Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells

The hierarchical reconstruction (HR) [Y.-J. Liu, C.-W. Shu, E. Tadmor, M.-P. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal. 45 (2007) 2442-2467] is applied to the piecewise quadratic discontinuous Galerkin method on two-dimensional unstructured triangular grids. A variety of limiter functions have been explored in the construction of piecewise linear polynomials in every hierarchical reconstruction stage. We show that on triangular grids, the use of center biased limiter functions is essential in order to recover the desired order of accuracy. Several new techniques have been developed in the paper: (a) we develop a WENO-type linear reconstruction in each hierarchical level, which solves the accuracy degeneracy problem of previous limiter functions and is essentially independent of the local mesh structure; (b) we find that HR using partial neighboring cells significantly reduces over/under-shoots, and further improves the resolution of the numerical solutions. The method is compact and therefore easy to implement. Numerical computations for scalar and systems of nonlinear hyperbolic equations are performed. We demonstrate that the procedure can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions.     

Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations

In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the LDG solution and the exact solution is ($k$+2)-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is ($k$+2)-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for $P^k$ polynomials with arbitrary $k$ ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.     

Total variation diminishing Runge-Kutta schemes

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

     

Genetically encoded fluorescent reporters of histone methylation in living cells

We report the design and characterization of two genetically encoded fluorescent reporters of histone protein methylation. The reporters are four-part chimeric proteins consisting of a substrate peptide from the N-terminus of histone H3 fused to a chromodomain (a natural methyllysine-specific recognition domain), sandwiched between a fluorescence resonance energy transfer (FRET)-capable pair of fluorophores, cyan fluorescent protein (CFP) and yellow fluorescent protein (YFP). Enzymatic methylation by a methyltransferase induces complexation of the methylated substrate peptide to the chromodomain, changing the FRET level between the flanking CFP and YFP domains. Reporters developed using the chromodomains from HP1 and Polycomb respond to enzymatic methylation at the lysine 9 and lysine 27 positions of histone H3, respectively, giving 60% and 28% YFP/CFP emission ratio increases in vitro or in single living cells. These reporters should be useful for studying gene silencing and X-chromosome inactivation with high spatial and temporal resolution in intact cells and may also aid in the search for conjectured histone demethylase activity.     

Spectrophotometric Studies of Nickel(II) Chelate of 2-Picolylamine

Nickel(II) chelate of 2–picolylamine has been studied spectrophotometrically in aqueous solution at 25°C and at an ionic strength of 0.3 M. The formation of pink color chelate was pH dependent, and the optimum pH range was between 7.0 to 8.5. Its mole ratio of ligand to nickel(II) ion was found to be 3 to 1 stoichiometry and the formation constant, logK, was determined as 13.31 ± 0.10. By using the wavelength 535 run, determination of trace amount of nickel(II) ion with the sensitivity of 5.28 τ/Cm² was possible. Enthalpy and entropy changes characterizing the formation of the chelate have been calculated as follows: ΔG°=–8.15Kcal mole^-1, ΔH°=–9.65 Kcal mole^-1, ΔS°=28.5eu mole^-1.     

Analysis of the local discontinuous Galerkin method for the drift-diffusion model of semiconductor devices

We consider the drift-diffusion(DD) model of one dimensional semiconductor devices, which is a system involving not only first derivative convection terms but also second derivative diffusion terms and a coupled Poisson potential equation. Optimal error estimates are obtained for both the semi-discrete and fully discrete local discontinuous Galerkin(LDG) schemes with smooth solutions. In the fully discrete scheme, we couple the implicit-explicit(IMEX) time discretization with the LDG spatial discretization, in order to allow larger time steps and to save computational cost. The main technical difficulty in the analysis is to treat the inter-element jump terms which arise from the discontinuous nature of the numerical method and the nonlinearity and coupling of the models. A simulation is also performed to validate the analysis.     

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology.The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical fluxes.By using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth enough.The theoretical results are proved for piecewise polynomials with any degree k 1.Finally,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy.Numerical results are provided to demonstrate these RKDG methods.     

On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes

We construct uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynamics. The same framework also applies to high order accurate finite volume (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO)) schemes. Motivated by Perthame and Shu (1996) [20] and Zhang and Shu (2010) [26], a general framework, for arbitrary order of accuracy, is established to construct a positivity preserving limiter for the finite volume and DG methods with first order Euler forward time discretization solving one-dimensional compressible Euler equations. The limiter can be proven to maintain high order accuracy and is easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the positivity property. Following the idea in Zhang and Shu (2010) [26], we extend this framework to higher dimensions on rectangular meshes in a straightforward way. Numerical tests for the third order DG method are reported to demonstrate the effectiveness of the methods.     

A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry

We develop a new cell-centered control volume Lagrangian scheme for solving Euler equations of compressible gas dynamics in cylindrical coordinates. The scheme is designed to be able to preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid. Unlike many previous area-weighted schemes that possess the spherical symmetry property, our scheme is discretized on the true volume and it can preserve the conservation property for all the conserved variables including density, momentum and total energy. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry, accuracy and non-oscillatory properties.     

Numerical simulations of compressible mixing layers with a discontinuous Galerkin method

Discontinuous Galerkin(DG) method is known to have several advantages for flow simulations,in particular,in fiexible accuracy management and adaptability to mesh refinement. In the present work,the DG method is developed for numerical simulations of both temporally and spatially developing mixing layers. For the temporally developing mixing layer,both the instantaneous fiow field and time evolution of momentum thickness agree very well with the previous results. Shocklets are observed at higher convective Mach numbers and the vortex paring manner is changed for high compressibility. For the spatially developing mixing layer,large-scale coherent structures and self-similar behavior for mean profiles are investigated. The instantaneous fiow field for a three-dimensional compressible mixing layer is also reported,which shows the development of largescale coherent structures in the streamwise direction. All numerical results suggest that the DG method is effective in performing accurate numerical simulations for compressible shear fiows.