Discontinuous Galerkin approximation for excitatory-inhibitory networks with delay and refractory periods

In this study, we consider the network of noisy leaky integrate-and-fire (NNLIF) model, which governs by a second-order nonlinear time-dependent partial differential equation (PDE). This equation uses the probability density approach to describe the behavior of neurons with refractory states and the transmission delays. A numerical approximation based on the discontinuous Galerkin (DG) method is used for the spatial discretization with the analysis of stability. The strong stability-preserving explicit Runge–Kutta (SSPERK) method is performed for the temporal discretization. Finally, some test examples and numerical simulations are given to examine the behavior of the solution. The execution of the constructed scheme is measured by the quantitative comparison with the existing finite difference technique, namely weighted essentially nonoscillatory (WENO) scheme.

A new type of finite difference WENO schemes for Hamilton–Jacobi equations

In this paper, we propose a new type of finite difference weighted essentially nonoscillatory (WENO) schemes to approximate the viscosity solutions of the Hamilton–Jacobi equations. The new scheme has three properties: (1) the scheme is fifth-order accurate in smooth regions while keep sharp discontinuous transitions with no spurious oscillations near discontinuities; (2) the linear weights can be any positive numbers with the symmetry requirement and that their sum equals one; (3) the scheme can avoid the clipping of extrema. Extensive numerical examples are provided to demonstrate the accuracy and the robustness of the proposed scheme.