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Influence of the Forward Difference Scheme for the Time Derivative on the Stability of Wave Equation Numerical Solution

Research on numerical stability of difference equations has been quite intensive in the past century. The choice of difference schemes for the derivative terms in these equations contributes to a wide range of the stability analysis issues - one of which is how a chosen scheme may directly or indirectly contribute to such stability. In the present paper, how far the forward difference scheme for the time derivative in the wave equation influences the stability of the equation numerical solution, is particularly investigated. The stability analysis of the corresponding difference equation involving four schemes, namely Lax's, central, forward, and rearward differences, were carried out, and the resulting stability criteria were compared. The results indicate that the instability of the solution of wave equation is not always due to the forward difference scheme for the time derivative. Rather, it is shown in this paper that the stability criterion is still possible when the spatial derivative is represented by an appropriate difference scheme. This sheds light on the degree of applicability of a difference scheme for a hyperbolic equation.


INTRODUCTION
Finite difference is one of the numerical methods used in the discretization of partial differential equations such as Poisson's equation, wave equation, and Benjamin-Bona-Mahony (BBM) equation [1]- [3]. Following the discretization process, the stability analysis is necessary, in particular when it involves an explicit difference equation. This is to ensure whether or not the equation leads to a stable solution. Works done by, for instance, [4] and [5] serve as excellent materials for the analysis of stability.

A. Discretization of the Hyperbolic Equation
Without losing the generality, we begin with the first-order partial differential wave equation; in Cartesian coordinates and tensor notation following the Einstein convention, the equation is where is a constant. The corresponding solution domain is covered by a grid shown in Figure 1. It is worth to note here that the grid expansion factor = 1. The solution involving ≠ 1 has been considered in [6].

B. Lax Method
Replacing time and spatial derivatives with Lax's difference [7] and central difference schemes, respectively, we have, after some rearrangement ( )  (3) where and are finite accuracy numerical solution from a real computer and exact solution of difference equation, respectively. Note that the numerical solution satisfies the difference equation (2). Replacing in (2) with in (3), one has since clearly satisfies (2). It can be proven that also satisfies the equation; subtracting (5) from (4), one has ( )

III. THE STABILITY
The solution of (2) is stable if and only if We use a Fourier series to analytically represent the random variation of with respect to space and time; is the wave amplitude, is the wave number, and is a constant.
Note that the difference equation (2) is linear. Furthermore, it is satisfied by as shown by (6). Thus, if the series (8) is substituted into (6), each term of the series and the series itself behave the same. This allows us to handle just one term of the series, namely x t e e ε = (9) in order to probe the stability of (2) without loss in generality. The stability of the difference equation depends on how grows in time steps. Substituting (9) into (6) and (7) gives ( ) Combining (11) and (12); Here we have two possible conditions which must hold simultaneously: A. Condition a ( ) ( ) Setting 1 1 x = , and 2 1, The stability of the difference equation, (2), is therefore proven. Moreover, the stability requirement is given by (16) or (18) called the Courant-Friedrichs-Lewy (CFL) condition [8]- [10]. Any number given by the LHS term in (18) It is important to note that the Lax's difference scheme used to represent the time derivative in (1) is accurate to the firstorder. The main reason a simpler first-order difference is not applicable is due to the instability of the resulting difference equations, an issue that will be discussed further in the next section. The more complex analysis of stability can be found in [11]- [15].

IV. SIMPLE FIRST-ORDER DIFFERENCE FOR THE TIME DERIVATIVE
Replacing the time derivative with a simple first-order forward difference instead of Lax's difference schemes while keeping the central difference scheme to represent the spatial derivative in (1), we have, after some rearrangement The numerical solution satisfies the difference (20); replacing in (20) with in (3), one has Similarly, replacing in (20) with gives which proves that also satisfies (20). The solution of (20) is stable if and only if the requirement in (7) is fulfilled.
Following the same argument as in the previous section, we are allowed to just handle one term of the Fourier series, (9), in order to probe the stability of (20) without loss in generality. Substituting (9) into (20) gives Since both conditions, (29) and (31), must hold simultaneously, then not only the CFL criterion is not fulfilled but also (20) is found to be unconditionally unstable; the situation which justify the selection of Lax's difference scheme in the previous section as an option for the time derivative in (1). We will further examine whether such instability is solely due to the simple first-order forward difference scheme for the time derivative or the combination of schemes that represent both derivatives (i.e. time and spatial derivatives), in the following section.

V. SIMPLE FIRST-ORDER DIFFERENCES FOR THE TIME AND SPATIAL DERIVATIVES
In this section we consider two combinations involving simple first-order forward and first-order rearward difference schemes.

A. Combination I
Replacing time and spatial derivatives with simple firstorder finite differences to represent the derivatives in (1), we have, after some rearrangement

B. Combination II
We now represent time and spatial derivatives in (1) with simple first-order forward and rearward finite-differences, respectively. The error satisfies the difference equation; 1 1 It can be shown that

1)
Condition a x = in (43) results in a trivial solution.

2)
Condition b ( ) Thus the solution of (39) is conditionally stable, where the stability criterion is that of CFL.

VI. CONCLUSION
We summarize all four cases of the finite-difference representations for the first-order hyperbolic wave equation in Table I.   TABLE I It has been shown in Case 1 and Case 2, that the first-order accurate LD is a suitable candidate for the FD replacement to represent the time derivative in the first-order wave equation while maintaining the CD for the spatial derivative, where the CFL is met. However, the FD seems to contribute to the stability of the corresponding difference equation solution if the spatial derivative is represented by the RD as shown in Case 4. This suggests the applicability of the FD in solving a hyperbolic partial differential equation.