A Multigrid Block LU-SGS Algorithm for Euler Equations on Unstructured Grids

We propose an efficient and robust algorithm to solve the steady Euler equa- tions on unstructured grids.The new algorithm is a Newton-iteration method in which each iteration step is a linear multigrid method using block lower-upper symmetric Gauss-Seidel(LU-SGS)iteration as its smoother To regularize the Jacobian matrix of Newton-iteration,we adopted a local residual dependent regularization as the replace- ment of the standard time-stepping relaxation technique based on the local CFL number The proposed method can be extended to high order approximations and three spatial dimensions in a nature way.The solver was tested on a sequence of benchmark prob- lems on both quasi-uniform and local adaptive meshes.The numerical results illustrated the efficiency and robustness of our algorithm.

Screening and Identification of Tobacco Mutants Resistant to Potato Virus Y( PVY)

[ Objective] This study aimed to screen and identify tobacco mutants resistant to potato virus Y （PVY）, thus laying the foundation for obtaining PVY resistance genes. [ Method ] At seedling stage, tobacco mutant materials were inoculated with PVY virus and preliminarily screened by naked-eye observation. Enzyme-linked immunosorbent assay （ELISA）, real-time fluorescence quantitative PCR and test strip assay were performed to further identify the pre-screened PVY- resistant seedlings. [ Result ] In 2011, two highly PVY-resistant tobacco mutants （MZE2-15 and MZE2-16） and three PVY-tolerant mutants （MZE2-70, MZE2- 207 and MZE2-228） were obtained, which were further screened and identified in 2012. According to the results, tobacco mutant materials MZE2-407 and MZE2- 428 were susceptible to PVY; mutant materials MZE2-16 and MZE2-15 were resistant to PVY. [ Conclusion] This study provide theoretical basis for the control of tobacco PVY disease.

Numerical Validation for High Order Hyperbolic Moment System of Wigner Equation

A globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in[6].For numerically solving the high order hyperbolic moment system therein,we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in[8],to validate the moment system of the Wigner equation.The method developed can keep both mass and momentum conserved,and the variation of the total energy under control though it is not strictly conservative.We systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion,and the convergence of the numerical solution of the moment system to the numerical solution of the Wigner equation using the discrete velocity method.The numerical results indicate that the high order moment system in[6]is a valid model for the Wigner equation,and the proposed numerical method for the moment system is quite promising to carry out the simulation of the Wigner equation.

A Hybrid Fluid-Solid Interaction Scheme Combining the Multi-Component Diffuse Interface Method and the Material Point Method

We propose a hybrid scheme combing the diffuse interface method and the material point method to simulate the complex interactions between the multiphase compressible flow and elastoplastic solid.The multiphase flow is modelled by the multi-component model and solved using a generalized Godunov method in the Eulerian grids,while the elastoplastic solid is solved by the classical material point method in a combination of Lagrangian particles and Eulerian background grids.In order to facilitate the simulation of fluid-solid interactions,the solid variables are further interpolated to the cell center and coexist with the fluid in the same cell.An instantaneous relaxation procedure of velocity and pressure is adopted to simulate the momentum and energy transfers between various materials,and to keep the system within a tightly coupled interaction.Several numerical examples,including shock tube problem,gasbubble problem,air blast,underwater explosion and high speed impact applications are presented to validate the numerical scheme.

Dimension-Reduced Hyperbolic Moment Method for the Boltzmann Equation with BGK-Type Collision

We develop the dimension-reduced hyperbolic moment method for the Boltzmann equation,to improve solution efficiency using a numerical regularized moment method for problems with low-dimensional macroscopic variables and highdimensional microscopic variables.In the present work,we deduce the globally hyperbolic moment equations for the dimension-reduced Boltzmann equation based on the Hermite expansion and a globally hyperbolic regularization.The numbers of Maxwell boundary condition required for well-posedness are studied.The numerical scheme is then developed and an improved projection algorithm between two different Hermite expansion spaces is developed.By solving several benchmark problems,we validate the method developed and demonstrate the significant efficiency improvement by dimension-reduction.

A General Moving Mesh Framework in 3D and its Application for Simulating the Mixture of Multi-Phase Flows

In this paper, we present an adaptive moving mesh algorithm for meshesof unstructured polyhedra in three space dimensions. The algorithm automaticallyadjusts the size of the elements with time and position in the physical domain to resolvethe relevant scales in multiscale physical systems while minimizing computationalcosts. The algorithm is a generalization of the moving mesh methods basedon harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To make 3D moving mesh simulations possible,the key is to develop an efficient mesh redistribution procedure so that this part willcost as little as possible comparing with the solution evolution part. Since the meshredistribution procedure normally requires to solve large size matrix equations, wewill describe a procedure to decouple the matrix equation to a much simpler blocktridiagonaltype which can be efficiently solved by a particularly designed multi-gridmethod. To demonstrate the performance of the proposed 3D moving mesh strategy,the algorithm is implemented in finite element simulations of fluid-fluid interface interactionsin multiphase flows. To demonstrate the main ideas, we consider the formationof drops by using an energetic variational phase field model which describesthe motion of mixtures of two incompressible fluids. Numerical results on two- andthree-dimensional simulations will be presented.

ARobust Riemann Solver for Multiple Hydro-Elastoplastic Solid Mediums

We propose a robust approximate solver for the hydro-elastoplastic solid material,a general constitutive law extensively applied in explosion and high speed impact dynamics,and provide a natural transformation between the fluid and solid in the case of phase transitions.The hydrostatic components of the solid is described by a family of general Mie-Gruneisen equation of state(EOS),while the deviatoric component includes the elastic phase,linearly hardened plastic phase and fluid phase.The approximate solver provides the interface stress and normal velocity by an iterative method.The well-posedness and convergence of our solver are proved with mild assumptions on the equations of state.The proposed solver is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian girds.Several numerical examples,including Riemann problems,shock-bubble interactions,implosions and high speed impact applications,are presented to validate the approximate solver.

An Edge-Based Anisotropic Mesh Refinement Algorithm and its Application to Interface Problems

Based on an error estimate in terms of element edge vectors on arbitrary unstructured simplex meshes,we propose a new edge-based anisotropic mesh refinement algorithm.As the mesh adaptation indicator,the error estimate involves only the gradient of error rather than higher order derivatives.The preferred refinement edge is chosen to reduce the maximal term in the error estimate.The algorithm is implemented in both two-and three-dimensional cases,and applied to the singular function interpolation and the elliptic interface problem.The numerical results demonstrate that the convergence order obtained by using the proposed anisotropic mesh refinement algorithm can be higher than that given by the isotropic one.

RECONSTRUCTED DISCONTINUOUS APPROXIMATION TO STOKES EQUATION IN A SEQUENTIAL LEAST SQUARES FORMULATION

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps.The approximation spaces are constructed by the patch reconstruction with one unknown per element.For the first step,we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace.By this space,we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure.In the second step,we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space.We derive error estimates for all unknowns under both L 2 norms and energy norms.Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.

A Reconstructed Discontinuous Approximation to Monge-Ampère Equation in Least Square Formulation

We propose a numerical method to solve the Monge-Ampère equation which admits a classical convex solution.The Monge-Ampère equation is reformulated into an equivalent first-order system.We adopt a novel reconstructed discontinuous approximation space which consists of piecewise irrotational polynomials.This space allows us to solve the first-order system in two sequential steps.In the first step,we solve a nonlinear system to obtain the approximation to the gradient.A Newton iteration is adopted to handle the nonlinearity of the system.The approximation to the primitive variable is obtained from the approximate gradient by a trivial least squares finite element method in the second step.Numerical examples in both two and three dimensions are presented to show an optimal convergence rate in accuracy.It is interesting to observe that the approximation solution is piecewise convex.Particularly,with the reconstructed approximation space,the proposed method numerically demonstrates a remarkable robustness.The convergence of the Newton iteration does not rely on the initial values.The dependence of the convergence on the penalty parameter in the discretization is also negligible,in comparison to the classical discontinuous approximation space.

An Extension of the Order-Preserving Mapping to the WENO-Z-Type Schemes

In the present study,we extend the order-preserving(OP)criterion proposed in our latest studies to the WENO-Z-type schemes.Firstly,we innovatively present the concept of the generalized mapped WENO schemes by rewriting the Ztype weights in a uniform formula from the perspective of the mapping relation.Then,we naturally introduce the OP criterion to improve the WENO-Z-type schemes,and the resultant schemes are denoted as MOP-GMWENO-X,where the notation“X”is used to identify the version of the existing WENO-Z-type scheme in this paper.Finally,extensive numerical experiments have been conducted to demonstrate the benefits of these new schemes.We draw the conclusion that,the convergence properties of the proposed schemes are equivalent to the corresponding WENO-X schemes.The major benefit of the new schemes is that they have the capacity to achieve high resolutions and simultaneously remove spurious oscillations for long simulations.The new schemes have the additional benefit that they can greatly decrease the post-shock oscillations on solving 2D Euler problems with strong shock waves.

An Arbitrary Order Reconstructed Discontinuous Approximation to Biharmonic Interface Problem

We present an arbitrary order discontinuous Galerkin finite element method for solving the biharmonic interface problem on the unfitted mesh.The approximation space is constructed by a patch reconstruction process with at most one degrees of freedom per element.The discrete problem is based on the symmetric interior penalty method and the jump conditions are weakly imposed by the Nitsche's technique.The C^(2)-smooth interface is allowed to intersect elements in a very general fashion and the stability near the interface is naturally ensured by the patch reconstruction.We prove the optimal a priori error estimate under the energy norm and the L^(2) norm.Numerical results are provided to verify the theoretical analysis.

Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions

We study the stationary Wigner equation on a bounded, one- dimensional spatial domain with inflow boundary conditions by using the parity decomposition of L. Barletti and P. F. Zweifel [Transport Theory Statist. Phys., 2001, 30（4-6）： 507-520]. The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd L2-spaee by analyzing the moment system. An example is provided to show that how to use the analysis to obtain the solution of the stationary Wigner equation with inflow boundary conditions.

ADAPTIVE FINITE ELEMENT APPROXIMATION FOR A CLASS OF PARAMETER ESTIMATION PROBLEMS

In this paper, we study adaptive finite element discretisation schemes for a class of parameter estimation problem. We propose to efficient algorithms for the estimation problem use adaptive multi-meshes in developing We derive equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.

High order moment closure for Vlasov-Maxwell equations

The extended magnetohydrodynamic models are derived based on the moment closure of the Vlasov-Maxwell （VM） equations. We adopt the Grad type moment expansion which was firstly proposed for the Boltzmann equation. A new regularization method for the Grad＇s moment system was recently proposed to achieve the globally hyperbolicity so that the local well-posedness of the moment system is attained. For the VM equations, the moment expansion of the convection term is exactly the same as that in the Boltzmann equation, thus the new developed regularization applies. The moment expansion of the electromagnetic force term in the VM equations turns out to be a linear source term, which can preserve the conservative properties of the distribution function in the VM equations perfectly.

A Multi-Mesh Adaptive Finite Element Approximation to Phase Field Models

In this work,we propose an efficient multi-mesh adaptive finite element method for simulating the dendritic growth in two-and three-dimensions.The governing equations used are the phase field model,where the regularity behaviors of the relevant dependent variables,namely the thermal field function and the phase field function,can be very different.To enhance the computational efficiency,we approximate these variables on different h-adaptive meshes.The coupled terms in the system are calculated based on the implementation of the multi-mesh h-adaptive algorithm proposed by Li(J.Sci.Comput.,pp.321-341,24(2005)).It is illustrated numerically that the multi-mesh technique is useful in solving phase field models and can save storage and the CPU time significantly.

Numerical Regularized Moment Method for High Mach Number Flow

This paper is a continuation of our earlier work[SIAM J.Sci.Comput.,32(2010),pp.2875-2907]in which a numericalmoment method with arbitrary order of moments was presented.However,the computation may break down during the calculation of the structure of a shock wave with Mach number M_(0)≥3.In this paper,we concentrate on the regularization of the moment systems.First,we apply the Maxwell iteration to the infinite moment system and determine the magnitude of each moment with respect to the Knudsen number.After that,we obtain the approximation of high order moments and close the moment systems by dropping some high-order terms.Linearization is then performed to obtain a very simple regularization term,thus it is very convenient for numerical implementation.To validate the new regularization,the shock structures of low order systems are computed with different shock Mach numbers.

A DISCONTINUOUS GALERKIN METHOD BY PATCH RECONSTRUCTION FOR BIHARMONIC PROBLEM

We propose a new discontinuous Galerkin method based on the least-squares patch reconstruction for the biharmonic problem. We prove the optimal error estimate of the proposed method. The two-dimensional and three-dimensional numerical examples are presented to confirm the accuracy and efficiency of the method with several boundary conditions and several types of polygon meshes and polyhedral meshes.

An Efficient Mapped WENO Scheme Using Approximate Constant Mapping

We present a novel mapping approach for WENO schemes through the use of an approximate constant mapping function which is constructed by employing an approximation of the classic signum function.The new approximate constant mapping function is designed to meet the overall criteria for a proper mapping function required in the design of the WENO-PM6 scheme.The WENO-PM6 scheme was proposed to overcome the potential loss of accuracy of the WENO-M scheme which was developed to recover the optimal convergence order of the WENO-JS scheme at critical points.Our new mapped WENO scheme,denoted as WENOACM,maintains almost all advantages of the WENO-PM6 scheme,including low dissipation and high resolution,while decreases the number of mathematical operations remarkably in every mapping process leading to a significant improvement of efficiency.The convergence rates of the WENO-ACM scheme have been shown through one-dimensional linear advection equation with various initial conditions.Numerical results of one-dimensional Euler equations for the Riemann problems,the Mach 3 shock-density wave interaction and the Woodward-Colella interacting blastwaves are improved in comparison with the results obtained by the WENO-JS,WENO-M and WENO-PM6 schemes.Numerical experiments with two-dimensional problems as the 2D Riemann problem,the shock-vortex interaction,the 2D explosion problem,the double Mach reflection and the forward-facing step problem modeled via the two dimensional Euler equations have been conducted to demonstrate the high resolution and the effectiveness of the WENO-ACM scheme.The WENOACM scheme provides significantly better resolution than the WENO-M scheme and slightly better resolution than the WENO-PM6 scheme,and compared to the WENOM and WENO-PM6 schemes,the extra computational cost is reduced by more than 83%and 93%,respectively.

AN IMPROVED ERROR ANALYSIS FOR FINITE ELEMENT APPROXIMATION OF BIOLUMINESCENCE TOMOGRAPHY

This paper is concerned with an ill-posed problem which results from the area of molecular imaging and is known as BLT problem. Using Tikhonov regularization technique, a quadratic optimization problem can be formulated. We provide an improved error estimate for the finite element approximation of the regularized optimization problem. Some numerical examples are presented to demonstrate our theoretical results.