Prefactored optimized compact finite-difference schemes for second spatial derivatives

Abstract

We have described systematically the processes of developing prefactored optimized compact schemes for second spatial derivatives. First, instead of emphasizing high resolution of a single monochromatic wave, we focus on improving the representation of the compact finite difference schemes over a wide range of wavenumbers. This leads to the development of the optimized compact schemes whose coefficients will be determined by Fourier analysis and the least-squares optimization in the wavenumber domain. The resulted optimized compact schemes provide the maximum resolution in spatial directions for the simulation of wave propagations. However, solving for each spatial derivative using these compact schemes requires the inversion of a band matrix. To resolve this issue, we propose a prefactorization strategy that decomposes the original optimized compact scheme into forward and backward biased schemes, which can be solved explicitly. We achieve this by ensuring a property that the real numerical wavenumbers of both the forward and backward biased stencils are the same as that of the original central compact scheme, and their imaginary numerical wavenumbers have the same values but with opposite signs. This property guarantees that the original optimized compact scheme can be completely recovered after the summation of the forward and backward finite difference operators. These prefactored optimized compact schemes have smaller stencil sizes than even those of the original compact schemes, and hence, they can take full advantage of the computer caches without sacrificing their resolving power. Comparisons were made throughout with other well-known schemes.