arXiv subjects

Ruo Li

Publications and source records attributed to Ruo Li.

At least 73 records · Page 4Linked to original sources

arXiv2019

A finite element method by patch reconstruction for the Stokes problem using mixed formulations

In this paper, we develop a patch reconstruction finite element method for the Stokes problem. The weak formulation of the interior penalty discontinuous Galerkin is employed. The proposed method has a great flexibility in velocity-pressure space pairs whose stability properties are confirmed by the inf-sup tests. Numerical examples show the applicability and efficiency of the proposed method.

math.NA↗

arXiv2018

A Nonlinear Moment Model for Radiative Transfer Equation in Slab Geometry

This paper is concerned with the approximation of the radiative transfer equation for a grey medium in the slab geometry by the moment method. We develop a novel moment model inspired by the classical $P_N$ model and $M_N$ model. The new model takes the ansatz of the $M_1$ model as the weight function and follows the primary idea of the $P_N$ model to approximate the specific intensity by expanding it around the weight function in terms of orthogonal polynomials. The weight function uses the information of the first two moments, which brings the new model the capability to approximate an anisotropic distribution. Mathematical properties of the moment model are investigated, and particularly the hyperbolicity and the characteristic structure of the Riemann problem of the model with three moments are studied in detail. Some numerical simulations demonstrate its numerical efficiency and show its superior in comparison to the $P_N$ model.

physics.class-ph↗

arXiv2018

Approximation to Singular Quadratic Collision Model in Fokker-Planck-Landau Equation

We propose a Hermite-Galerkin spectral method to numerically solve the spatially homogeneous Fokker-Planck-Landau equation with singular quadratic collision model. To compute the collision model, we adopt a novel approximation formulated by a combination of a simple linear term and a quadratic term very expensive to evaluate. Using the Hermite expansion, the quadratic term is evaluated exactly by calculating the spectral coefficients. To deal with singularities, we make use of Burnett polynomials so that even very singular collision model can be handled smoothly. Numerical examples demonstrate that our method can capture low-order moments with satisfactory accuracy and performance.

math.NA↗

arXiv2018

A Discontinuous Galerkin Method for the Stokes Equation by Divergence-free Patch Reconstruction

A discontinuous Galerkin method by patch reconstruction is proposed for Stokes flows. A locally divergence-free reconstruction space is employed as the approximation space, and the interior penalty method is adopted which imposes the normal component penalty terms to cancel out the pressure term. Consequently, the Stokes equation can be solved as an elliptic system instead of a saddle-point problem due to such weak form. The number of degree of freedoms of our method is the same as the number of elements in the mesh for different order of accuracy. The error estimations of the proposed method are given in a classical style, which are then verified by some numerical examples.

math.NA↗

arXiv2018

Filtered Hyperbolic Moment Method for the Vlasov Equation

In this paper, we investigate the effect of the filter for the hyperbolic moment equations(HME) [15] of the Vlasov-Poisson equations and propose a novel quasi time-consistent filter to suppress the numerical recurrence effect. By taking properties of HME into consideration, the filter preserves a lot of physical properties of HME, including Galilean invariance and the conservation of mass, momentum and energy. We present two viewpoints, collisional viewpoint and dissipative viewpoint, to dissect the filter, and show that the filtered hyperbolic moment method can be treated as a solver of Vlasov equation. Numerical simulations of the linear Landau damping and two stream instability are tested to demonstrate the effectiveness of the filter in restraining recurrence arising from particle streaming. Both the analysis and the numerical results indicate that the filtered HME can capture the evolution of the Vlasov equation, even when phase mixing and filamentation are dominant.

math.NA↗

arXiv2018

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.

math.NA↗

arXiv2018

An Approximate Solver for Multi-medium Riemann Problem with Mie-Grüneisen Equations of State

We propose an approximate solver for multi-medium Riemann problems with materials described by a family of general Mie-Grüneisen equations of state, which are widely used in practical applications. The solver provides the interface pressure and normal velocity by an iterative method. The well-posedness and convergence of the solver is verified with mild assumptions on the equations of state. To validate the solver, it is employed in computing the numerical flux on phase interfaces of a numerical scheme on Eulerian grids that was developed recently for compressible multi-medium flows. Numerical examples are presented for Riemann problems, air blast and underwater explosion applications.

math.NA↗

arXiv2018

Integrated Linear Reconstruction for Finite Volume Scheme on Arbitrary Unstructured Grids

In [L. Chen and R. Li, Journal of Scientific Computing, Vol. 68, pp. 1172--1197, (2016)], an integrated linear reconstruction was proposed for finite volume methods on unstructured grids. However, the geometric hypothesis of the mesh to enforce a local maximum principle is too restrictive to be satisfied by, for example, locally refined meshes or distorted meshes generated by arbitrary Lagrangian-Eulerian methods in practical applications. In this paper, we propose an improved integrated linear reconstruction approach to get rid of the geometric hypothesis. The resulting optimization problem is a convex quadratic programming problem, and hence can be solved efficiently by classical active-set methods. The features of the improved integrated linear reconstruction include that i). the local maximum principle is fulfilled on arbitrary unstructured grids, ii). the reconstruction is parameter-free, and iii). the finite volume scheme is positivity-preserving when the reconstruction is generalized to the Euler equations. A variety of numerical experiments are presented to demonstrate the performance of this method.

math.NA↗

arXiv2018

An Arbitrary-Order Discontinuous Galerkin Method with One Unknown Per Element

We propose an arbitrary-order discontinuous Galerkin method for second-order elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error estimates for the energy norm and for the L$^2$ norm. The accuracy and the efficiency of the method up to order six on several polygonal meshes are illustrated by a set of benchmark problems.

math.NA↗

arXiv2017

3D B_2 Model for Radiative Transfer Equation Part I: Modelling

We extend to three-dimensional space the approximate M_2 model for the slab geometry studied in our previous paper. The B_2 model therein, as a special case of the second order extended quadrature method of moments (EQMOM), is proved to be globally hyperbolic. The model we proposed here extends EQMOM to multiple dimensions following the idea to approximate the maximum entropy closure for the slab geometry case. Like the M_2 closure, the ansatz of the new model has the capacity to capture both isotropic and beam-like solutions, while the new model has fluxes in closed-form, thus is applicable to practical numerical simulations. The rotational invariance, realizability, and hyperbolicity of the model are studied.

math.NA↗

arXiv2017

Resolving Knudsen Layer by High Order Moment Expansion

We model the Knudsen layer in Kramers' problem by linearized high order hyperbolic moment system. Due to the hyperbolicity, the boundary conditions of the moment system is properly reduced from the kinetic boundary condition. For Kramers' problem, we give the analytical solutions of moment systems. With the order increasing of the moment model, the solutions are approaching to the solution of the linearized BGK kinetic equation. The velocity profile in the Knudsen layer is captured with improved accuracy for a wide range of accommodation coefficients.

math.NA↗

arXiv2017

Linear Stability of Hyperbolic Moment Models for Boltzmann Equation

Grad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems, and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collision terms. It is proved that the regularized models are linear stability at the local equilibrium and satisfy Yong's first stability condition with commonly used approximate collision terms, and particularly with Boltzmann's binary collision model.

math-ph↗

arXiv2017

13-Moment System with Global Hyperbolicity for Quantum Gas

We point out that the quantum Grad's 13-moment system [R. Yano, Physica A: Statistical Mechanics and its Applications, 416:231-241, 2014] is lack of global hyperbolicity, and even worse, the thermodynamic equilibrium is not an interior point of the hyperbolicity region of the system. To remedy this problem, by fully considering Grad's expansion, we split the expansion into the equilibrium part and the non-equilibrium part, and propose a regularization for the system with the help of the new theory developed in [Z. Cai et al., SIAM J. Appl. Math., 75(5):2001-2023, 2015, Y. Fan, J. Stat. Phys., 161(4), 2015]. This provides us a new model which is hyperbolic for all admissible thermodynamic states, and meanwhile preserves the approximate accuracy of the original system. It should be noted that this procedure is not a trivial application of the theory in [Z. Cai et al., SIAM J. Appl. Math., 75(5):2001-2023, 2015, Y. Fan, J. Stat. Phys., 161(4), 2015].

math-ph↗

arXiv2016

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 in (Barletti and weifel, Trans. Theory Stat. Phys., 507--520, 2001). 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 $L^2$-space 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.

math.AP↗

arXiv2016

Acceleration for Microflow Simulations of High-Order Moment Models by Using Lower-Order Model Correction

We study the acceleration of steady-state computation for microflow, which is modeled by the high-order moment models derived recently from the steady-state Boltzmann equation with BGK-type collision term. By using the lower-order model correction, a novel nonlinear multi-level moment solver is developed. Numerical examples verify that the resulting solver improves the convergence significantly thus is able to accelerate the steady-state computation greatly. The behavior of the solver is also numerically investigated. It is shown that the convergence rate increases, indicating the solver would be more efficient, as the total levels increases. Three order reduction strategies of the solver are considered. Numerical results show that the most efficient order reduction strategy would be $m_{l-1} = \lceil m_{l} / 2 \rceil$.

math.NA↗

arXiv2016

A Second Order Time Homogenized Model for Sediment Transport

A multi-scale method for the hyperbolic systems governing sediment transport in subcritical case is developed. The scale separation of this problem is due to the fact that the sediment transport is much slower than flow velocity. We first derive a zeroth order homogenized model, and then propose a first order correction. It is revealed that the first order correction for hyperbolic systems has to be applied on the characteristic speed of slow variables in one dimensional case. In two dimensional case, besides the characteristic speed, the source term is also corrected. We develop a second order numerical scheme following the framework of heterogeneous multi-scale method. The numerical results in both one and two dimensional cases demonstrate the effectiveness and efficiency of our method.

math.NA↗

arXiv2016

Preserving Hyperbolicity in Stochastic Galerkin Method for Uncertainty Quantification

We first investigate the structure of the systems derived from the gPC based stochastic Galerkin method for the nonlinear hyperbolic systems with random inputs. This method adopts a generalized Polynomial Chaos (gPC) approximations in the stochastic Galerkin framework, but such approximations to the nonlinear hyperbolic systems do not necessarily yield hyperbolic systems \cite{Lucor2013}. Thus based on the work in \cite{framework}, we propose a framework to carry out the model reduction for the general nonlinear hyperbolic system to derive a final global system. Within this framework, the nonlinear hyperbolic system in one space dimension and the symmetric hyperbolic system in multiple space dimensions are reduced into a symmetric hyperbolic system based on the stochastic Galerkin method. We note that the basis functions in the expansion are not restricted to the random-dependent polynomials as that in gPC method and there is no restriction on the dimensions of the random variables neither.

math.NA↗

arXiv2015

Model Reduction of Kinetic Equations by Operator Projection

By a further study of the mechanism of the hyperbolic regularization of the moment system for Boltzmann equation proposed in [Z. Cai, Y. Fan, R. Li, Comm. Math. Sci. 11(2): 547-571, 2013], we point out that the key point is treating the time and space derivative in the same way. Based on this understanding, a uniform framework to derive globally hyperbolic moment systems from kinetic equations using an operator projection method is proposed. The framework is so concise and clear that it can be treated as an algorithm with four inputs to derive hyperbolic moment system by routine calculations. Almost all existing globally hyperbolic moment system can be included in the framework, as well as some new moment system including globally hyperbolic regularized versions of Grad ordered moment system and a multidimensional extension of the quadrature-based moment system.

math-ph↗