One of the main challenges in computational simulations of gas detonation propagation is that negative density or negative pressure may emerge during the time evolution, which will cause blow-ups. Therefore, schemes with provable positivity-preserving of density and pressure are desired. First order and second order positivity-preserving schemes were well studied. For high order discontinuous Galerkin (DG) method, even though the characteristicwise TVB limiter can kill oscillations, it is not sufficient to maintain the positivity. A simple solution for arbitrarily high order positivity-preserving schemes solving Euler equations was proposed recently. In this paper, we first discuss an extension of the technique to design arbitrarily high order positivity-preserving DG schemes for reactive Euler equations. We then present a simpler and more robust implementation of the positivity-preserving limiter than the one before. Numerical tests, including very demanding examples in gaseous detonations, indicate that the third order DG scheme with the new positivity-preserving limiter produces satisfying results even without the TVB limiter.

In this paper, we study the superconvergence behavior of the semi-discrete discontinuous Galerkin (DG) method for {\color{red} scalar} nonlinear hyperbolic equations in one spatial dimension. Superconvergence results for problems with fixed and alternating wind directions are established. On the one hand, we prove that, if the wind direction is fixed (i.e., the derivative of the flux function is bounded away from zero), both the cell average error and numerical flux error at cell interfaces converge at a rate of $2k+1$ when upwind fluxes and piecewise polynomials of degree $k$ are used. Moreover, we also prove that the function value approximation of the DG solution is superconvergent at interior right Radau points, and the derivative value approximation is superconvergent at interior left Radau points, with an order of $k+2$ and $k+1$, respectively. As a byproduct, we show a $k+2$-th order superconvergence of the DG solution towards the Gauss-Radau projection of the exact solution. On the other hand, superconvergence results for problems with alternating wind directions (i.e., the derivative of the flux function either changes sign or otherwise achieves the value zero in the domain) are also established. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order of $k+2$, when Godunov fluxes are used. We then prove that the highest superconvergence rate of the DG solution itself is $k+\frac32$ when sonic points (i.e., the derivative of the flux function achieves zero) appear in the computational domain. As byproducts, we obtain superconvergence properties for the DG solution and the DG flux function at special points and for cell average. Numerical experiments demonstrate that most of our results are optimal, i.e., the superconvergence rates are sharp.

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Peskin's Immersed Boundary (IB) method is one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, fluid mechanics, material sciences, and many other areas. Peskiness IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the $L^{\infty}$ norm. But almost no rigorous proof could be found in the literature until recently [Mori, Comm. Pure. Appl. Math: 61:2008] in which the author showed that the velocity is indeed first order accurate for the Stokes equations with a periodic boundary condition. In this paper, we show first order convergence with a $\log h$ factor of the IB method for elliptic interface problems with Dirichlet boundary conditions.

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