In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson–Nernst–Planck (PNP) equations, which have found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.

We consider the problem of reconstructing unknown inclusions inside a thermal conductor from boundary measurements, which arises from active thermography and is formulated as an inverse boundary value problem for the heat equation. In our previous works, we proposed a sampling-type method for reconstructing the boundary of the unknown inclusion and gave its rigorous mathematical justification. In this paper, we continue our previous works and provide a further investigation of the reconstruction method from both the theoretical and numerical points of view. First, we analyze the solvability of the Neumann-to-Dirichlet map gap equation and establish a relation of its solution to the Green function of an interior transmission problem for the inclusion. This naturally provides a way of computing this Green function from the Neumann-to-Dirichlet map. Our new findings reveal the essence of the reconstruction method. A convergence result for noisy measurement data is also proved. Second, based on the heat layer potential argument, we perform a numerical implementation of the reconstruction method for the homogeneous inclusion case. Numerical results are presented to show the efficiency and stability of the proposed method.

In this paper, a new type of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are constructed for solving Hamilton-Jacobi (HJ) equations. Point values of both the solution and its first derivatives are used in the HWENO reconstruction and evolved via time advancing. While the evolution of the solution is still through the classical numerical fluxes to ensure convergence to weak solutions, the evolution of the first derivatives of the solution is through a simple dimension-by-dimension non-conservative procedure to gain efficiency. The main advantages of this new scheme include its compactness in the spatial field and its simplicity in the reconstructions. Extensive numerical experiments in one and two dimensional cases are performed to verify the accuracy, high resolution and efficiency of this new scheme.

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In this paper, we consider multiple signals sharing same instantaneous frequencies. This kind of data is very common in scientic and engineering problems. To take ad- vantage of this special structure, we modify our data-driven time-frequency analysis by updating the instantaneous frequencies simultaneously. Moreover, based on the simul- taneously sparsity approximation and fast Fourier transform, some ecient algorithms is developed. Since the information of multiple signals is used, this method is very ro- bust to the perturbation of noise. And it is applicable to the general nonperiodic signals even with missing samples or outliers. Several synthetic and real signals are used to test this method. The performances of this method are very promising.