The Best Finite-Difference Scheme for the Helmholtz Equation

The best finite-difference scheme for the Helmholtz equation is suggested. A method of solving obtained finite-difference scheme is developed. The efficiency and accuracy of method were tested on several examples.


Introduction
The finite-difference method is a standard numerical method for solving boundary value problems.Recently, considerable attention has been attracted to construct a best (or exact) difference approximation for some ordinary and partial differential equations [1][2][3].In this paper a best finite-difference method is developed for Helmholtz equation with general boundary conditions on the rectangular domain in R 2 .The method proposed here comes out from [4] and is based on separation of variables method and expansion of one-dimensional threepoint difference operators for sufficiently smooth solution.The paper is organized as follows.The statement of problem and the separation of variables method are considered in Section 2. A detailed description of the best difference approximation to the Helmholtz equation in rectangular domain is given in Section 3.
Section 4 is devoted to derive the best approximation for the given third kind boundary conditions.The method of solution for the obtained difference equations is considered in Section 5 and numerical examples are given last Section 6.

Statement of Problem
Let be an open rectangular domain in Euclidean R 2 space with boundary given by    It is well known that the stabilized oscillation problems and diffusing processes in gas lead to the so called Helmholtz Equation (2.1) with a positive coefficient 2 .

C  
The diffusing process in the moving field leads to the Equation (2.1) with negative coefficient If C = 0 the Equation (2.1) leads to Laplace's ones.Obviously, the properties of the solution of Equation (2.1) depend essentially upon the sign of the coefficient C in (2.1).We will assume that the problem (2.1), (2.2) has an unique and sufficiently smooth solution.
By virtue of variables method looking for the solution   , u x y of Equation (2.1), (2.2) in the form we arrive at equation which is splitted into two independing equations where the unknown separation constant ω is to be found.By virtue of (2.3) the boundary condition (2.2) is splitted info ones for The solution of boundary value problem (2.4a), (2.5) is founded in a closed from where When ω < 0 the functions sh and ch in (2.7) are to be replaced by sin and cos respectively and ω replaced by -ω.Analogously, we can find the solutions of boundary value problem (2.4b) and (2.6) in closed form.Then from (2.3) and (2.7) clear, that the problem consists in determining the separation constant ω.

Construction of the Best Finite-Difference Equations
For the numerical solution of problem (2.1), (2.2) is introduced the uniform rectangular grid : are the mesh sizes in the x and y directions respectively.Usually, the Equation (2.1) is approximated by the five-point difference equation The local discretization error of the Equation (3.1) is of order.Now we describe how to derive the best difference scheme for Equation (2.1).To this end, we consider expression U y respectively, the using (2.3) the Equation (3.2) may be written as Due to smoothness assumption of solution   , , u x y as well as, functions the Taylor series expansion yields Because of (2.4) we have where E is unit operator.The difference Equation (3.6) contains unknown nonzero parameter ω and therefore it may be considered as a nonlinear equation with respect to the parameter ω and The series in (3.6) may be expressed through analytical functions depending on the sign of quantities ω and β and thereby the Equation (3.6) can be rewritten as There are three cases: In this case D is given by Thus we obtain the best (or exact) five-point difference Equation (3.7) for the Equation (2.1) (see, for example, Mickens [2] and Agarwal [1]).The function   D  in (3.7) can be presented as a sum of two ones,

The Best Finite-Difference Boundary Condition
Now we will derive a best difference boundary condition for (2.5), (2.6).Using (2.4) in the Taylor series expansion   in (2.5), then we have Analogously, it is easy to verify that the exact difference boundary condition for at point where In the same way, as before, one can construct the best difference boundary conditions for .We omit the evaluation and present only the final results: , where

Method for Solution of Finite-Difference Equations
In this section we consider a method for solving the finite-difference Equations (3.7).For this purpose we rewrite it in the from in which we have used (3.11)From this it is clear, that Equation (5.1) will be fulfilled if we choose and 1i (5.2a) (5.2b) The last weakly coupled system of Equation (5.2) is splitted into two equations with corresponding boundary conditions.First, we consider the Equation (5.2a) subject to boundary conditions (4.2) and (4.4).
According to (2. The nonlinear Equation (5.3) can be solved by Newton's method: (5.5) The value in the dominator of (5.5) is found by differentiating the Equation (5.4) and (5.2a) with respect to ω.The iteration process (5.5) is terminated by criterion where is a reassigned accuracy.
causes some difficulty we can use secant method instead of Newton's ones.After finding ω the three-point difference equations (5.2b) with boundary conditions (4.6), (4.7) can be solved by elimination method.

Numerical Results
We have tested the efficiency and accuracy of finitedifference scheme (3.7) on the several examples.
Example 1. 0, 0 , The exact solution is given by In Table 1 we present the computed values of brackets) for N = 5 and M = 4.In order to use secant method we need two first approximations 0  and 1  to ω.The iteration was terminated by criterion (5.6) with .

    was tabulated in
The exact solution is given by The exact solution is given by In Table 2 we present the computed values of In Table 4 we present the computed values of 1 2 ij i j (exact values of present in brackets) for N = 6 and M = 6.In order to use secant where C in (2.1) is a given number and n   is the outward normal on  .

2 
U  from (2.5) and substituting it in (4.1) we get  are given by

 and 1  2   . The exact value of ω is 1 
to ω.In this example we were choose 0 3   and 1  .The convergence of k  was tabulated in Table5.The iteration was terminated by criterion (5.6) with ε = 10 −7 .

Table 3 .
The iteration was termi-