High Order Compact Explicit Difference Schemes For Solving The Wave Equations

This paper is aiming at developing finite difference method for the initial and boundary value problems of the wave equations. Firstly, for the one dimensional wave equation, we utilize Taylor series expansion formula and the original equation to establish a difference scheme of the unknown function value on the first time layer; then we utilize the Pade approximation to discrete second order spatial derivative term on the internal grid nodes and use the central difference in time direction, which results in a compact explicit difference scheme with truncation error of O(τ2+h4). Owing to the mismatch of time accuracy and space accuracy of the above scheme, we utilize the remainder of the truncation error correction method to improve the time accuracy. The truncation error of the improved scheme is O(τ4+τ2h2+h4), which means the scheme has overall fourth-order accuracy. Then, using the Fourier method, we analysis the stability conditions of the two schemes. The former stability condition is |a|λ≤√2/3 and the latter is |a|λ∈[0,1](?)[√2,√3]. Since the schemes of this paper belong to explicit difference schemes, they just need only once Thomas algorithm and once explicit recursive calculation. At last, we verify the accuracy and the stability of the present schemes by some numerical experiments and comparisons with the results in the literature.Secondly, we extend the above two schemes of the one dimensional wave equations, to the two dimensional cases. The truncation error of the former schemes is O(τ2+h4), and the latter one is O(τ4+τ2h2+h4). We utilize the Fourier analysis to analysis the stability of the two schemes.The former stability condition is|a|λ≤√1/3, and the latter one is|a|λ≤∈[0, √2/2](?)[1, √6/2]. Since the schemes of this paper belong to the explicit difference schemes, they just need only twice Thomas algorithms and once explicit recursive calculation without iterations. At last, we verify the accuracy and the stability of the present schemes by some numerical experiments and comparisons with the results in the literature.Finally, we extend the four-order accuracy scheme for the two-dimensional wave equation to the three dimensional cases. The truncation error of the scheme is O(τ4+τ2h2+h4).And then using the Fourier method, we get the stability condition is|a|λ∈[0, √3/3](?)[√6/3,1].Since the scheme of this paper belongs to the explicit difference scheme, it just needs only three times Thomas algorithms and once explicit recursive calculation without iterations. At last, we verify the accuracy and the stability of the present schemes by some numerical experiments and comparisons with the results in the literature.