A local discontinuous Petrov-Galerkin method for numerical simulation of two kinds of air pollution models is constructed. Firstly, air pollution model equations are transformed into equivalent first-order differential equations with variable substitution. Secondly, discontinuous Petrov-Galerkin method is used to solve the differential equations. The method can choose different test function and trial function space, and maintains advantages of the intermittent Petrov-Galerkin method. Compared with local discontinuous finite element method, calculation formula of the method is simpler. Numerical examples show that the method has third-order accuracy and less error than the finite volume method. The algorithm provides a practical tool for numerical simulation of air pollution models.

Interactions among parallel screw dislocations and a Griffith crack in magnetoelectroelastic media are investigated in terms of fundamental equations. Combining Muskhelishvili techniques and operator theory, analytic solutions for stress fields, electric fields and magnetic fields induced by screw dislocation and crack in magnetoelectroelastic solid are derived. Numerical examples show that singularity of stress occurs at crack tip and dislocation core. Away from dislocation point, the generalized stress is smaller, which is verified with existing results. As dislocation point approaches crack tip, stress fields between crack and dislocation tends to zero.

A local discontinuous Petrov-Galerkin method is proposed for solving three types of partial differential equations with second, third and fourth order derivatives, respectively. They are Burgers type equations, KdV type equations and bi-harmonic type equations. The method extends discontinuous Petrov-Galerkin method for conservation laws by rewriting corresponding equations into a first order system and solving the system instead of the original equation. The method has a fourth order accuracy and maintains advantages of discontinuous Petrov-Galerkin method. Numerical simulations verify that the method reaches optimal convergence order and simulates well complex wave interaction such as soliton propagation and collision.

A cell-centered scheme is constructed for two-dimensional gas dynamics equations in Lagrangian coordinates on rectangular grids. Spacial discretizations are accomplished by control volume discontinuous Petrov-Galerkin method and temporal discretization is accomplished by second order total variation diminishing Runge-Kutta method. A limiter is used to maintain stability and non-oscillatory property of Runge-Kutta control volume (RKCV) method. The method preserves local conservation of physical variables. Compared with Runge-Kutta discontinuous Galerkin (RKDG) method, computational formula of RKCV method is simpler since it contains no volume quadrature in RKDG method. Numerical examples are given to demonstrate reliability and efficiency of the algorithm.

Runge-Kutta control volume (RKCV) discontinuous finite element method for multi-medium fluid simulations is constructed. Linear and nonlinear Riemann solvers are used for numerical flux at fluid interfaces. The method preserves local conservation and high-resolution. Numerical results show that even with a linear Riemann solver the schemes works well. Comparisons with Runge-Kutta discontinuous Galerkin method show advantages of RKCV method.

We present a Lagrangian scheme for one-dimensional Euler equations.The scheme uses Runge-Kutta discontinuous Galerkin (RKDG) finite element method to solve Euler equations under Lagrangian framework.The mesh moves with fluid flow.The scheme is conservative for density,momentum and total energy.It achieves second-order accuracy both in space and time.Numerical tests are presented to demonstrate accuracy and non-oscillatory properties of the scheme.

We discuss numerical simulation of one-and two-dimensional nonlinear Schrödinger (NLS) equations (NLS).With numerical flux of diffusive generalized Riemann problem,a direct discontinuous Galerkin (DDG) method is proposed.L² stability of the DDG scheme is proved and it is shown that it is a conservative numerical scheme.The one-dimensional case indicates that the DDG scheme simulates various kinds of soliton propagations and it has excellent long-time numerical behaviors.Two-dimensional numerical results demonstrate that the method has high accuracy and is capable of capturing strong gradients.

We construct a Runge-Kutta discontinuous Galerkin (RKDG) finite element method for two-dimensional compressible gas dynamic equations in Lagrangian coordinate.The equations for fluid dynamics and geometry conservation laws are solved simultaneously.All calculations can be done on fixed meshes.Information of grid velocities are not needed in calculation.Several numerical examples are used to evaluate efficiency and reliability of the scheme.It shows that the algorithm works well.