The convergence of a fourth order finite difference method for the 2-D unsteady, viscous incompressible Boussinesq equations, based on the vorticity-stream function formulation, is established in this article. A compact fourth order scheme is used to discretize the momentum equation, and long-stencil fourth order operators are applied to discretize the temperature transport equation. A local vorticity boundary condition is used to enforce the no-slip boundary condition for the velocity. One-sided extrapolation is used near the boundary, dependent on the type of boundary condition for the temperature, to prescribe the temperature at ‘‘ghost’’ points lying outside of the computational domain. Theoretical results of the stability and accuracy of the method are also provided. In numerical experiments the method has been shown to be capable of producing highly resolved solutions at a reasonable computational cost.

In this paper, a mathematical analysis of the popular carpet cloak model is carried out. The well-posedness of the model is first proved, and then a finite element time-domain method is developed for solving this model. Three carpet cloak simulations are demonstrated to show the effectiveness of our model and the developed algorithm. To the best of our knowledge, this is the first mathematical analysis carried out for the carpet cloak model. The numerical simulation using edge elements for the carpet cloak model is also original.

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In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the $j$th component, $c_j$, should be between 0 and 1. There are three main difficulties. Firstly, $c_j$ does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all $c_j's$ and enforce $\sum_jc_j=1$ simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure $dp/dt$ as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to $c_j$. To construct the BP technique, we will not approximate $c_j$ directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.

In this paper, we present entropy satisfying schemes for solving an integro-differential equation that describes the evolution of a population structured with respect to a continuous trait. In [P.-E. Jabin and G. Raoul, J. Math. Biol., 63 (2011), pp. 493–517] solutions are shown to converge toward the so-called evolutionary stable distribution (ESD) as time becomes large, using the relative entropy. At the discrete level, the ESD is shown to be the solution to a quadratic programming problem and can be computed by any well-established nonlinear programing algorithm. The schemes are then shown to satisfy the entropy dissipation inequality on the set where initial data are positive and the numerical solutions tend toward the discrete ESD in time. An alternative algorithm is presented to capture the global ESD for nonnegative initial data, which is made possible due to the mutation mechanism built into the modified scheme. A series of numerical tests are given to confirm both accuracy and the entropy satisfying property and to underline the efficiency of capturing the large time asymptotic behavior of numerical solutions in various settings.