ABSTRACT

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.

1. 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 [1-3]. In this paper a best finite-difference method is developed for Helmholtz equation with general boundary conditions on the rectangular domain in R². 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.

2. Statement of Problem

Let be an open rectangular domain in Euclidean R² space with boundary given by. The aim is to determine a function, satisfying equation

(2.1)

with boundary condition

(2.2)

where C in (2.1) is a given number and is the outward normal on.

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 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 of Equation (2.1), (2.2) in the form    

(2.3)

we arrive at equation

which is splitted into two independing equations

(2.4a)

and

(2.4b)

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 and

(2.5)

and

(2.6)

The solution of boundary value problem (2.4a), (2.5) is founded in a closed from

(2.7)

where

and

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.

3. Construction of the Best Finite-Difference Equations

For the numerical solution of problem (2.1), (2.2) is introduced the uniform rectangular grid

where and are the mesh sizes in the x and y directions respectively. Usually, the Equation (2.1) is approximated by the five-point difference equation

(3.1)

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

(3.2)

where. If we denote by the values of the values of and respectively, the using (2.3) the Equation (3.2) may be written as

(3.3)

Due to smoothness assumption of solution as well as, functions and the Taylor series expansion yields

(3.4a)

(3.4b)

Because of (2.4) we have

(3.5)

Taking into account (3.4), (3.5) in (3.3) it follows that

(3.6)

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

(3.7)

There are three cases:

1) Let Then it is easy to show that

(3.8)

2) Let In this case D is given by

(3.9)

3) Let In this case D is given by

(3.10)

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 in (3.7) can be presented as a sum of two ones, i.e.,

(3.11)

where and correspond to the first and second terms in (3.8), (3.9) and (3.10) respectively.

4. 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

we obtain

(4.1)

If in (2.5), then we have

(4.2a)

If then finding from (2.5) and substituting it in (4.1) we get

(4.2b)

where and are given by

(4.3a)

and

(4.3b)

Analogously, it is easy to verify that the exact difference boundary condition for at point is given by

when         (4.4a)

when   (4.4b)

where and are given by

(4.5a)

and

(4.5b)

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:

, when          (4.6a)

when     (4.6b)

and

when         (4.7a)

when    (4.7b)

where are defined by

(4.8a)

(4.8b)

and

(4.9a)

(4.9b)

5. 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

(5.1)

in which we have used (3.11) From this it is clear, that Equation (5.1) will be fulfilled if we choose and such that

(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 (2.1), (2.2) and (2.3) the function will be defermined within an arbitrary multiplicative constant. Therefore the three-point finite-difference Equation (5.2a) can be solved by shooting method starting with, and which are required to be known. Thanks to (4.2) it is possible to find or depending on the. For example, if then is determined by (4.2a) and and ω to be chosen arbitrary. Otherwise, is determined by (4.2b) and and ω to be chosen arbitrary.

Note, that when one of the boundary conditions (2.5), (2.6) is assumed to be homogeneous. For Laplace’s equation we always can leads to equation with homogeneous boundary conditions by change of variables. The exact value of parameter ω must satisfy

(5.3)

where, for examples, when defined by

(5.4)

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

(5.6)

where is a reassigned accuracy.

If the evaluation of 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.

6. Numerical Results

We have tested the efficiency and accuracy of finitedifference scheme (3.7) on the several examples.

Example 1.

with boundary condition

The exact solution is given by

In Table 1 we present the computed values of (exact values of present in brackets) for N = 5 and M = 4. In order to use secant method we need two first approximations and to ω. The iteration was terminated by criterion (5.6) with.

Example 2.

with boundary condition

The exact solution is given by

In Table 2 we present the computed values of

(exact values of present in brackets) for N = 6 and M = 6. In order to use secant method we need two first approximations and to ω. In this example choose and. The exact value of ω is The convergence of was tabulated in Table 3. The iteration was terminated by criterion (5.6) with.

Example 3.

with boundary condition

The exact solution is given by

In Table 4 we present the computed values of

(exact values of present in brackets) for N = 6 and M = 4. In order to use secant

method we need two first approximations and to ω. In this example we were choose and. The exact value of ω is. The convergence of was tabulated in Table 5. The iteration was terminated by criterion (5.6) with ε = 10⁻⁷.

REFERENCES