In this paper, we develop a discontinuous Galerkin (DG) method to solve the ideal special relativistic hydrodynamics (RHD) and design a bound-preserving (BP) limiter for this scheme by extending the idea in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 8918-8934). For RHD, the density and pressure are positive and the velocity is bounded by the speed of light. One difficulty in numerically solving the RHD in its conservative form is that the failure of preserving these physical bounds will result in ill-posedness of the problem and blowup of the code, especially in extreme relativistic cases. The standard way in dealing with this difficulty is to add extra numerical dissipation, while in doing so there is no guarantee in maintaining the high order of accuracy. Our BP limiter has the following features. It can theoretically guarantee to preserve the physical bounds for the numerical solution and maintain its designed high order accuracy. The limiter is local to the cell and hence is very easy to implement. Moreover, it renders $L^1$-stability to the numerical scheme. Numerical experiments are performed to demonstrate the good performance of this bound-preserving DG scheme. Even though we only discuss the BP limiter for DG schemes, it can be applied to high order finite volume schemes, such as weighted essentially non-oscillatory (WENO) finite volume schemes as well.

This paper is devoted to developing the Il'in-Allen-Southwell (IAS) parameter-uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection-diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct the IAS scheme for 1-D systems and then extend the scheme to 2-D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme is formally second order accurate in the sense that it is second order convergent when the perturbation parameter $\varepsilon$ is not too small and when the $\varepsilon$ is sufficiently small, the scheme is first order convergent in the discrete maximum norm uniformly in $\varepsilon$.

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For every two-dimensional random walk on the square lattice$Z$^{2}having zero mean and finite variance we obtain fine asymptotic estimates of the probability that the walk hits the negative real line for the first time at a site ($s$,0), when it is started at a site far from both (0,$s$) and the origin.