On Compact 4th Order Finite-Difference Schemes for the Wave Equation

Abstract

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the n-dimensional nonhomogeneous wave equation, n≥ 1. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for n≥ 2. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schro¨dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for n≥ 2. For n = 1 and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.

On properties of an explicit in time fourth-order vector compact scheme for the multidimensional wave equation

“…Such compact schemes are implicit and most often conditionally stable. Among them, the first type of schemes is constituted by implicit schemes which require application of FFT (for constant coefficients) or iterative methods (for variable coefficients) for their efficient implementation, see, in particular, [2,8,18,22,24], where additional references are contained.…”

“…More important compact ADI schemes with the 4th approximation order, i.e. O(h 4 t + |h| 4 ), for such 2D and 3D equations were constructed and studied, in particular, in [4,5,12,19,24]. For the variable coefficient 2D wave equation, the compact ADI scheme having the approximation order O(h 2 t + |h| 6 ) has recently been constructed in [3], and the 4th order compact ADI-like schemes were proposed and studied in the 2D and 3D cases in [10,13,23].…”

“…They are similar to the main equations and exploit only second order finite-differences of the initial functions. Notice that we do not use derivatives of the free term and initial functions in contrast to [7] that is essential for applications of the scheme in the non-smooth case, for example, see [8,10,22,24]. The scheme is new even in the simplest 1D case.…”

“…Next, new results on the conditional stability of the scheme in the mesh energy norm and the discrete energy conservation law are derived from the corresponding general results in [24,26] (proved by the energy method) by excluding the auxiliary sought functions. The case of general free terms in the main and auxiliary equations of the scheme is treated that is essential for the completeness of the stability analysis.…”

“…Finally, the generalizations of the scheme to the cases of the non-uniform rectangular mesh in space and the non-uniform mesh in time are presented following [24], and the important case of the wave equation with variable coefficients is covered too. For general non-uniform meshes, the scheme has the 3th approximation order as other compact schemes.…”