Discontinuous Galerkin (DG) methods are a class of finite element methods using discontinuous basis functions, which are usually chosen as piecewise polynomials. Since the basis functions can be discontinuous, these methods have the flexibility which is not shared by typical finite element methods, such as the allowance of arbitrary triangulation with hanging nodes, less restriction in changing the polynomial degrees in each element independent of that in the neighbors (p adaptivity), and local data structure and the resulting high parallel efficiency. In this paper, we give a general review of the local DG (LDG) methods for solving high-order time-dependent partial differential equations (PDEs). The important ingredient of the design of LDG schemes, namely the adequate choice of numerical fluxes, is highlighted. Some of the applications of the LDG methods for high-order time-dependent PDEs are also be discussed.

Borehole acoustic logging-while-drilling (LWD) for formation evaluation has become an indispensable part of hydrocarbon reservoir assessment [F. Cittá, C. Russell, R. Deady and D. Hinz, The Leading Edge, 23 (2004), pp. 566-573]. However, the detection of acoustic formation arrivals over tool mode contamination has been a challenging problem in acoustic LWD technology. In this paper we propose a new method for separating tool waves from formation acoustic waves in acoustic LWD. This method is to measure the seismoelectric signal excited by the LWD acoustic waves. The LWD tool waves which propagate along the rigid tool rim can not excite any electric signal. This is due to the effectively grounding of the drill string during the LWD process makes it impossible to accumulate any excess charge at the conductive tool — borehole fluid interface. Therefore, there should be no contribution by the tool modes to the recorded seismoelectric signals. To theoretically understand the seismoelectric conversion in the LWD geometry, we calculate the synthetic waveforms for the multipole LWD seismoelectric signals based on Pride's theory [S. R. Pride, Phys. Rev. B, 50 (1994), pp. 15678-15696]. The synthetic waveforms for the electric field induced by the LWD-acoustic-wave along the borehole wall demonstrate the absence of the tool mode. We also designed the laboratory experiments to collect simulated LWD monopole and dipole acoustic and seismoelectric signals in a borehole in sandstone. By analyzing the spectrum of acoustic and electric signals, we can detect and filter out the difference between the two signals, which are the mainly tool modes and noise.

The complete analytical solution of the Riemann problem for the homogeneous Dispersive Nonlinear Shallow Water Equations [Antuono, Liapidevskii and Brocchini, Stud. Appl. Math., 122 (2009), pp. 1-28] is presented, for both wet-bed and dry-bed conditions. Moreover, such a set of hyperbolic and dispersive depth-averaged equations shows an interesting resonance phenomenon in the wave pattern of the solution and we define conditions for the occurrence of resonance and present an algorithm to capture it. As an indirect check on the analytical solution we have carried out a detailed comparison with the numerical solution of the government equations obtained from a dissipative method that does not make explicit use of the solution of the local Riemann problem.

High order finite difference approximations are derived for a one-dimensional model of the shifted wave equation written in second-order form. The domain is discretized using fully compatible summation by parts operators and the boundary conditions are imposed using a penalty method, leading to fully explicit time integration. This discretization yields a strictly stable and efficient scheme. The analysis is verified by numerical simulations in one-dimension. The present study is the first step towards a strictly stable simulation of the second-order formulation of Einstein's equations in three spatial dimensions.

Adaptive moving mesh research usually focuses either on analytical derivations for prescribed solutions or on pragmatic solvers with challenging physical applications. In the latter case, the monitor functions that steer mesh adaptation are often defined in an ad-hoc way. In this paper we generalize our previously used monitor function to a balanced sum of any number of monitor components. This avoids the trial-and-error parameter fine-tuning that is often used in monitor functions. The key reason for the new balancing method is that the ratio between the maximum and average value of a monitor component should ideally be equal for all components. Vorticity as a monitor component is a good motivating example for this. Entropy also turns out to be a very informative monitor component. We incorporate the monitor function in an adaptive moving mesh higher-order finite volume solver with HLLC fluxes, which is suitable for nonlinear hyperbolic systems of conservation laws. When applied to compressible gas flow it produces very sharp results for shocks and other discontinuities. Moreover, it captures small instabilities (Richtmyer-Meshkov, Kelvin-Helmholtz). Thus showing the rich nature of the example problems and the effectiveness of the new monitor balancing.

Thanks to the Cagniard-de Hoop's method we derive the solution to the problem of wave propagation in an infinite bilayered acoustic/poroelastic media, where the poroelastic layer is modelled by the biphasic Biot's model. This first part is dedicated to solution to the two-dimensional problem. We illustrate the properties of the solution, which will be used to validate a numerical code.

A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of evolving family of plane curves. A curve can be evolved in the normal direction by a combination of fourth order terms related to the intrinsic Laplacian of the curvature, second order terms related to the curvature, first order terms related to anisotropy and by a given external velocity field. The evolution is numerically stabilized by an asymptotically uniform tangential redistribution of grid points yielding the first order intrinsic advective terms in the governing system of equations. By using a semi-implicit in time discretization it can be numerically approximated by a solution to linear penta-diagonal systems of equations (in presence of the fourth order terms) or tri-diagonal systems (in the case of the second order terms). Various numerical experiments of plane curve evolutions, including, in particular, nonlinear, anisotropic and regularized backward curvature flows, surface diffusion and Willmore flows, are presented and discussed.

Natural cavitation is defined as the phenomenon of the formation of vapor bubbles in a flow due to the pressure falls below the liquid's vapor pressure. The inception of the cavitation bubble is influenced by many factors, such as impurities, turbulence, liquid thermal properties etc. In this paper, we simulate a 2D cavitation "bubble" growth under shear flow in the inception stage by Single-Component-Multiphase Lattice Boltzmann Model (SCMP LBM). An empirical boundary condition sensitive 2D bubble growth rate, R ∼ e^t , is postulated. Furthermore, the comparison is conducted for bubble behavior under different shear rates. The results show that the cavitation bubble deformation is coincident with prior droplet theories and the bubble growth decreases slightly with the flow shear rate.

The characteristic of individual is described by the Penna model. Based on information entropy and the Penna model we define the entropy in the Penna model to discuss common ancestors and genetic diversity for two reproduction modes with and without mutation. About the problem of common ancestor, we find that all living individuals at any time step have a set of common ancestors which belongs to different times. They have the most recent common ancestor and earlier common ancestors. The analysis of genetic diversity shows that the complexity of ecosystems is caused by mutations and there is no contribution of sexual reproduction to conserving genetic diversity at long time scales. Moreover, in stable environment genetic diversity in asexual reproduction mode is higher than that in the sexual case.