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

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Journal of Computational Mathematics 2009年第Z1期 280-298页

作者： Ziqing Xie Zuozheng Zhang Zhimin Zhang Key Laboratory of Computational and Stochastic Mathematics and Its Applications,Universities of Hunan Province,Hunan Normal University,Changsha 410081,China

In this paper,we consider the local discontinuous Galerkin method (LDG) for solv-ing singularly perturbed convection-diffusion problems in one- and two-dimensional *** existence and uniqueness of the LDG solutions are *** ex-periments demonstrate that it seems impossible to obtain uniform superconvergence fornumerical fluxes under uniform *** to the implementation of two-type dif-ferent anisotropic meshes,i.e.,the Shishkin and an improved grade meshes,the uniform2p+1-order superconvergence is observed numerically for both one-dimensional and two-dimensional cases.

关键词： Singularly perturbed problems Local discontinuous Galerkin method numerical fluxes Uniform superconvergence

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

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Advances in Applied Mathematics and Mechanics 2022年第1期14卷 275-298页

作者： Uttam Singh Rajput Krishna Mohan Singh Department of Mechanical and Industrial Engineering Indian Institute of Technology RoorkeeRoorkee 247667India

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.(***.,207(2005),pp.542-567)for solving hyperbolic conservation *** reconstruction of numerical flux is done using primitive variables instead of conservative *** 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(***.,228(1996),pp.202-228).The third-order Runge-Kutta method has been used for solution advancement in *** Harten-Lax-van Leer-Contact(HLLC)shock-capturing method is used to provide necessary upwinding into 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.

关键词： High resolution scheme unsteady non-linear weights numerical fluxes alternative WENO scheme hyperbolic equations

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

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Communications in Computational Physics 2012年第9期12卷 1183-1214页

作者： Guglielmo Stecca Annunziato Siviglia Eleuterio F.Toro Department of Civil and Environmental Engineering University of TrentoVia Mesiano 77I-38100 TrentoItaly Laboratory of AppliedMathematics University of TrentoVia Mesiano 77I-38100 TrentoItaly.

We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative *** applies in multidimensional structured and unstructured *** 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 *** proposed first-order method is shown to be identical to the Godunov upwindmethod in applications to a 2×2 linear hyperbolic *** method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water *** to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured ***,numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.

关键词： Conservative hyperbolic systems centred schemes unstructured meshes numerical fluxes shallow water equations FORCE upwind-biased

Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System

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Communications on Applied Mathematics and Computation 2022年第2期4卷 381-416页

作者： Jiawei Sun Shusen Xie Yulong Xing Department of Mathematics The Ohio State UniversityColumbusOH 43210USA School of Mathematical Sciences Ocean University of ChinaQingdao 266100ShandongChina

Boussinesq type equations have been widely studied to model the surface water *** 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,*** propose local discontinuous Galerkin(LDG)methods,with carefully chosen numerical fluxes,to numerically solve this abcd Boussinesq *** 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,*** 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.

关键词： Local discontinuous Galerkin methods Boussinesq equations Coupled BBM equations Error estimate numerical fluxes Head-on collision

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

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Communications on Applied Mathematics and Computation 2024年第3期6卷 1832-1859页

作者： Vitaly Gyrya Mikhail Shashkov Alexei Skurikhin Svetlana Tokareva Los Alamos National Laboratory Applied Mathematics and Plasma Physics(T-5)Los AlamosUSA Los Alamos National Laboratory Continuum Models and Numerical Methods(XCP-4)Los AlamosUSA Los Alamos National Laboratory Space Data Science and Systems(ISR-3)Los AlamosUSA

We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid *** Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock *** solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics,such as finite-volume or discontinuous Galerkin ***,a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial *** exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear *** 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 *** compare two approaches:(i)Gaussian process(GP)regressions,and(ii)neural network(NN)*** 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.

关键词： Machine learning(ML) Neural network(NN) Gaussian process(GP) Riemann problem numerical fluxes Finite-volume method