computational Mechanics

2018

tom Vol. 17, nr 4

73--84

In this paper, explicit Improved Runge-Kutta (IRK) methods with two, three and four stages have been analyzed in detail to derive the error estimates inherent in them whereas their convergence, order of local accuracy, stability and arithmetic complexity have been proved in the relevant literature. Using single and multivariate Taylor series expansion for a mathematical function of one and two variables respectively, slopes involved in the IRK methods have been expanded in order to obtain the general expression for the leading or principal term in the local truncation error of the methods. In addition to this, principal error functions of the methods have also been derived using the idea of Lotkin bounds which consequently gave rise to the error estimates for the IRK methods. Later, these error estimates were compared with error estimates of the two, three, and four-stage standard explicit Runge-Kutta (RK) methods to show the better performance of the IRK methods in terms of the error bounds on the constant step-size h used for solving the initial value problems in ordinary differential equations. Finally, a couple of initial value problems have been tested to determine the maximum absolute global errors, absolute errors at the final nodal point of the integration interval and the CPU times (seconds) for all the methods under consideration to get a better idea of how the methods behave in a particular situation especially when it comes to analyzing the error terms.