A fourth order orthogonal spline collocation method Interface boundary value problem

Objective: A higher order numerical scheme for two-point boundary interface problem with Dirichlet and Neumann boundary condition on two diﬀerent sides is propounded. Methods: Orthogonal cubic spline collocation techniques have been used (OSC) for the two-point interface boundary value problem. To approximate the solution a piecewise Hermite cubic basis functions have been used. Findings: Remarkable features of the OSC are accounted for the numerous applications, theoretical clarity, and convenient execution. The stability and eﬃciency of orthogonal spline collocation methods over B-splines have made the former more preferable than the latter. As against ﬁnite element methods, determining the approximate solution and the coeﬃcients of stiﬀness matrices and mass is relatively fast as the evaluation of integrals is not a requirement. The systematic incorporation of boundary and interface conditions in OSC adds to the list of advantages of preferring this method. Novelty: As against the existing methodologies it becomes clear from our ﬁndings that OSC is dominantly computationally superior. A computational treatment has been implemented on the two-point interface boundary value problem with super-convergent results of derivative at the nodal points, being the noteworthy ﬁnding of the study.


Introduction
The1D-Helmholtz equation under consideration is: with boundary conditions α a y (a) + β a y ′ (a) = g 0 , α b y (b) + β b y ′ (b) = g 1 (1.2) where α a , β a ,α b , β b , g 0 , g 1 are known constants and k 2 is the wave number. We assume the coefficient q(x) to be piece-wise constant or piece-wise continuous with finite jump across the interface = x i , x i ∈ (a, b). For convenience, we assume that ω 2 = k 2 q(x) which is a piece-wise constant or piece-wise continuous across the interface x = x i , and assume that solution y(x) satisfies the natural jump conditions across the interface (y] = 0, ( y ′ ] = 0. The jump conditions across the interface are: denote a partition of I, and set I j = ( x j−1 , x j ] , j = 1, . . . , N + 1, h j = x j − x j−1 and h = max j h j . The Helmholtz equation is used in many physical applications such as acoustics, elastic waves, and electromagnetic waves. The present study intends to provide an efficient numerical skill for Helmholtz problem. Existing literature of theoretical and numerical treatment to Helmholtz equation using finite difference methods (1)(2)(3) , finite element methods (4) and for existence uniqueness results can be found at (5)(6)(7) . 1D and 2D Helmholtz equation have been treated by Xiufang Feng (8) and Xiufang Feng et al. (9) respectively by using high order compact finite difference methods.
This paper treats 1D-Helmholtz equation with piece-wise constant or piecewise continuous functions by employing OSC to it. The stability and efficiency of orthogonal spline collocation methods over B-splines have made the former more preferable than the latter. As against finite element methods, determining the approximate solution and the coefficients of stiffness matrices and mass is relatively fast as the evaluation of integrals is not a requirement. The systematic incorporation of boundary and interface conditions in OSC adds to the list of advantages of preferring this method.
We show that the OSC handle the interface conditions effectively with less discretization. To accomplish the fourth-order accuracy, we utilize piece-wise Hermite cubic basis functions for approximating the solution. This article can be outlined as: Section 2 uses OSC to approximate the solution. Section 3 deals with numerical experiments. Discontinuous data has been used and the solution has been approximated using piece-wise Hermite cubic basis functions. Grid refinement analysis is performed and the order of convergence for −-norm and −-norm is found. Section 4 hosts the conclusion.

Orthogonal spline collocation methods
Here, we employ OSC to approximate the solutions of interface boundary value problem (1.1).
Let H m (I) = {v : v ∈ C m−1 (I) and v m is a piecewise continuous f unction on I}, with norm where, Also set represents a partition of I, and set we assume thatx i ∈ π. In the OSC, the approximate solution, y h , lies in a space C 1 piece-wise polynomials of degree ≥ 3.
Here we choose the space of piece-wise Hermite cubics, M 3 1 (π) as: where C 1 (I) denotes the space of functions which are one times continuously differentiable on I,P 3 represents the set of polynomials of degree ≤ 3 and (v| I j denotes the restriction of the function v to the interval I j .
We denote by M 3,0 . It is to see that M 3 1 (π) and M 3,0 1 (π) are linear spaces of dimensions 2N + 2 and 2N , respectively. We consider the collocation points(ξ j } 2N j=1 , where These are the composite two-point Gauss quadrature points. We now introduce the standard basis for the space The function v i and s i are called value function and slope function, respectively, associated with the node x i ∈ π. With the basis The coefficients are then evaluated with the restriction that y h satisfies (1.1) at the collocation points(ξ j } 2N j=1 , and the boundary conditions (1.2) so that The orthogonal spline collocation approximation for problem (1.1) -(1.2) is stated as: Approximate y h ∈M 3 1 (I) so that α a y (a) + β a y ′ (a) = g a , As only four basis , and, for j = 1, 2, . . . . . . .., N, ] . https://www.indjst.org/

Numerical experiments
In order to arrive at an approximate solution, piece-wise Hermite cubic basis functions will be considered for the experimentation and as for the determination of the order of convergence of the numerical method we will emphasize on grid refinement analysis. The approximate solution y h (x) ∈ M 3 1 on each subinterval (x i−1 , x i ] , i = 1, 2, . . . , N is: where δ i j is the kronecker delta function with δ i j = 1, i f i = j and δ i j = 0, i f i ̸ = j. The expressions for value functions and slope functions, we refer to (3) .
Taking derivatives of (3.1) wrt x, we have, . . , N are two-point Gauss-Legendre quadrature points defined by Similarly, at x = ξ 2 we have the following expression

can be expressed in the matrix form as
where A i and B i structure out as: ] , and where A i and B i structure out as: ] , ] .
Combining (3.5) -(3.6), we obtain an ABD linear system of order 2 N + 2 for  where L b and R b are contributed by lateral boundaries, left and right. The matrix system has been solved using the almost block diagonal (ABD) solver of MATLAB. While existing methods require more number operations (≈ n 3 ) to achieve fourth-order accuracy, the OSC requires only 'n' operations.

Numerical Example
Example 1: The problem under consideration is as follows: with boundary conditions i.e., Dirichlet in one side and Neumann on the other side y (−π) = 0, y The exact solution is given by y (x) = (x − π) 2 sinx. The order of convergence computes out to be: , i = 1, 2, . . . , 5 where y: exact solution, y h i : numerical solution with step size h i . The following table describes the errors in max-norm and order of convergence at nodal points.
https://www.indjst.org/ N.B: Since it is a numerical scheme, so the convergence depends upon the large value N. Initially it may get some deviation but at the higher value of N, it will converge to 4 th order, which has been inferred from the above mentioned table.

Conclusion
An OSC to 1D-Helmholtz equation with discontinuous coefficients has been established in the study. Discontinuous data has been experimented on, using numerical methodologies. Fourth-order convergence at the grid points for ∥y − y h ∥ L ∞ -norm https://www.indjst.org/ and y ′ − y ′ h L ∞ -norm has been found. As against the methods that exist, OSC handles the discontinuous coefficients potently and gives optimal order of convergence for ∥y − y h ∥ L ∞ -norm and super-convergent result for ∥y − y h ∥ L ∞ -norm. Despite having theorized and having computed the OSC for a single point in the interface we can extend our theory to a finite set of points. .