A numerical simulation method for one-dimensional chemical reaction fluid mechanics equations is studied. Combining ideal gas state equation and using numerical flux of HLLC solution at boundary of each element, we present an ALE discontinuous finite element method. In high-order calculations, TVD slope limiter is used to suppress non-physical oscillation that may result from numerical solutions. Numerical results show that the algorithm maintains conservation and high precision of physical quantity. It captures clearly structural characteristics of detonation waves.

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.

We present DG method for solving two-way coupling compressible gas-solid two-phase flow. Equations of both phases are discreted simultaneously, including convection term and source term. Splitting technique to discretize governing equations separately is avoided. Numerical flux of both phases is based on approximate Riemann solver. Dusty-gas shock tube problem with particles in low pressure section is simulated. Comparisons of equilibrium flow and frozen flow are made. Influence of particles in gas and interaction rules between two phases in relaxation zone behind shock are studied. It found that mass ratio of particles determines last equilibrium state and particles diameter determines transition process of two-phase flow from nonequilibrium to equilibrium flow. Namely, different diameter particles correspond to different relaxation time and distance. It shows that the numerical method proposed is reliable. It lays a foundation for more complicated gas-solid two-phase flow problems.

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 present a Lax-Wendroff discontinuous Galerkin (LWDG) method combining with adaptive mesh refinement (AMR) to solve three-dimensional hyperbolic conservation laws. Compared with Runge-Kutta discontinuous finite element method (RKDG) the method has higher efficiency. We give an effective adaptive strategie. Equidistribution strategy is easily implemented on nonconforming tetrahedral mesh. Error indicator is introduced to solve three-dimensional Euler equations. Numerical experiments demonstrate that the method has satisfied numerical efficiency.

Four semi-implicit discretization schemes are used to construct preconditioners.And preconditioned Jacobian-free Newton-Krylov (JFNK) are presented to solve one-dimensional problems.Numerical results show that the preconditioning methods improve the convergence behavior of JFNK method dramatically.

A numerical method is developed for two-dimensional nonequilibrium radiation diffusion equations.Discontinuous Galerkin method is applied in spatial diseretization in which numerical flux is constructed with weighted flux averages.Implicit-explicit integration factor method for time discretization is applied to nonlinear ordinary differential equations which is obtained with discontinuous Galerkin method. Radiation diffusion equations with multiple materials are solved on unstructured grids in numerical tests.It demonstrates that the method is effective for high nonlinear and tightly coupled radiation diffusion equations.

We discuss numerical simulation of one-dimensional non-equilibrium radiation diffusion equations.A weighted numerical flux between adjacent grid cells is obtained by solving heat conduction equation with discontinuous coefficient.With this numerical flux of diffusive generalized Riemann problem(dGRP),a discontinuous finite element method is proposed for radiation diffusion equations. A backward Euler time diseretization is applied for semi-discrete form and a Picard iteration is used to solve nonlinear system of equations.Numerical results demonstrate that the method has a capability of capturing strong gradients and can be accommodated to discontinuous diffusion coefficient.

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.

A study on oblique shock wave reflection in condensed matter is carried out by means of numerical simulation and theoretical analysis.Runge-Kutta control volume discontinuous finite element method is used to solve Euler equations.Equations of state for condensed matter adopt "stiffen gas" formulas.Patterns of oblique shock wave reflection in condensed matter are discussed.A shock polar theory is employed in analyzing critical agles of transition from regular reflection to irregular reflection.It gives states of reflected shock wave.Numerical results and shock polar solutions are compared and typical oblique shock wave reflections are obtained.

We use discontinuous finite element method to solve three-dimensional hydrodynamic equations.The domain is divided with an unstructured tetrahedral mesh.In order to overcome low efficiency of explicit scheme,especially as sizes of cells vary strongly,we use a local time stepping technique(LTS).We integrate control equations in space and time to obtain a single-step scheme.The calculation of each grid cell can be localized.It avoids excessive memory difficulties as dealing with three-dimensional problem with high order Runge-Kutta method.ADER method is used to calculate numerical flux across element boundary to improve accuracy of the hydrodynamic equations.Finally,numerical examples demonstrate stability and effectiveness of the method.

We combine Runge-Kutta discontinuous finite element method(RKDG) with adaptive method to solve Euler equations.Domain is divided into unstructured tetrahedral meshes.Local mesh refinement technique is used.According to changes in numerical solution,mesh is refined or coarsened locally.Therefore,number of overall grids is reduced and computational efficiency is increased.We give four different adaptive strategies and analyze advantages and disadvantages.Finally,several examples validate the method.

A conservative remapping algorithm based on donor-cell method for multiscale dynamic simulation is proposed which couples micro molecular dynamics (MD) simulation with macro finite element (FE) method. Since physical quantities are obtained with integral reconstruction from information of FE nodes and their underlying MD atoms, the algorithm can be applied to both structured and unstructured meshes. An auxiliary mesh is introduced for vertex-centered unknowns. Accuracy and efficiency of the method are validated with numerical experiments.

We present an explicit compatible finite element method for fluid dynamics problems in three-dimensional Cartesian geometry.Trilinear brick elements with a staggered-grid placement of the spatial variables are used to discretize fluid equations.An edge-centered artificial viscosity whose magnitude is regulated by local velocity gradients is used to capture shocks.Subzonal perturbed pressure is adopted to resist spurious grid motions.Artificial viscosity forces and subzonal pressure forces agree well with general compatibility.Numerical examples show accuracy and robustness of the method.

Runge-Kutta control volume discontinuous finite element method (RKCVDFEM) is proposed to solve numerically hyperbolic conservation laws,in which space discretization is based on control volume finite element method (CVFEM) while time discretization is based on a second order accurate TVB Runge-Kutta technique.Piecewise linear function space is chosen as finite element space.The scheme is total variation bounded (TVB) and is formally second order accurate in space and time.Numerical examples show that numerical solution converges to the entropy solution,and order of convergence is optimal for smooth solution.Compared with numerical results of Runge-Kutta discontinuous Galerkin method (RKDGM) those of RKCVDFEM are better.

For systems of nonlinear hyperbolic conservation laws,two adaptive discontinuous Galerkin finite element methods(ADGM) generating conforming unstructured triangular meshes are proposed.The first one is for structured mesh. It is simple and fast.The second one is for both structured and unstructured meshes.Based on posteriori error estimation of nonlinear hyperbolic conservation laws,a discontinuous interfacial mesh refinement indicator is shown in generating adaptive meshes. It is shown that the methods are flexible and reliable. Computation cost is decreased.

ALE (Arbitrary Lagrangian Eulerian) finite volume method for compressible fluid flows on moving quadrilateral meshes in two dimensional planar coordinates and axisymmetric coordinates is studied.A vertex-centered finite volume method and an HLLC numerical flux adapted to various equations of state are employed.A second order accuracy in space is achieved by using a reconstruction of primitive variables based on WENO approach.An explicit two-stage Runge-Kutta time-stepping scheme is used in discretization of time.The method offers accurate and robust solutions in capturing strong shock,contact discontinuities and material interface on arbitrarily moving grids.

A scheme is outlined for solving hyperbolic conservation laws by finite element method of piecewise linear interpolations. It is different from the discontinuous finite element on the boundaries of neighboring cells to solve Riemann problems that the scheme is designed to solve hyperbolic conservation laws based on the Hamilton Jacobi equations. Under the CFL condition, the scheme is proved that it satisfies the maximal principle and is a TVD scheme. Numerical examples are given and discussed.

The adaptive finite element methods are very effective for solving partial differential equations in scientific researches and engineering designs.By using these methods the best possible results can be obtained at less computational costs.A posteriori error estimates serve as a key to realize the adaptive finite element computation.This paper surveys the progress in the adaptive finite element methods and a posteriori error estimates for solving elliptic equations,parabolic equations and hyperbolic equations.