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Since the development of Yee scheme back in 1966, it has become one of the most popular simulation tools for modeling electromagnetic wave propagation in various situations.However, its rigorous error analysis on nonuniform rectangular type gridswas carried out until 1994 by Monk and Süli. They showed that theYee scheme is still second-order convergent on a nonuniform mesh even though the local truncation error is only of first order. In this paper, we extend their results to Maxwell’s equations in metamaterials by a simpler proof, and show the second-order superconvergence in space for the trueYee scheme instead of the only semi-discrete form discussed in Monk and Süli’s original work. Numerical results supporting our analysis are presented.

Wormhole propagation, arising in petroleum engineering, is used to describe the distribution of acid and the increase of porosity in carbonate reservoir under dissolution of injected acid. The important physical features of porosity and acid concentration include their boundedness between $0$ and $1$, as well as the monotone increasing for porosity. How to keep these properties in the simulation is crucial to the robustness of the numerical algorithm. In this paper, we propose high-order bound-preserving discontinuous Galerkin methods to keep these important physical properties. The main technique is to introduce a new variable $r$ to replace the original acid concentration and use a consistent flux pair to deduce a ghost equation such that the positive-preserving technique can be applied on both original and deduced equations. A high-order slope limiter is used to keep a polynomial upper bound which changes over time for $r$. Moreover, the high-order accuracy is attained by the flux limiter. Numerical examples are given to demonstrate the high-order accuracy and bound-preserving property of the numerical technique.

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.

To easily generalize the maximum-principle-satisfying schemes for scalar conservation laws to convection diffusion equations, we propose a non-conventional high order finite volume weighted essentially non-oscillatory (WENO) scheme which can be proven maximum-principle-satisfying. Two-dimensional extensions are straightforward. We also show that the same idea can be used to construct high order schemes preserving the maximum principle for two-dimensional incompressible Navier-Stokes equations in the vorticity stream-function formulation. Numerical tests for the fifth order WENO schemes are reported.