Development and Comparison of Numerical Fluxes for LWDG Methods

The discontinuous Galerkin （DO） or local discontinuous Galerkin （LDG） method is a spatial discretization procedure for convection-diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters. The Lax- Wendroff time discretization procedure is an altemative method for time discretization to the popular total variation diminishing （TVD） Runge-Kutta time discretizations. In this paper, we develop fluxes for the method of DG with Lax-Wendroff time discretization procedure （LWDG） based on different numerical fluxes for finite volume or finite difference schemes, including the first-order monotone fluxes such as the Lax-Friedfichs flux, Godunov flux, the Engquist-Osher flux etc. and the second-order TVD fluxes. We systematically investigate the performance of the LWDG methods based on these different numerical fluxes for convection terms with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one-dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.

A NUMERICAL STUDY FOR THE PERFORMANCE OF THE WENO SCHEMES BASED ON DIFFERENT NUMERICAL FLUXES FOR THE SHALLOW WATER EQUATIONS

In this paper we investigate the performance of the weighted essential non-oscillatory （WENO） methods based on different numerical fluxes, with the objective of obtaining better performance for the shallow water equations by choosing suitable numerical fluxes. We consider six numerical fluxes, i.e., Lax-Friedrichs, local Lax-Friedrichs, Engquist-Osher, Harten-Lax-van Leer, HLLC and the first-order centered fluxes, with the WENO finite volume method and TVD Runge-Kutta time discretization for the shallow water equations. The detailed numerical study is performed for both one-dimensional and two-dimensional shallow water equations by addressing the property, and resolution of discontinuities. issues of CPU cost, accuracy, non-oscillatory

The Corrected Simulation Method of Critical Heat Flux Prediction for Water-Cooled Divertor Based on Euler Homogeneous Model

An accurate critical heat flux（CHF） prediction method is the key factor for realizing the steady-state operation of a water-cooled divertor that works under one-sided high heating flux conditions.An improved CHF prediction method based on Euler＇s homogeneous model for flow boiling combined with realizable k-ε model for single-phase flow is adopted in this paper in which time relaxation coefficients are corrected by the Hertz-Knudsen formula in order to improve the calculation accuracy of vapor-liquid conversion efficiency under high heating flux conditions.Moreover,local large differences of liquid physical properties due to the extreme nonuniform heating flux on cooling wall along the circumference direction are revised by formula IAPWSIF97.Therefore,this method can improve the calculation accuracy of heat and mass transfer between liquid phase and vapor phase in a CHF prediction simulation of water-cooled divertors under the one-sided high heating condition.An experimental example is simulated based on the improved and the uncorrected methods.The simulation results,such as temperature,void fraction and heat transfer coefficient,are analyzed to achieve the CHF prediction.The results show that the maximum error of CHF based on the improved method is 23.7%,while that of CHF based on uncorrected method is up to 188%,as compared with the experiment results of Ref.[12].Finally,this method is verified by comparison with the experimental data obtained by International Thermonuclear Experimental Reactor（ITER）,with a maximum error of 6% only.This method provides an efficient tool for the CHF prediction of water-cooled divertors.

Machine Learning Approaches for the Solution of the Riemann Problem in Fluid Dynamics:a Case Study

We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid dynamics.The Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock tube.The solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics,such as finite-volume or discontinuous Galerkin methods.Therefore,a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial importance.The exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear equation.Prior to delving into the complexities of ML for the Riemann problem,we consider a much simpler formulation,yet very informative,problem of learning roots of quadratic equations based on their coefficients.We compare two approaches:(i)Gaussian process(GP)regressions,and(ii)neural network(NN)approximations.Among these approaches,NNs prove to be more robust and efficient,although GP can be appreciably more accurate(about 30\%).We then use our experience with the quadratic equation to apply the GP and NN approaches to learn the exact solution of the Riemann problem from the initial data or coefficients of the gas equation of state(EOS).We compare GP and NN approximations in both regression and classification analysis and discuss the potential benefits and drawbacks of the ML approach.

Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System

Boussinesq type equations have been widely studied to model the surface water wave.In this paper,we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system,the BBM-BBM system,the Bona-Smith system,etc.We propose local discontinuous Galerkin(LDG)methods,with carefully chosen numerical fluxes,to numerically solve this abcd Boussinesq system.The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a,b,c,d.Numerical experiments are shown to test the convergence rates,and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.

A carbuncle cure for the Harten-Lax-van Leer contact(HLLC)scheme using a novel velocity-based sensor

A hybrid numerical flux scheme is proposed by adapting the carbunclefree modified Harten-Lax-van Leer contact(HLLCM) scheme to smoothly revert to the Harten-Lax-van Leer contact(HLLC) scheme in regions of shear. This hybrid scheme, referred to as the HLLCT scheme, employs a novel, velocity-based shear sensor. In contrast to the non-local pressure-based shock sensors often used in carbuncle cures, the proposed shear sensor can be computed in a localized manner meaning that the HLLCT scheme can be easily introduced into existing codes without having to implement additional data structures. Through numerical experiments, it is shown that the HLLCT scheme is able to resolve shear layers accurately without succumbing to the shock instability.

Discontinuous Galerkin Method for Macroscopic Traffic Flow Models on Networks

In this paper,we describe a numerical technique for the solution of macroscopic traffic flow models on networks of roads.On individual roads,we consider the standard Lighthill-Whitham-Richards model which is discretized using the discontinuous Galerkin method along with suitable limiters.To solve traffic flows on networks,we construct suitable numerical fluxes at junctions based on preferences of the drivers.We prove basic properties of the constructed numerical flux and the resulting scheme and present numerical experiments,including a junction with complicated traffic light patterns with multiple phases.Differences with the approach to numerical fluxes at junctions fromČanićet al.(J Sci Comput 63:233-255,2015)are discussed and demonstrated numerically on a simple network.

An Approximate Riemann Solver for Advection-Diffusion Based on the Generalized Riemann Problem

We construct an approximate Riemann solver for scalar advection-diffusion equations with piecewise polynomial initial data.The objective is to handle advection and diffusion simultaneously to reduce the inherent numerical diffusion produced by the usual advection flux calculations.The approximate solution is based on the weak formulation of the Riemann problem and is solved within a space-time discontinuous Galerkin approach with two subregions.The novel generalized Riemann solver produces piecewise polynomial solutions of the Riemann problem.In conjunction with a recovery polynomial,the Riemann solver is then applied to define the numerical flux within a finite volume method.Numerical results for a piecewise linear and a piecewise parabolic approximation are shown.These results indicate a reduction in numerical dissipation compared with the conventional separated flux calculation of advection and diffusion.Also,it is shown that using the proposed solver only in the vicinity of discontinuities gives way to an accurate and efficient finite volume scheme.

A NUMERICAL STUDY OF UNIFORM SUPERCONVERGENCE OF LDG METHOD FOR SOLVING SINGULARLY PERTURBED PROBLEMS

In this paper, we consider the local discontinuous Galerkin method （LDG） for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform 2p ＋ i-order superconvergence is observed numerically for both one-dimensional and twodimensional cases.

Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Differential Equations

Discontinuous Galerkin (DG) methods are a class of finite element methodsusing discontinuous basis functions, which are usually chosen as piecewise polynomi-als. Since the basis functions can be discontinuous, these methods have the flexibilitywhich is not shared by typical finite element methods, such as the allowance of ar-bitrary triangulation with hanging nodes, less restriction in changing the polynomialdegrees in each element independent of that in the neighbors (p adaptivity), and localdata structure and the resulting high parallel efficiency. In this paper, we give a generalreview of the local DG (LDG) methods for solving high-order time-dependent partialdifferential equations (PDEs). The important ingredient of the design of LDG schemes,namely the adequate choice of numerical fluxes, is highlighted. Some of the applica-tions of the LDG methods for high-order time-dependent PDEs are also be discussed.

The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections

Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution,a direct discontinuous Galerkin(DDG)method for diffusion problems was introduced in[H.Liu and J.Yan,SIAM J.Numer.Anal.47(1)(2009),475-698].In this work,we show that higher order(k≥4)derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method;still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all p^(k) elements.The refined DDG method with such numerical fluxes enjoys the optimal(k+1)th order of accuracy.The developed method is also extended to solve convection diffusion problems in both one-and two-dimensional settings.A series of numerical tests are presented to demonstrate the high order accuracy of the method.

Study of accelerator neutrino detection at a spallation source

We study the detection of accelerator neutrinos produced at the China Spallation Neutron Source （CSNS）. Using the code FLUKA, we have simulated the production of neutrinos in a proton beam on a tungsten target and obtained the yield efficiency, numerical flux, and average energy of different flavors of neutrinos. Furthermore, detection of these accelerator neutrinos is investigated in two reaction channels： neutrino-electron reactions and neutrino-carbon reactions. The expected numbers of different flavors of neutrinos have also been calculated.

A Fifth Order Alternative Mapped WENO Scheme for Nonlinear Hyperbolic Conservation Laws

In this work,we have developed a fifth-order alternative mapped weighted essentially nonoscillatory(AWENO-M)finite volume scheme using non-linear weights of mapped WENO reconstruction scheme of Henrick et al.(J.Comput.Phys.,207(2005),pp.542-567)for solving hyperbolic conservation laws.The reconstruction of numerical flux is done using primitive variables instead of conservative variables.The present scheme results in less spurious oscillations near discontinuities and shows higher-order accuracy at critical points compared to the alternative WENO scheme(AWENO)based on traditional non-linear weights of Jiang and Shu(J.Comput.Phys.,228(1996),pp.202-228).The third-order Runge-Kutta method has been used for solution advancement in time.The Harten-Lax-van Leer-Contact(HLLC)shock-capturing method is used to provide necessary upwinding into the solution.The performance of the present scheme is evaluated in terms of accuracy,computational cost,and resolution of discontinuities by using various one and two-dimensional test cases.

An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation

An iterative discontinuous Galerkin(DG)method is proposed to solve the nonlinear Poisson Boltzmann(PB)equation.We first identify a function space inwhich the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations,while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space.For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations.More precisely,we use one initial guess when the Debye parameter l=O(1),and a special initial guess for l≪1 to ensure convergence.The iterative parameter is carefully chosen to guarantee the existence,uniqueness,and convergence of the iteration.In particular,iteration steps can be reduced for a variable iterative parameter.Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases of l=O(1)and l≪1.The(m+1)th order of accuracy for L2 and mth order of accuracy for H1 for Pm elements are numerically obtained.

A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws:Application to Shallow Water Equations

We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form.It applies in multidimensional structured and unstructured meshes.The proposed method is an extension of the UFORCEmethod developed by Stecca,Siviglia and Toro[25],in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan.The proposed first-order method is shown to be identical to the Godunov upwindmethod in applications to a 2×2 linear hyperbolic system.The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations.Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes.Finally,numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.