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A new class of highly accurate differentiation schemes based on the prolate spheroidal wave functions

机译:基于扁球面波函数的一类新型高精度微分方案

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We introduce a new class of numerical differentiation schemes constructed via the prolate spheroidal wave functions (PSWFs). Compared to existing differentiation schemes based on orthogonal polynomials, the new class of differentiation schemes requires fewer points per wavelength to achieve the same accuracy when it is used to approximate derivatives of bandlimited functions. In addition, the resulting differentiation matrices have spectral radii that grow asymptotically as m for the case of first derivatives, and m~2 for second derivatives, with m being the dimensions of the matrices. The results mean that the new class of differentiation schemes is more efficient in the solution of time-dependent PDEs involving bandlimited functions when compared to existing schemes such as the Chebyshev collocation method. The improvements are particularly prominent in large-scale time-dependent PDEs whose solutions contain large numbers of wavelengths in the computational domains.
机译:我们介绍了一类新的通过微分球面波函数(PSWF)构造的数值微分方案。与现有的基于正交多项式的微分方案相比,新的微分方案在用于近似带限函数的导数时,每个波长需要更少的点才能达到相同的精度。另外,对于一阶导数,所得的微分矩阵的谱半径渐近地增长为m,对于二阶导数,其渐近增长为m〜2,其中m为矩阵的维数。结果表明,与现有方案(例如Chebyshev搭配方法)相比,新型微分方案在涉及带限函数的时间相关PDE的解决方案中更为有效。这些改进在大规模时变PDE中尤为突出,其解决方案在计算域中包含大量波长。

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