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Staggered grid techniques are attractive ideas for flow problems due to their more enhanced conservation properties. Recently, a staggered discontinuous Galerkin method is developed for the Stokes system. This method has several distinctive advantages, namely high order optimal convergence as well as local and global conservation properties. In addition, a local postprocessing technique is developed, and the postprocessed velocity is superconvergent and pointwisely divergence-free. Thus, the staggered discontinuous Galerkin method provides a convincing alternative to existing schemes. For problems with corner singularities and flows in porous media, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, we will derive a computable error indicator for the staggered discontinuous Galerkin method and prove that this indicator is both efficient and reliable. Moreover, we will present some numerical results with corner singularities and flows in porous media to show that the proposed error indicator gives a good performance.

This paper proposes a two-level continuum transportation system approach to modeling a dynamic taxi traffic assignment (DTTA) problem in a dense network with real-time traffic information provision and three types of vehicles, including private cars, occupied taxis, and vacant taxis. The proposed approach treats the dense network as a continuum in the first level, in which private car and occupied taxi drivers are free to choose their paths in a two-dimensional continuous space. The proposed approach also divides the modeling region into many identical squares to form a cell-based network in the second level, in which the cells are classified into two categories: target cell with an acceptable expected rate of return (EROR) to vacant taxi drivers and non-target cell with an unacceptable EROR. The EROR associated with a cell is the ratio of the cumulative expected profit of a taxi driver who successfully picks up a customer during the customer search that starts from that cell to the sum of expected search time for this customer and expected occupied travel time to serve this customer. Based on the cell-based network, we develop a cell-based intervening opportunity model to capture the fact that vacant taxi drivers can meet a customer on the way to their destination zones and estimate the EROR. Each vacant taxi driver has a mixed strategy to determine his/her customer-search direction according to the EROR: Each vacant taxi driver in a target cell selects its neighbor cells with maximum EROR, and each vacant taxi driver in a non-target cell selects the travel time-based shortest path to his/her target cell. Meanwhile, each private car driver chooses the path that minimizes his/her own generalized travel cost, and each occupied taxi driver chooses the path that minimizes his/her customer's generalized in-vehicle travel cost. In our model, traffic density in the system is governed by the conservation law (CL), and the flow directions of different vehicles are determined by the path-choice strategies of their drivers, which are captured by Hamilton–Jacobi (HJ) equations. Both the proposed CL and HJ equations can be solved by the Lax–Friedrichs scheme, which forms the backbone of the developed solution algorithm. Finally, numerical examples and a case study are used to demonstrate the properties of the model, the performance of the solution algorithm, and the value of using our methodology for estimating network performance.

In this paper, we will develop a new staggered hybridization technique for discontinuous Galerkin methods to discretize linear elastodynamic equations. The idea of hybridization is used extensively in many discontinuous Galerkin methods, but the idea of staggered hybridization is new. Our new approach offers several advantages, namely energy conservation, high-order optimal convergence, preservation of symmetry for the stress tensor, block diagonal mass matrices as well as low dispersion error. The key idea is to use two staggered hybrid variables to enforce the continuity of the velocity and the continuity of the normal component of the stress tensor on a staggered mesh. We prove the stability and the convergence of the proposed scheme in both the semi-discrete and the fully-discrete settings. Numerical results confirm the optimal rate of convergence and show that the method has a superconvergent property for dispersion. Furthermore, an application of this method to Rayleigh wave propagation is presented.

In this paper, we present a discontinuous Galerkin method with staggered hybridization to discretize a class of nonlinear Stokes equations in two dimensions. The utilization of staggered hybridization is new and this approach combines the features of traditional hybridization method and staggered discontinuous Galerkin method. The main idea of our method is to use hybrid variables to impose the staggered continuity conditions instead of enforcing them in the approximation space. Therefore, our method enjoys some distinctive advantages, including mass conservation, optimal convergence and preservation of symmetry of the stress tensor. We will also show that, one can obtain superconvergent and strongly divergence-free velocity by applying a local postprocessing technique on the approximate solution. We will analyze the stability and derive a priori error estimates of the proposed scheme. The resulting nonlinear system is solved by using the Newton’s method, and some numerical results will be demonstrated to confirm the theoretical rates of convergence and superconvergence.

The simulation of Rayleigh waves is important in a variety of geophysical applications. The computational challenge is the fact that very fine mesh is necessary as the waves are concentrated at the free surface and decay exponentially away from the free surface. To overcome this challenge and to develop a robust high order scheme for the simulation of Rayleigh waves, we develop a mortar discontinuous Galerkin method with staggered hybridization. The use of the mortar technique allows one to use fine mesh in only a local region near the free surface, and use coarse mesh in most of the domain. This approach reduces the computational cost significantly. The staggered hybridization allows the preservation of the strong symmetry of the stress tensor without complicated construction of basis functions. In particular, the basis functions are piecewise polynomial without any continuity requirement, and the coupling of the basis functions is performed by using carefully chosen hybridized variables. The resulting scheme is explicit in time, and only local saddle point system is solved for each time step. We will present several benchmark problems to demonstrate the performance of the proposed method.

This paper proposes a second-order pedestrian model that comprises two types of equations: continuity equation and a set of transport equations. To complete the model, we develop Eikonal equations to explicitly consider the effects of the collective decisions of individuals and crowd pressure on pedestrian dynamics. Then, the crowd movement is simulated using a set of partial differential equations under appropriate initial and boundary conditions. Based on the stability requirements derived by performing a standard linear stability analysis, suitable parameters are selected to test the model in a numerical example. The proposed second-order system is then solved using the characteristic-wise third-order weighted essentially non-oscillatory (WENO3) scheme, and the Eikonal equations are solved using the fast sweeping method. The numerical results indicate the effectiveness of the model because the derived local flow-density relationship produces a second peak in the high-density region, which is consistent with previous empirical studies. Besides, the applicability of the model to an unstable condition is verified through the simulation of complex phenomena such as stop-and-go waves. Furthermore, the estimate of crowd pressure in the simulation results can be used as a risk-level indicator for crowd management and control.

In this paper, we study the classical Allen-Cahn equations and investigate the maximum-principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen-Cahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its flexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stiff source given in the equation, we use the conservative modified exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

In this paper, we apply high-order finite difference (FD) schemes for multispecies and multireaction detonations (MMD). In MMD, the density and pressure are positive and the mass fraction of the ith species in the chemical reaction, say zi, is between 0 and 1, with ∑z_i=1. Due to the lack of maximum-principle, most of the previous bound-preserving technique cannot be applied directly. To preserve those bounds, we will use the positivity-preserving technique to all the z′is and enforce ∑z_i=1 by constructing conservative schemes, thanks to conservative time integrations and consistent numerical fluxes in the system. Moreover, detonation is an extreme singular mode of flame propagation in premixed gas, and the model contains a significant stiff source. It is well known that for hyperbolic equations with stiff source, the transition points in the numerical approximations near the shocks may trigger spurious shock speed, leading to wrong shock position. Intuitively, the high-order weighted essentially non-oscillatory (WENO) scheme, which can suppress oscillations near the discontinuities, would be a good choice for spatial discretization. However, with the nonlinear weights, the numerical fluxes are no longer “consistent”, leading to nonconservative numerical schemes and the bound-preserving technique does not work. Numerical experiments demonstrate that, without further numerical techniques such as subcell resolutions, the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.