Orthogonal spline collocation methods for 1D-parabolic problems with interfaces

Document Type : Original Article

Authors

Santosh Kumar Bhal ¹

Snehalata Behura ²

Ashish Kumar Nandi ³

¹ Department of Mathematics, School of Applied Sciences, Centurion University of Technology and Management, Paralakhemundi, India

² Department of Mathematics, CIPET:IPT, Bhubaneswar-751024, India.

³ Mathematics Division, School of Advanced Sciences, Vellore Institute of Technology, Chennai, 600127, Tamil Nadu, India.

10.22034/kjm.2022.286589.2263

Abstract

Orthogonal spline collocation methods (OSC) are used to solve one dimensional heat conduction problems with interfaces. Cubic monomial basis functions are used to approximate the solution for spatial discretization and the Crank-Nicolson method for time stepping. Existence and uniqueness results are established for a discrete problem. This method is easily extended to monomials of higher degree. We present the results of experiments involving several examples which show the efficiency of OSC method. For both cubic and quartic basis functions, the results of numerical experiments demonstrate fourth-order accuracy in L∞ and L2 norms, and third-order accuracy in the
H1 norm. Moreover, sixth order superconvergence in nodal error of derivative of the OSC approximation for quartics is observed. OSC approach gives rise to almost block diagonal linear systems which are solved using standard software.

Keywords

Heat conduction equation

Orthogonal cubic spline collocation methods

Discontinuous data

Cubic monomial basis functions

Almost block diagonal (ABD) matrix

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