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Bhal, Santosh Kumar; Mohanty, Sahasransu; Nandi, Ashish Kumar

Fourth order orthogonal spline collocation methods for two-point boundary value problems with interfaces. (English) Zbl 1507.65127

J. Anal. 31, No. 1, 791-819 (2023).
Summary: Orthogonal spline collocation methods (OSC) find application in solving two-point boundary value problems (BVPs) with interfaces. We find a solution to the one-dimensional Helmholtz equation with discontinuous wave coefficients by the standard OSC approach with piece-wise Hermite bases of cubic behaviour. For the self-adjoint two-point BVPs with interfaces, we incorporate cubic monomial bases to OSC. Results of numerical experiments demonstrate fourth-order accuracy in the \(L^{\infty}\) and \(L^2\) norms, and third-order accuracy in the \(H^1\) norm. Moreover, we observe fourth order super-convergence in the nodal error of the derivative of the OSC approximation. We observe super-convergence at nodal points when we use quartics. Almost block diagonal linear systems that structure out from each OSC approach are computed with the help of MATLAB.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations

Keywords:

two-point BVPs; Helmholtz equation; interface conditions; orthogonal spline collocation methods; monomial bases; almost block diagonal linear systems

Software:

ARCELO; ABDPACK; COLROW; Matlab
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Full Text: DOI

References:

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