Abstract In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are shown to be conditionally $\ell^1$ contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, $ (\Delta t)^{\alpha} = O(\Delta x)$, where $\alpha \in (0,1]$ is the fractional exponent in the derivative. When $\alpha$ small, such strong constraint makes the numerical implementation extremely impractical. We have then proposed the implicit upwind scheme to overcome this issue, which is proved to be unconditionally $\ell^1$ contracting and TVD. Various numerical tests are presented to validate the properties of the methods and provide more numerical evidence in interpreting the memory effect in conservation laws.

In this paper we study the stochastic Galerkin approximation for the linear transport equation with random inputs and diffusive scal- ing. We first establish uniform (in the Knudsen number) stability results in the random space for the transport equation with uncertain scattering coefficients, and then prove the uniform spectral conver- gence (and consequently the sharp stochastic Asymptotic-Preserving property) of the stochastic Galerkin method. A micro-macro decom- position based fully discrete scheme is adopted for the problem and proved to have a uniform stability. Numerical experiments are conducted to demonstrate the stability and asymptotic properties of the method.

In this paper, we consider the time dependent Maxwell’s equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell’s equations when dispersive media are involved.

In this paper, we consider the time-dependent Maxwell’s equations when Cole–Cole dispersive medium is involved. The Cole–Cole model contains a fractional time derivative term, which couples with the standard Maxwell’s equations in free space and creates some challenges in developing and analyzing time-domain finite element methods for solving this model as mentioned in our earlier work [J. Li, J. Sci. Comput., 47 (2001), pp. 1–26]. By adopting some techniques developed for the fractional diffusion equations [V.J. Ervin, N. Heuer, and J.P. Roop, SIAM J. Numer. Anal., 45 (2007), pp. 572–591], [Y. Lin and C. Xu, J. Comput. Phys., 225 (2007), pp. 1533–1552], [F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Appl. Math. Comput., 191 (2007), pp. 12–20], we propose two fully discrete mixed finite element schemes for the Cole–Cole model. Numerical stability and optimal error estimates are proved for both schemes. The proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis.

In this paper we present a rigorous derivation of the material parameters for both the cylinder and rectangle cloaking structures. Numerical results using these material parameters are presented to demonstrate the cloaking effect.