This paper is concerned with fifth-order weighted essentially non-oscillatory (WENO) scheme with a new smoothness indicator. As the so-called WENO-JS scheme (Jiang and Shu in J Comput Phys 126:202-228, 1996) provides the third-order accuracy at critical points where the first and third order derivatives do not becomes zero simultaneously, several fifth-order WENO scheme have been developed through modifying the known smoothness indicators of WENO-JS. Recently a smoothness indicator based on L 1 -norm has been proposed by Ha et al. (J Comput Phys 232:68-86, 2013) (denoted by WENO-NS). The aim of this paper is twofold. Firstly, we further improve the smoothness indicator of WENO-NS and secondly, using this measurement, we suggest new nonlinear weights by simplifying WENO-NS weights. The proposed WENO scheme provides the fifth-order accuracy, even at critical points. Some numerical experiments are provided to demonstrate that the present scheme performs better than other WENO schemes of the same order.

We propose and analyze the spectral collocation approximation for the partial integrodifferential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomial and the Sobolev regularity of the solution.

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