The1D Helmholtz equation under consideration is:
with boundary conditions
For convenience, we assume that
The jump conditions across the interface are:
Let
denote a partition of
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
This paper treats 1DHelmholtz equation with piecewise constant or piecewise continuous functions by employing OSC to it. The stability and efficiency of orthogonal spline collocation methods over Bsplines 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 fourthorder accuracy, we utilize piecewise 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 piecewise Hermite cubic basis functions. Grid refinement analysis is performed and the order of convergence for
Here, we employ OSC to approximate the solutions of interface boundary value problem (1.1).
Let
with norm
,
where,
Also set
Let
represents a partition of
we assume that
where
We denote by
It is to see that
We consider the collocation points
These are the composite twopoint Gauss quadrature points.
We now introduce the standard basis for the space
where
The function
where,
The coefficients are then evaluated with the restriction that
The orthogonal spline collocation approximation for problem (1.1)  (1.2) is stated as:
Approximate
As only four basis functions
where
In order to arrive at an approximate solution, piecewise 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
where,
Since
where
Taking derivatives of (3.1) wrt x, we have,
and
Now substituting equations (3.1)  (3.2) in (1.1) and the resulting equation, we calculate on
Similarly, at
Let
where
In the similar manner, the collocation equations on the subintervals [
where
Combining (3.5)  (3.6), we obtain an ABD linear system of order 2 N + 2 for
where
Example 1: The problem under consideration is as follows:
with boundary conditions i.e., Dirichlet in one side and Neumann on the other side
where
The exact solution is given by
The order of convergence computes out to be:
where
The following table describes the errors in maxnorm and order of convergence at nodal points.





N 

order 

order 
8 
2.2685e01 

5.5156e02 

16 
1.3846e03 
7.3561e+00 
3.3897e03 
4.0243e+00 
32 
1.0897e04 
3.6674e+00 
1.6551e04 
4.3562e+00 
64 
6.8095e06 
4.0003e+00 
1.1538e05 
3.8424e+00 
128 
4.2561e07 
3.9999e+00 
7.2997e07 
3.9824e+00 
256 
2.6628e08 
3.9985e+00 
4.5619e08 
4.0001e+00 
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.





N 

order 

order 
8 
3.3212e01 

8.6859e02 

16 
6.7995e03 
5.6102e+00 
7.2906e03 
3.5746e+00 
32 
4.2674e04 
3.9940e+00 
1.1388e03 
2.6786e+00 
64 
2.7118e05 
3.9761e+00 
8.8430e05 
3.6868e+00 
128 
1.7192e06 
3.9794e+00 
5.6226e06 
3.9752e+00 
256 
1.0749e07 
3.9994e+00 
3.5136e07 
4.0002e+00 
Example 2: We consider the following problem
With boundary conditions i.e., Dirichlet on one side and Neumann on the other side
where,
The exact solution is given by
The expression for





N 

order 

order 
8 
1.4615e05 

1.1332e05 

16 
8.7474e07 
4.0624e+00 
6.7401e07 
4.0714e+00 
32 
5.3506e08 
4.0311e+00 
4.1092e08 
4.0358e+00 
64 
3.3080e09 
4.0157e+00 
2.5361e09 
4.0182e+00 
128 
2.0561e10 
4.0079e+00 
1.5749e10 
4.0092e+00 
256 
1.2815e11 
4.0040e+00 
9.8099e12 
4.0049e+00 
An OSC to 1D Helmholtz equation with discontinuous coefficients has been established in the study. Discontinuous data has been experimented on, using numerical methodologies. Fourthorder convergence at the grid points for
.