Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition

There are many problems in the field of science, engineering and technology which can be solved by differential equations formulation. The wave equation is a second order linear hyperbolic partial differential equation that describes the propagation of variety of waves, such as sound or water waves. In this paper we consider the convergence analysis of the explicit schemes for solving one dimensional, time-dependent wave equation with Drichlet and Neumann boundary condition. Taylor's series expansion is used to expand the finite difference approximations in the explicit scheme. We present the derivation of the schemes and develop a computer program to implement it We use spectral radius of Matrix obtained from discretization and Von Neumann stability condition to determine stability, and consistence of the method from truncated error from discretized method. Using Lax Equivalence Theorem, convergence of the methods was described by testing consistency and stability of the methods. And it is found out that the scheme is stable with the Drichlet boundary and conditionally stable with Derivative boundary condition.


Introduction
The wave equation is a second order linear hyperbolic partial differential equation that describes the propagation of variety of waves, such as sound or water waves. It arises in different fields such as an acoustics, electromagnetic or fluid dynamics [6,7]. In many situations finding analytic solutions to partial differential equation is unrealistic or even impossible. Numerical methods that utilize computer algorithms are then used to find approximate solution [2,3,10]. The focus of this paper is to determine the stability and convergence of finite difference schemes that approximates a solution of wave equation. Suppose that an elastic string of length L is tightly stretched between two supports at the same horizontal level. So that the x-axis lies along the string. The elastic string may be thought of as violin sting, guy wire or possibly an electric power line. Suppose that the string is set in motion so it vibrates in vertical plane and let , denote the vertical displacement experienced by the string at the point at time if damping effects, such as air resistance are neglected. If the amplitude of the motion is not too large, then , satisfies the equation. = On domain 0 < < , > 0 the equation is one dimensional wave equation. [11]

The coefficient in the equation is
= where T is the tension (force) in the string and m is the mass per unit length of the string material. To describes the motion of the string completely it is necessary also to specify suitable initial and boundary conditions for displacement , . The ends are assumed to remain fixed and therefore the boundary condition are 0, =0 , = 0 ≥ 0 since the wave Equation with Drichlet and Neumann Boundary Condition equation is of second order with respect to t is plausible to prescribe two initial conditions those are the initial position of the string , 0 = . 0 ≤ ≤ And its initial velocity , 0 = 0 ≤ ≤ where and are given functions. The finite difference methods are the techniques for numerical solution to the wave equation by the discretization of space and time Let the string in the deformed state coincide with the interval [0, ] on the x axis, and let ; be the displacement at time in the y direction of a point initially at . the displacement function is governed by the mathematical model

(Governing equation)
, 0 = , ,! = ∈ 0, (Initial condition) Since this PDE Contains a second-order derivative in time, we need two initial conditions, here , 0 = specifying the initial shape of the string , , and ,! = In addition, PDEs need boundary conditions, here 0, = , = 0, specifying that the string is fixed at the ends, that ist the Displacement is zero. The solution , varies in space and time and describes waves that are moving with velocity to the left and right.

Finite Difference Methods
The finite difference techniques are based up on the approximations that permit replacing differential equation by finite difference equation. There finite difference approximations are algebraic in form, and the solutions are related to grid points. Thus, a finite difference solution basically involves three steps: Dividing the solution into grids of notes. Approximating the given differential equation by finite difference equivalence that relates the solutions to grid points.
Solving the difference equations subject to the prescribed boundary conditions and or Initial conditions. [2,5,9]

Explicit Finite Difference Method
The numerical solution of one dimensional wave equation using explicit scheme is obtained and the error calculated. To determine the stability and convergence, we will consider the simplified form of the wave equation (1) Common form.

Matrix Form of Explicit Scheme
Referring to equation (1), we discretize in space, using n nodes the temperature at time j ∆ t is given by

One Dimensional Wave Equation with Derivative Boundary Condition
The boundary condition , = makes change sign at the boundary, while the condition , = 0 perfectly reflects the wave. Our next task is to determine the stability with boundary condition , = 0, which is more complicated to express numerically. We shall present two methods for implementing , = 0 in a finite difference scheme, one based on deriving a modified stencil at the boundary, and another one bead on extending the mesh with ghost cells and ghost points.
Neumann boundary condition: The derivative . is in the outward normal direction from a general boundary.
For one dimension domain [0; ], we have that: Boundary conditions that specify the value of . are known as Neumann conditions, while refer to specifications of . When ( . = 0 or = 0 ) it is Dirichler boundary condition The Discretization of derivatives at the boundary in the finite difference scheme is used central differences in all the other approximations to derivatives in the scheme, it is tempting to implement (12) at = 0 and = . by the difference The problem is that (/,. is not a value that is being computed since the point is outside the mesh. However, if we combine 13 with the scheme for j = 0,

Fourier Method (Von Neumann Stability)
Fourier stability analysis allows determining appropriate step sizes for an accurate solution when the wavelength or decay constant (which is given in terms of parameters such as a diffusion constant or wave velocity in the (PDE) has a certain value. Fourier stability analysis does not take boundary conditions for a specific problem into account. And it performed by substituting the analytic solution to a partial differential equation into the numerical finite difference equation. [8, [12][13][14] Assume Y for stability we need G < 1 for all frequencies k. Conditional stability means we only have stability on a certain condition. Usually the condition limits ∆t in function of ∆x

Matrix Method to Determine Stability
The condition for stability of methods is determined by finding the spectral radiu which is [ \ =max (] C ) where ] C is an eigenvalue of matrix A, as illustrated below.
(i) If [ \ <1 then the system is stable.
(iii) If [ \ >1 then the system is unstable. Tridiagonal matrices are often found in connection with finite differences. Tridiagonal matrices are easy to deal with since there exists ancient numerical methods both for solving their linear systems of equations and eigenvalue problem. Here we consider the eigenvalue problem for a general tridiagonal matrix of the form [1,4,13,15] Lax Equivalence theorem: The Lax-Richtmyer Equivalence Theorem is often called the Fundamental Theorem of Numerical Analysis, even though it is only applicable to the small subset of linear numerical methods for well-posed, linear partial di erential equations. Along with Dahl Quist's equivalence theorem for ordinary di erential equations, the notion that the relationship consistency + stability ⇒ convergence always holds has caused a great deal of confusion in the numerical analysis of di erential equations. In the case of PDEs, mathematicians are most often interested in nonlinear phenomena, for which Lax-Richtmyer does not apply. More damningly, the forward implication that Consistent +stability ⇒ convergence Theorem: Gerschgorin's theorem: Consider a square matrix A = (_ C-), for row i the disk `C have centre _ CC and radius∑ a_ C-a b -N/ -cC . Then the theorem states that; Every eigenvalue of A lies in some `C. If S is the union of s disks `C such that S is disjoint from all other disks of this type, then S contains precisely m eigenvalues of A.

Consistence of the Explicit Method
The truncation error of the approximation of the time derivative is / given by Where j is a bound for | , e |andj is a bound for | f, | Hence h → 0 as k, h → 0 that is the difference approximation is consistent

Stability For Explicit with Drichlet Boundary
Condition The stability of this numerical scheme proceeds as follows We first need to recast the second order iteration system (14) in to a fist order system This places a restriction on the relative sizes of the time and space steps. We conclude that the numerical scheme is conditionally stable. The stability criterion (23) is known as the Courant condition, The Courant condition requires that the mesh slope, which is defined to be the ratio of the space step size to the time step size, namely ℎ/X, must be strictly greater than the characteristic slope c

Stability of the Explicit Method with Derivative Boundary Condition
The difference equation that we will use to time stepping the numerical scheme

Numerical Experiment
Numerical examples are presented to verify stability and convergence of the method Example 1 Use the explicit scheme to solve the one dimensional wave equation

Discussion
I used one dimensional wave equation by considering the space domain to have a length of M that is = + , with drichelet and derivative boundary conditions. The explicit finite differences were considered. To determine its stability by Von Neumann stability condition and Eigen value of tridiagonal matrix obtained from discretized scheme of the equation was used to develop the analysis. To produce values for the problem variable finite difference mesh point were used for the space domain. A mat lab code was written for explicit methods with drichlet and derivative boundary condition The numerical scheme is said to be stable when an error is introduced at certain stage, then remains bounded as time approaches infinity. It so happen that the error propagates in the same way as the problem variable, so an unstable process can be observed by the solution growing beyond any bounds. With small r the method performed well. The explicit scheme has an advantage that it is easy to set up, and disadvantage that it is unstable for r greater than one with drichilet and unstable for 5 ≤ √OE with derivative boundary

Conclusion
This study has considered the explicit finite difference schemes for solving one dimensional time dependent wave equation with drichlet and Neumann boundary conditions. The difference schemes are derived. Using Lax Equivalence Theorem convergence of the method was described by testing consistency and stability of the methods. Stability was discussed by using Gerschgorin's Theorem and Von Neumann stability condition. And the stability of the Explict method is shown by the table below Table 3. Summary of the finding. " ## = ' " " "" + •(", #) Matrix method and Fourier method Methods Drichilet Derivative Explicit scheme stable 5 ≤ 1 conditionally stable 5 ≤ √3/2 In the above table the Drichilet boundary condition are (0, ) = ( , ) = 0 and the diivative boundary condition are ( , ) = ( ), ( , ) = ( ) 1 is the critical value such that for 5 ≤ 1 scheme is stable.
A systematic study was applied to the two test numerical problems and the schemes have been successfully applied. The performance of the schemes for the considered problems was measured by calculating the error.