Propagation characteristics of a wave are defined by the dispersion relationship, from which the governing partial differential equation (PDE) can be recovered. PDEs are commonly solved numerically using the finite-difference (FD) method, with stencils constructed from truncated Taylor series expansions which, whilst typically providing good approximation of the PDE in the space-time domain, often differ considerably from the original partial differential in the wavenumber-frequency domain where the dispersion relationship is defined. Consequentially, stable, high-order FD schemes may not necessarily result in realistic wave behavior, commonly exhibiting numerical dispersion: lagging high-frequency components as a product of discretization. A method for optimizing FD stencil weightings via constrained minimization to better approximate the partial derivative in the wavenumber domain is proposed, allowing for accurate propagation with coarser grids than would be otherwise possible. This was applied to second derivatives on a standard grid and first derivatives on a staggered grid. To evaluate the efficacy of the method, a pair of numerical simulations were devised to compare spatially-optimized stencils with conventional formulations of equivalent extent. A spatially-optimized formulation of the 1D acoustic wave equation with Dirichlet boundary conditions is presented, evaluating performance at a range of grid spacings, examining the interval between the theoretical maximum grid spacings for the conventional and optimized schemes in finer detail. The optimized scheme was found to offer superior performance for undersampled wavefields and heavily oversampled wavefields. Staggered-grid first derivative stencils were then applied to the P-SV elastic wave formulation, simulating seismic wave propagation for a two-layer, water-over-rock model.
翻译:分散关系定义了波浪的映射特性,从中可以找到调节部分差异方程式(PDE)的分布式关系。 PDE通常使用有限差异(FD)方法进行数字解析,由短短泰勒序列扩展制成的电磁键,虽然通常在时空域内提供对 PDE 的精确近似,但通常与确定分散关系的波数-频域内原部分差异大不相同。因此,稳定、高顺序的流动法办法不一定导致现实的波态行为,通常表现出数字分散:高频构件滞后作为离散产物。提出了一种通过限制最小化优化调高频加权法,以更好地接近波数域内部分衍生物的方法,允许在时空网域内精确地接近PDE,这适用于标准网格上的第二衍生物,在交错式网格上的第一个衍生物。为了评估方法的功效,设计了一组数字模拟,将首次应用的空间优化的电动波变速预变速法组合:高频组合的高频组合,Stencial-realal-dealalal-dealalal-deal develild dalmaild dal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deal deald.