NUMERICAL SIMULATION OF UNSTEADY-STATE UNDEREXPANDED JET USING DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD

A discontinuous Galerkin finite element method （DG-FEM） is developed for solving the axisymmetric Euler equations based on two-dimensional conservation laws. The method is used to simulate the unsteady-state underexpanded axisymmetric jet. Several flow property distributions along the jet axis, including density, pres- sure and Mach number are obtained and the qualitative flowfield structures of interest are well captured using the proposed method, including shock waves, slipstreams, traveling vortex ring and multiple Mach disks. Two Mach disk locations agree well with computational and experimental measurement results. It indicates that the method is robust and efficient for solving the unsteady-state underexpanded axisymmetric jet.

NUMERICAL INVESTIGATION OF TOROIDAL SHOCK WAVES FOCUSING USING DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD

A numerical simulation of the toroidal shock wave focusing in a co-axial cylindrical shock tube is inves- tigated by using discontinuous Galerkin （DG） finite element method to solve the axisymmetric Euler equations. For validating the numerical method, the shock-tube problem with exact solution is computed, and the computed results agree well with the exact cases. Then, several cases with higher incident Mach numbers varying from 2.0 to 5.0 are simulated. Simulation results show that complicated flow-field structures of toroidal shock wave diffraction, reflection, and focusing in a co-axial cylindrical shock tube can be obtained at different incident Mach numbers and the numerical solutions appear steep gradients near the focusing point, which illustrates the DG method has higher accuracy and better resolution near the discontinuous point. Moreover, the focusing peak pres- sure with different grid scales is compared.

A hybridized weak Galerkin finite element scheme for the Stokes equations

In this paper a hybridized weak Galerkin(HWG) finite element method for solving the Stokes equations in the primary velocity-pressure formulation is introduced.The WG method uses weak functions and their weak derivatives which are defined as distributions.Weak functions and weak derivatives can be approximated by piecewise polynomials with various degrees.Different combination of polynomial spaces leads to different WG finite element methods,which makes WG methods highly flexible and efficient in practical computation.A Lagrange multiplier is introduced to provide a numerical approximation for certain derivatives of the exact solution.With this new feature,the HWG method can be used to deal with jumps of the functions and their flux easily.Optimal order error estimates are established for the corresponding HWG finite element approximations for both primal variables and the Lagrange multiplier.A Schur complement formulation of the HWG method is derived for implementation purpose.The validity of the theoretical results is demonstrated in numerical tests.

Weak Galerkin finite element method for valuation of American options

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.

Galerkin finite element analysis of magneto-hydrodynamic natural convection of Cu-water nanoliquid in a baffled U-shaped enclosure

In this paper,single-phase homogeneous nanofluid model is proposed to investigate the natural convection of magneto-hydrodynamic(MIID)flow of Newtonian Cu—H20 nanoli­quid in a baffled U-shaped enclosure.The Brinkman model and Wasp model are considered to measure the effective dynamic viscosity and effective thermal conductivity of the nanoliquid coreespondingly.Nanoliquid's effective properties such as specific heat,density and thermal expansion coefficient are modeled using mixture theory.The complicated PDS(partial differ­ential system)is treated for numeric solutions via the Galerkin finite element method.The perti­nent parameters Hartmann number(1≤Ha≤60),Rayleigh number(10^(3)≤Ra≤10^(6))and nanoparticles volume fraction (0% ≤Ф≤4%) are taken for the parametric analysis, and it is conducted via streamlines and isotherms. Excellent agreement between numerical results and open literature. It is ascertained that heat transfer rate enhances with Rayleigh number Ra and volume fraction 0, however it is diminished for laiger Hartmann number Ha.

Galerkin Finite Element Approximation for Semilinear Stochastic Time-Tempered Fractional Wave Equations with Multiplicative Gaussian Noise and Additive Fractional Gaussian Noise

To model wave propagation in inhomogeneous media with frequency dependent power-law attenuation,it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time.The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative Gaussian noise and additive fractional Gaussian noise,because of the potential fluctuations of the external sources.The purpose of this work is to discuss the Galerkin finite element approximation for the semilinear stochastic fractional wave equation.First,the space-time multiplicative Gaussian noise and additive fractional Gaussian noise are discretized,which results in a regularized stochastic fractional wave equation while introducing a modeling error in the mean-square sense.We further present a complete regularity theory for the regularized equation.A standard finite element approximation is used for the spatial operator,and a mean-square priori estimates for the modeling error and the approximation error to the solution of the regularized problem are established.Finally,numerical experiments are performed to confirm the theoretical analysis.

A WEAK GALERKIN FINITE ELEMENT METHOD FOR THE LINEAR ELASTICITY PROBLEM IN MIXED FORM

In this paper, we use the weak Galerkin （WG） finite element method to solve the mixed form linear elasticity problem. In the mixed form, we get the discrete of proximation of the stress tensor and the displacement field. For the WG methods, we define the weak function and the weak differential operator in an optimal polynomial approximation spaces. The optimal error estimates are given and numerical results are presented to demonstrate the efficiency and the accuracy of the weak Galerkin finite element method.

ANALYSIS OF A MULTI-TERM VARIABLE-ORDER TIME-FRACTIONAL DIFFUSION EQUATION AND ITS GALERKIN FINITE ELEMENT APPROXIMATION

In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution.We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t=0.More precisely,we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C 2([0,T])in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness,otherwise the solution exhibits the same singular behavior like its constant-order counterpart.Based on these regularity results,we prove optimalorder convergence rate of the Galerkin finite element scheme.Furthermore,we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives.Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.

A symplectic finite element method based on Galerkin discretization for solving linear systems

We propose a novel symplectic finite element method to solve the structural dynamic responses of linear elastic systems.For the dynamic responses of continuous medium structures,the traditional numerical algorithm is the dissipative algorithm and cannot maintain long-term energy conservation.Thus,a symplectic finite element method with energy conservation is constructed in this paper.A linear elastic system can be discretized into multiple elements,and a Hamiltonian system of each element can be constructed.The single element is discretized by the Galerkin method,and then the Hamiltonian system is constructed into the Birkhoffian system.Finally,all the elements are combined to obtain the vibration equation of the continuous system and solved by the symplectic difference scheme.Through the numerical experiments of the vibration response of the Bernoulli-Euler beam and composite plate,it is found that the vibration response solution and energy obtained with the algorithm are superior to those of the Runge-Kutta algorithm.The results show that the symplectic finite element method can keep energy conservation for a long time and has higher stability in solving the dynamic responses of linear elastic systems.

A MODIFIED WEAK GALERKIN FINITE ELEMENTMETHOD FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION-REACTION PROBLEMS

In this work,a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations.The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators,respectively.We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh.The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods.Optimal order of convergences are obtained in suitable norms.We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method.Various numerical examples are presented to support the theoretical results.It is theoretically and numerically shown that the method is quite stable.

Alternating Direction Implicit Galerkin Finite Element Method for the Two-Dimensional Time Fractional Evolution Equation

New numerical techniques are presented for the solution of the twodimensional time fractional evolution equation in the unit square.In these methods,Galerkin finite element is used for the spatial discretization,and,for the time stepping,new alternating direction implicit(ADI)method based on the backward Euler method combined with the first order convolution quadrature approximating the integral term are considered.The ADI Galerkin finite element method is proved to be convergent in time and in the L2 norm in space.The convergence order is O(k|ln k|+h^(r)),where k is the temporal grid size and h is spatial grid size in the x and y directions,respectively.Numerical results are presented to support our theoretical analysis.

AStabilizer-FreeWeak Galerkin Finite Element Method for the Stokes Equations

A stabilizer-free weak Galerkin finite element method is proposed for the Stokes equations in this paper.Here we omit the stabilizer term in the new method by increasing the degree of polynomial approximating spaces for the weak gradient operators.The new algorithm is simple in formulation and the computational complexity is also reduced.The corresponding approximating spaces consist of piecewise polynomials of degree k≥1 for the velocity and k-1 for the pressure,respectively.Optimal order error estimates have been derived for the velocity in both H^(1) and L^(2) norms and for the pressure in L^(2) norm.Numerical examples are presented to illustrate the accuracy and convergency of the method.

TheWeak Galerkin Finite Element Method for Solving the Time-Dependent Integro-Differential Equations

In this paper,we solve linear parabolic integral differential equations using the weak Galerkin finite element method(WG)by adding a stabilizer.The semidiscrete and fully-discrete weak Galerkin finite element schemes are constructed.Optimal convergent orders of the solution of the WG in L^(2) and H^(1) norm are derived.Several computational results confirm the correctness and efficiency of the method.

A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

In this article,a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed.The idea of using the P_(k)-harmonic polynomial space instead of the full polynomial space P_(k)is to use a much smaller number of basis functions to achieve the same accuracy when k≥2.The optimal rate of convergence is derived in both H^(1)and L^(2)norms.Numerical experiments have been conducted to verify the theoretical error estimates.In addition,numerical comparisons of using the P_(2)-harmonic polynomial space and using the standard P_(2)polynomial space are presented.

Membrane finite element method for simulating fluid flow in porous medium

A new membrane finite element method for modeling fluid flow in a porous medium is presented in order to quickly and accurately simulate the geo-membrane fabric used in civil engineering. It is based on discontinuous finite element theory, and can be easily coupled with the normal Galerkin finite element method. Based on the saturated seepage equation, the element coefficient matrix of the membrane element method is derived, and a geometric transform relation for the membrane element between a global coordinate system and a local coordinate system is obtained. A method for the determination of the fluid flux conductivity of the membrane element is presented. This method provides a basis for determining discontinuous parameters in discontinuous finite element theory. An anti-seepage problem regarding the foundation of a building is analyzed by coupling the membrane finite element method with the normal Galerkin finite element method. The analysis results demonstrate the utility and superiority of the membrane finite element method in fluid flow analysis of a porous medium.

A multithreaded parallel upwind sweep algorithm for the S_(N) transport equations discretized with discontinuous finite elements

The complex structure and strong heterogeneity of advanced nuclear reactor systems pose challenges for high-fidelity neutron-shielding calculations. Unstructured meshes exhibit strong geometric adaptability and can overcome the deficiencies of conventionally structured meshes in complex geometry modeling. A multithreaded parallel upwind sweep algorithm for S_(N) transport was proposed to achieve a more accurate geometric description and improve the computational efficiency. The spatial variables were discretized using the standard discontinuous Galerkin finite-element method. The angular flux transmission between neighboring meshes was handled using an upwind scheme. In addition, a combination of a mesh transport sweep and angular iterations was realized using a multithreaded parallel technique. The algorithm was implemented in the 2D/3D S_(N) transport code ThorSNIPE, and numerical evaluations were conducted using three typical benchmark problems:IAEA, Kobayashi-3i, and VENUS-3. These numerical results indicate that the multithreaded parallel upwind sweep algorithm can achieve high computational efficiency. ThorSNIPE, with a multithreaded parallel upwind sweep algorithm, has good reliability, stability, and high efficiency, making it suitable for complex shielding calculations.

Discontinuous-Galerkin-Based Analysis of Traffic Flow Model Connected with Multi-Agent Traffic Model

As the number of automobiles continues to increase year after year,the associated problem of traffic congestion has become a serious societal issue.Initiatives to mitigate this problem have considered methods for optimizing traffic volumes in wide-area road networks,and traffic-flow simulation has become a focus of interest as a technique for advance characterization of such strategies.Classes of models commonly used for traffic-flow simulations include microscopic models based on discrete vehicle representations,macroscopic models that describe entire traffic-flow systems in terms of average vehicle densities and velocities,and mesoscopic models and hybrid(or multiscale)models incorporating both microscopic and macroscopic features.Because traffic-flow simulations are designed to model traffic systems under a variety of conditions,their underlyingmodelsmust be capable of rapidly capturing the consequences of minor variations in operating environments.In other words,the computation speed of macroscopic models and the precise representation of microscopic models are needed simultaneously.Thus,in this study we propose a multiscale model that combines a microscopic model—for detailed analysis of subregions containing traffic congestion bottlenecks or other localized phenomena of interest-with a macroscopic model enabling simulation of wide target areas at a modest computational cost.In addition,to ensure analytical stability with robustness in the presence of discontinuities,we discretize our macroscopic model using a discontinuous Galerkin finite element method(DGFEM),while to conjoin microscopic and macroscopic models,we use a generating/absorbing sponge layer,a technique widely used for numerical analysis of long-wavelength phenomena in shallow water,to enable traffic-flow simulations with stable input and output regions.

A weak Galerkin-mixed finite element method for the Stokes-Darcy problem

In this paper,we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition.We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation.A discrete inf-sup condition is proved and the optimal error estimates are also derived.Numerical experiments validate the theoretical analysis.

Finite element solution of nonlinear convective flow of Oldroyd-B fluid with Cattaneo-Christov heat flux model over nonlinear stretching sheet with heat generation or absorption

In this study,a two-dimensional boundary layer flow of steady incompressible nonlinear convective flow of Oldroyd-B fluid over a nonlinearly stretching sheet with Cattaneo-Christov heat flux model and heat generation or absorption is examined.The governing equations of the boundary layer flow which are highly nonlinear partial differential equations are converted to the ordinary differential equations using similarity transformations and then the Galerkin finite element method(GFEM)is used to solve the proposed problem.The effect of local Deborah numbers 0,and ft.local buoyancy parameter z,Prandtl number Pr,Deborah number y,and heat generation/absorption parameter<5 on the temperature and the velocity as well as heat transfer rate and shear stress are discussed both in graphical and tabular forms.The result shows the enlargement in the local buoyancy parameter A will improve the velocity field and the heat transfer rate of the boundary layer flow.Moreover,our present work evinced both local skin friction coefficient and heat transfer rate step up if we add the values of non-linear stretching sheet parameter and local heat generation/absorption parameter has quite the opposite effect.The numerically computed values of local skin friction coefficient and local Nusselt number are validated with available literature and evinced excellent agreement.

ON THE STABILITY OF THE RESIDUAL-FREE BUBBLES FOR THE NAVIER-STOKES EQUATIONS

This paper considers the Calerkin finite element method for the incompressible Navier-Stokes equations in two dimensions, where the finite-dimensional spaces employed consist of piecewise polynomials enriched with residual-free bubble （RFB） functions. The stability features of the residual-free bubble functions for the linearized Navier-Stokes equations are analyzed in this work. It is shown that the enrichment of the velocity space by bubble functions stabilizes the numerical method for any value of the viscosity parameter for triangular elements and for values of the viscosity parameter in the vanishing limit case for quadrilateral elements.