We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski, Cheng and Shu 09}), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of Bokanowski et al. (Bokanowski_Forcadel_Zidani_2010), leading to a level set formulation driven by $\min(u_t + H(x,\nabla u), u-g(x))=0$, where $g(x)$ is an obstacle function. The DG scheme is motivated by the variational formulation when the Hamiltonian $H$ is a linear function of $\nabla u$, corresponding to linear convection problems in presence of obstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis are performed for the linear case with Euler forward, a Heun scheme and a Runge-Kutta third order time discretization using the technique proposed in Zhang and Shu 2010. Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost.

In this paper, we develop bound-preserving modified exponential Runge-Kutta (RK) discontinuous Galerkin (DG) schemes to solve scalar hyperbolic equations with stiff source terms by extending the idea in Zhang and Shu \cite{ZhSh:10max}. Exponential strong stability preserving (SSP) high order time discretizations are constructed and then modified to overcome the stiffness and preserve the bound of the numerical solutions. It is also straightforward to extend the method to two dimensions on rectangular and triangular meshes. Even though we only discuss the bound-preserving limiter for DG schemes, it can also be applied to high order finite volume schemes, such as weighted essentially non-oscillatory (WENO) finite volume schemes as well.

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.

Poisson equation on point cloud with Dirichlet boundary condition plays important role in many problems. In this paper, we use the volume constraint proposed by Du et.al to handle the Dirichlet boundary condition in the point integral method for Poisson equation on point cloud. We prove that the solution given by volume constraint converges to the true solution as the point cloud converges to the underlying smooth manifold.

In this paper, we give an integral approximation for the elliptic operators with anisotropic coecients on smooth manifold. Using the integral approximation, the elliptic equation is tranformed to an integral equation. The integral approximation preserves the symmetry and coercivity of the original elliptic operator. Based on these good properties, we prove the convergence between the solutions of the integral equation and the original elliptic equation.