In this paper, we analyze implicit-explicit (IMEX) Runge-Kutta (RK) time discretization methods for solving linear convection-diffusion equations. The diffusion operator is treated implicitly via the embedded discontinuous Galerkin (EDG) method and the convection operator explicitly via the upwinding discontinuous Galerkin method.

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.

We introduce a new adaptive method for analyzing nonlinear and nonstationary data. This method is inspired by the empirical mode decomposition (EMD) method and the recently developed compressed sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t) cos(θ(t))}, where a ≥ 0 is assumed to be smoother than cos(θ(t)) and θ is a piecewise smooth increasing function. We formulate this problem as a nonlinear L1 optimization problem. Further, we propose an iterative algorithm to solve this nonlinear optimization problem recursively. We also introduce an adaptive filter method to decompose data with noise. Numerical examples are given to demonstrate the robustness of our method and comparison is made with the EMD method. One advantage of performing such a decomposition is to preserve some intrinsic physical property of the signal, such as trend and instantaneous frequency. Our method shares many important properties of the original EMD method. Because our method is based on a solid mathematical formulation, its performance does not depend on numerical parameters such as the number of shifting or stop criterion, which seem to have a major effect on the original EMD method. Our method is also less sensitive to noise perturbation and the end effect compared with the original EMD method.

In this paper, we aim to solve one and two dimensional hyperbolic conservation laws on arbitrarily distributed point clouds. The initial condition is given on such a point cloud, and the algorithm solves for point values of the solution at later time also on this point cloud. By using the Voronoi technique and by introducing a grouping algorithm, we divide the computational domain into non-overlapping cells. Each cell is a polygon and contains a minimum number of the given points to ensure accuracy. We carefully select points in each cell during the grouping procedure, and hence are able to interpolate or fit the discrete initial values with piecewise polynomials. By adapting the traditional discontinuous Galerkin method on the constructed polygonal mesh, we obtain a stable, conservative and high order method. Numerical results for both one and two dimensional scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of our mesh generation algorithm as well as the numerical scheme.

In this paper, a novel penalty method based on the immersed boundary formulation is proposed for simulating the transient Stokes flow with an inextensible interface enclosing a suspended solid particle. The main idea of this approach relies on the penalty techniques by modifying the constitutive equation of Stokes flow to weaken the incompressibility condition, relating the surface divergence to the elastic tension $\sigma$ to relax the interface's inextensibility, and connecting the particle surface-velocity with the particle surface force ${\mbf F}$ to regularize the particle's rigid motion. The advantage of these regularized governing equations is that when they are discretized by the standard centered difference scheme on a staggered grid, the resulting linear system can easily be reduced by eliminating the unknowns $p_h$, $\sigma_h$ and ${\mbf F}_h$ directly, so that we just need to solve a smaller linear system of the velocity approximation ${\mbf u}_h$. This advantage is preserved and even enhanced when such approach is applied to the transient Stokes flow with multiple compound vesicles. Moreover, this smaller linear system is symmetric and negative-definite, which enables us to use efficient linear solvers. Another important feature of the proposed method is that the discretization scheme is unconditionally stable in the sense that an appropriately defined energy functional associated with the discrete system is decreasing and hence bounded in time. We numerically test the accuracy and stability of the immersed boundary discretization scheme. The tank-treading and tumbling motions of inextensible interface with a suspended solid particle in the simple shear flow will be studied extensively. The simulation of the motion of multiple compound vesicles will be performed as well. Numerical results illustrate the superior performance of the proposed penalty method.