55 results on '"Chi-Wang Shu"'
Search Results
2. L$^2$ Error Estimate to Smooth Solutions of High Order Runge--Kutta Discontinuous Galerkin Method for Scalar Nonlinear Conservation Laws with and without Sonic Points
- Author
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Jingqi Ai, Yuan Xu, Chi-Wang Shu, and Qiang Zhang
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2022
3. An Essentially Oscillation-Free Discontinuous Galerkin Method for Hyperbolic Systems
- Author
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Yong Liu, Jianfang Lu, and Chi-Wang Shu
- Subjects
Computational Mathematics ,Applied Mathematics - Published
- 2022
4. An Oscillation-free Discontinuous Galerkin Method for Scalar Hyperbolic Conservation Laws
- Author
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Yong Liu, Chi-Wang Shu, and Jianfang Lu
- Subjects
Numerical Analysis ,Computational Mathematics ,Conservation law ,Discontinuous Galerkin method ,Oscillation ,Applied Mathematics ,Scalar (mathematics) ,Mathematical analysis ,Superconvergence ,High order ,Spurious oscillations ,Mathematics - Abstract
In this paper, we propose a novel discontinuous Galerkin (DG) method to control the spurious oscillations when solving the scalar hyperbolic conservation laws. Usually, the high order linear numeri...
- Published
- 2021
5. A Discontinuous Galerkin Method for Stochastic Conservation Laws
- Author
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Chi-Wang Shu, Shanjian Tang, and Yunzhang Li
- Subjects
Computational Mathematics ,Conservation law ,Discontinuous Galerkin method ,Applied Mathematics ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Itō's lemma ,01 natural sciences ,Stability (probability) ,Mathematics - Abstract
In this paper we present a discontinuous Galerkin (DG) method to approximate stochastic conservation laws, which is an efficient high-order scheme. We study the stability for the semidiscrete DG me...
- Published
- 2020
6. Error Estimate of the Fourth-Order Runge--Kutta Discontinuous Galerkin Methods for Linear Hyperbolic Equations
- Author
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Chi-Wang Shu, Yuan Xu, and Qiang Zhang
- Subjects
Physics::Computational Physics ,Numerical Analysis ,Applied Mathematics ,010103 numerical & computational mathematics ,Computer Science::Numerical Analysis ,01 natural sciences ,Energy analysis ,Mathematics::Numerical Analysis ,Computational Mathematics ,Runge–Kutta methods ,Fourth order ,Discontinuous Galerkin method ,Applied mathematics ,Condensed Matter::Strongly Correlated Electrons ,0101 mathematics ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper we consider the Runge--Kutta discontinuous Galerkin (RKDG) method to solve linear constant-coefficient hyperbolic equations, where the fourth-order explicit Runge--Kutta time-marching...
- Published
- 2020
7. Existence and Computation of Solutions of a Model of Traffic Involving Hysteresis
- Author
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Chi-Wang Shu and Haitao Fan
- Subjects
Hysteresis ,Applied Mathematics ,Computation ,Total variation diminishing ,Applied mathematics ,Upwind scheme ,Two-phase flow ,Traffic flow ,Borel measure ,Hyperbolic systems ,Mathematics - Abstract
The meaning of weak solutions of a nonconservative hyperbolic system with discontinuous coefficients modeling traffic flows involving hysteresis is defined. An upwinding approximation scheme for th...
- Published
- 2020
8. On New Strategies to Control the Accuracy of WENO Algorithms Close to Discontinuities
- Author
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Chi-Wang Shu, Juan Ruiz, Sergio Amat, Fundación Séneca, Ministerio de Economía y Competitividad, and National Science Foundation (NSF)
- Subjects
Signal processing ,Numerical Analysis ,12 Matemáticas ,Applied Mathematics ,Order of accuracy ,Matemática Aplicada ,010103 numerical & computational mathematics ,Classification of discontinuities ,01 natural sciences ,Raising (metalworking) ,Computational Mathematics ,Nonlinear system ,Improved adaption to discontinuities ,New optimal weights ,WENO schemes ,0101 mathematics ,Control (linguistics) ,Algorithm ,Interpolation ,Mathematics - Abstract
This paper is devoted to the construction and analysis of new nonlinear optimal weights for weighted ENO (WENO) interpolation capable of raising the order of accuracy close to discontinuities. The new nonlinear optimal weights are constructed using a strategy inspired by the original WENO algorithm, and they work very well for corner or jump singularities, leading to optimal theoretical accuracy. This is the first part of a series of two papers. In this first part we analyze the performance of the new algorithms proposed for univariate function approximation in the point values (interpolation problem). In the second part, we will extend the analysis to univariate function approximation in the cell averages (reconstruction problem). Our aim is twofold: to raise the order of accuracy of the WENO type interpolation schemes both near discontinuities and in the interval which contains the singularity. The first problem can be solved using the new nonlinear optimal weights, but the second one requires a new strategy that locates the position of the singularity inside the cell in order to attain adaption. This new strategy is inspired by the ENO-SR schemes proposed by Harten [J. Comput. Phys., 83 (1989), pp. 148--184]. Thus, we will introduce two different algorithms in the point values. The first one can deal with corner singularities and jump discontinuities for intervals not containing the singularity. The second algorithm can also deal with intervals containing corner singularities, as they can be detected from the point values, but jump discontinuities cannot, as the information of their position is lost during the discretization process. As mentioned before, the second part of this work will be devoted to the cell averages and, in this context, it will be possible to work with jump discontinuities as well. The work of the authors was supported by the Programa de Apoyo a la Investigatión de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18, by the national research project MTM2015-64382-P (MINECO/FEDER), and by National Science Foundation grant DMS-1719410.
- Published
- 2019
9. Strong Stability of Explicit Runge--Kutta Time Discretizations
- Author
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Zheng Sun and Chi-Wang Shu
- Subjects
Physics::Computational Physics ,Numerical Analysis ,Conservation law ,Applied Mathematics ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Computer Science::Numerical Analysis ,01 natural sciences ,Stability (probability) ,Mathematics::Numerical Analysis ,Computational Mathematics ,Runge–Kutta methods ,FOS: Mathematics ,Energy method ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Abstract
Motivated by studies on fully discrete numerical schemes for linear hyperbolic conservation laws, we present a framework on analyzing the strong stability of explicit Runge-Kutta (RK) time discretizations for semi-negative autonomous linear systems. The analysis is based on the energy method and can be performed with the aid of a computer. Strong stability of various RK methods, including a sixteen-stage embedded pair of order nine and eight, has been examined under this framework. Based on numerous numerical observations, we further characterize the features of strongly stable schemes. A both necessary and sufficient condition is given for the strong stability of RK methods of odd linear order.
- Published
- 2019
10. Superconvergence of Discontinuous Galerkin Method for Scalar Nonlinear Hyperbolic Equations
- Author
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Waixiang Cao, Yang Yang, Chi-Wang Shu, and Zhimin Zhang
- Subjects
Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,Scalar (mathematics) ,010103 numerical & computational mathematics ,Superconvergence ,Computer Science::Numerical Analysis ,01 natural sciences ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Exact solutions in general relativity ,Discontinuous Galerkin method ,Bounded function ,Piecewise ,0101 mathematics ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper, we study the superconvergence behavior of the semi-discrete discontinuous Galerkin (DG) method for scalar nonlinear hyperbolic equations in one spatial dimension. Superconvergence results for problems with fixed and alternating wind directions are established. On the one hand, we prove that, if the wind direction is fixed (i.e., the derivative of the flux function is bounded away from zero), both the cell average error and numerical flux error at cell interfaces converge at a rate of $2k+1$ when upwind fluxes and piecewise polynomials of degree $k$ are used. Moreover, we also prove that the function value approximation of the DG solution is superconvergent at interior right Radau points, and the derivative value approximation is superconvergent at interior left Radau points, with an order of k+2 and k+1, respectively. As a byproduct, we show a (k+2)th order superconvergence of the DG solution towards the Gauss--Radau projection of the exact solution. On the other hand, superconvergence resu...
- Published
- 2018
11. Optimal Error Estimates of the Semidiscrete Central Discontinuous Galerkin Methods for Linear Hyperbolic Equations
- Author
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Yong Liu, Chi-Wang Shu, and Mengping Zhang
- Subjects
Numerical Analysis ,Conservation law ,Degree (graph theory) ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics::Numerical Analysis ,law.invention ,010101 applied mathematics ,Computational Mathematics ,Dimension (vector space) ,Discontinuous Galerkin method ,law ,Piecewise ,Applied mathematics ,Polygon mesh ,Cartesian coordinate system ,0101 mathematics ,Hyperbolic partial differential equation ,Mathematics - Abstract
We analyze the central discontinuous Galerkin method for time-dependent linear conservation laws. In one dimension, optimal a priori $L^2$ error estimates of order $k+1$ are obtained for the semidiscrete scheme when piecewise polynomials of degree at most $k$ ($k\geq0$) are used on overlapping uniform meshes. We then extend the analysis to multidimensions on uniform Cartesian meshes when piecewise tensor-product polynomials are used on overlapping meshes. Numerical experiments are given to demonstrate the theoretical results.
- Published
- 2018
12. Implicit Positivity-Preserving High-Order Discontinuous Galerkin Methods for Conservation Laws
- Author
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Chi-Wang Shu and Tong Qin
- Subjects
Conservation law ,Polynomial ,Discretization ,Generalization ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Backward Euler method ,010101 applied mathematics ,Computational Mathematics ,Tensor product ,Discontinuous Galerkin method ,Applied mathematics ,Polygon mesh ,0101 mathematics ,Mathematics - Abstract
Positivity-preserving discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws have been extensively studied in the last several years, but nearly all the developed schemes are coupled with explicit time discretizations. Explicit discretizations suffer from the constraint for the Courant--Friedrichs--Lewy (CFL) number. This makes explicit methods impractical for problems involving unstructured and extremely varying meshes or long-time simulations. Instead, implicit DG schemes are often popular in practice, especially in the computational fluid dynamics (CFD) community. In this paper we develop a high-order positivity-preserving DG method with the backward Euler time discretization for conservation laws. We focus on one spatial dimension. However, the result easily generalizes to multidimensional tensor product meshes and polynomial spaces. This work is based on a generalization of the positivity-preserving limiters in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 3091--312...
- Published
- 2018
13. Discontinuous Galerkin Methods for Weakly Coupled Hyperbolic MultiDomain Problems
- Author
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Chi-Wang Shu, Qingyuan Liu, and Mengping Zhang
- Subjects
0301 basic medicine ,030103 biophysics ,Applied Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Stability (probability) ,03 medical and health sciences ,Computational Mathematics ,Simple (abstract algebra) ,Discontinuous Galerkin method ,Biological cell ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In this paper, we develop and analyze the Runge--Kutta discontinuous Galerkin (RKDG) method to solve weakly coupled hyperbolic multidomain problems. Such problems involve transfer type boundary conditions with discontinuous fluxes between different domains, calling for special techniques to prove stability of the RKDG methods. We prove both stability and error estimates for our RKDG methods on simple models, and then apply them to a biological cell proliferation model [N. Echenim, D. Monniaux, M. Sorine, and F. Clement, Math. Biosci., 198 (2005), pp. 57--79]. Numerical results are provided to illustrate the good behavior of our RKDG methods.
- Published
- 2017
14. A High Order Stable Conservative Method for Solving Hyperbolic Conservation Laws on Arbitrarily Distributed Point Clouds
- Author
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Chi-Wang Shu and Jie Du
- Subjects
Conservation law ,Applied Mathematics ,Mathematical analysis ,Point cloud ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Discontinuous Galerkin method ,Mesh generation ,Polygon ,Piecewise ,Initial value problem ,0101 mathematics ,Voronoi diagram ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics - Abstract
In this paper, we aim to solve one- and two-dimensional hyperbolic conservation laws on arbitrarily distributed point clouds. The initial condition is given on such a point cloud, and the algorithm solves for point values of the solution at later time also on this point cloud. By using the Voronoi technique and by introducing a grouping algorithm, we divide the computational domain into nonoverlapping cells. Each cell is a polygon and contains a minimum number of the given points to ensure accuracy. We carefully select points in each cell during the grouping procedure and hence are able to interpolate or fit the discrete initial values with piecewise polynomials. By adapting the traditional discontinuous Galerkin method on the constructed polygonal mesh, we obtain a stable, conservative, and high order method. Numerical results for both one- and two-dimensional scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of our mesh generation algorithm as w...
- Published
- 2016
15. High Order Positivity-Preserving Discontinuous Galerkin Methods for Radiative Transfer Equations
- Author
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Chi-Wang Shu, Daming Yuan, and Juan Cheng
- Subjects
Photon ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,Discontinuous Galerkin method ,Radiative transfer ,0101 mathematics ,Scaling ,Intensity (heat transfer) ,Mathematics - Abstract
The positivity-preserving property is an important and challenging issue for the numerical solution of radiative transfer equations. In the past few decades, different numerical techniques have been proposed to guarantee positivity of the radiative intensity in several schemes; however it is difficult to maintain both high order accuracy and positivity. The discontinuous Galerkin (DG) finite element method is a high order numerical method which is widely used to solve the neutron/photon transfer equations, due to its distinguished advantages such as high order accuracy, geometric flexibility, suitability for $h$- and $p$-adaptivity, parallel efficiency, and a good theoretical foundation for stability and error estimates. In this paper, we construct arbitrarily high order accurate DG schemes which preserve positivity of the radiative intensity in the simulation of both steady and unsteady radiative transfer equations in one- and two-dimensional geometry by using a combined technique of the scaling positivi...
- Published
- 2016
16. Stability and Error Estimates of Local Discontinuous Galerkin Methods with Implicit-Explicit Time-Marching for Advection-Diffusion Problems
- Author
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Chi-Wang Shu, Haijin Wang, and Qiang Zhang
- Subjects
Numerical Analysis ,Discretization ,Advection ,Applied Mathematics ,Mathematical analysis ,Stability (probability) ,Square (algebra) ,Mathematics::Numerical Analysis ,Computational Mathematics ,Discontinuous Galerkin method ,Diffusion (business) ,Constant (mathematics) ,Convection–diffusion equation ,Mathematics - Abstract
The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with carefully chosen implicit-explicit (IMEX) Runge--Kutta time discretization up to third order accuracy for solving one-dimensional linear advection-diffusion equations. In the time discretization the advection term is treated explicitly and the diffusion term implicitly. There are three highlights of this work. The first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG methods. The second is that, by aid of the aforementioned relationship and the energy method, we show that the IMEX LDG schemes are unconditionally stable for the linear problems in the sense that the time-step $\tau$ is only required to be upper-bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent...
- Published
- 2015
17. An Alternative Formulation of Finite Difference Weighted ENO Schemes with Lax--Wendroff Time Discretization for Conservation Laws
- Author
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Chi-Wang Shu, Yan Jiang, and Mengping Zhang
- Subjects
Computational Mathematics ,Conservation law ,Monotone polygon ,Discretization ,Lax–Wendroff method ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Scalar (mathematics) ,Finite difference ,Stencil ,Mathematics - Abstract
We develop an alternative formulation of conservative finite difference weighted essentially nonoscillatory (WENO) schemes to solve conservation laws. In this formulation, the WENO interpolation of the solution and its derivatives are used to directly construct the numerical flux, instead of the usual practice of reconstructing the flux functions. Even though this formulation is more expensive than the standard formulation, it does have several advantages. The first advantage is that arbitrary monotone fluxes can be used in this framework, while the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. The second advantage, which is fully explored in this paper, is that a narrower effective stencil is used compared with previous high order finite difference WENO schemes based on the reconstruction of flux functions, with a Lax--Wendroff time discretization. We will describe the scheme formulation and present numerical tests for one- and two-dimensional scalar ...
- Published
- 2013
18. Optimal Error Estimates of the Semidiscrete Local Discontinuous Galerkin Methods for High Order Wave Equations
- Author
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Chi-Wang Shu and Yan Xu
- Subjects
Numerical Analysis ,Discretization ,Applied Mathematics ,Mathematical analysis ,Order of accuracy ,Wave equation ,Mathematics::Numerical Analysis ,Schrödinger equation ,Computational Mathematics ,symbols.namesake ,Third order ,Discontinuous Galerkin method ,symbols ,Jump ,Order (group theory) ,Mathematics - Abstract
In this paper, we introduce a general approach for proving optimal $L^2$ error estimates for the semidiscrete local discontinuous Galerkin (LDG) methods solving linear high order wave equations. The optimal order of error estimates holds not only for the solution itself but also for the auxiliary variables in the LDG method approximating the various order derivatives of the solution. Examples including the one-dimensional third order wave equation, one-dimensional fifth order wave equation, and multidimensional Schrodinger equation are explored to demonstrate this approach. The main idea is to derive energy stability for the various auxiliary variables in the LDG discretization by using the scheme and its time derivatives with different test functions. Special projections are utilized to eliminate the jump terms at the cell boundaries in the error estimate in order to achieve the optimal order of accuracy.
- Published
- 2012
19. Analysis of Optimal Superconvergence of Discontinuous Galerkin Method for Linear Hyperbolic Equations
- Author
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Chi-Wang Shu and Yang Yang
- Subjects
Numerical Analysis ,Conservation law ,Discretization ,Applied Mathematics ,Mathematical analysis ,Superconvergence ,Computer Science::Numerical Analysis ,Projection (linear algebra) ,Finite element method ,Mathematics::Numerical Analysis ,Computational Mathematics ,Discontinuous Galerkin method ,Piecewise ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper, we study the superconvergence of the error for the discontinuous Galerkin (DG) finite element method for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise $k$th degree polynomials, the error between the DG solution and the exact solution is ($k+2$)th order superconvergent at the downwind-biased Radau points with suitable initial discretization. Moreover, we also prove the DG solution is ($k+2$)th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution. The superconvergence result in this paper leads to a new a posteriori error estimate. Our analysis is valid for arbitrary regular meshes and for $\mathcal{P}^k$ polynomials with arbitrary $k\geq1$, and for both periodic boundary conditions and for initial-boundary value problems. We perform numerical experiments to demonstrate that the superconvergence rate proved in this paper is optimal.
- Published
- 2012
20. Maximum-principle-satisfying High Order Finite Volume Weighted Essentially Nonoscillatory Schemes for Convection-diffusion Equations
- Author
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Yuanyuan Liu, Xiangxiong Zhang, and Chi-Wang Shu
- Subjects
Computational Mathematics ,Conservation law ,Maximum principle ,Finite volume method ,Incompressible flow ,Applied Mathematics ,Scalar (mathematics) ,Mathematical analysis ,Vorticity ,Convection–diffusion equation ,Navier–Stokes equations ,Mathematics - Abstract
To easily generalize the maximum-principle-satisfying schemes for scalar conservation laws in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 3091-3120] to convection diffusion equations, we propose a nonconventional high order finite volume weighted essentially nonoscillatory (WENO) scheme which can be proved maximum-principle-satisfying. Two-dimensional extensions are straightforward. We also show that the same idea can be used to construct high order schemes preserving the maximum principle for two-dimensional incompressible Navier-Stokes equations in the vorticity stream-function formulation. Numerical tests for the fifth order WENO schemes are reported.
- Published
- 2012
21. Superconvergence of Discontinuous Galerkin Methods for Scalar Nonlinear Conservation Laws in One Space Dimension
- Author
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Chi-Wang Shu, Qiang Zhang, Xiong Meng, and Boying Wu
- Subjects
Numerical Analysis ,Conservation law ,Applied Mathematics ,Scalar (mathematics) ,Mathematical analysis ,Superconvergence ,Upper and lower bounds ,Mathematics::Numerical Analysis ,Computational Mathematics ,Nonlinear system ,Exact solutions in general relativity ,Discontinuous Galerkin method ,Piecewise ,Mathematics - Abstract
In this paper, an analysis of the superconvergence property of the semidiscrete discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves $\left(k + \frac32\right)$th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree $k$ ($k \ge 1$), under the condition that $|f'(u)|$ possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on $f(u)$ is artificial.
- Published
- 2012
22. Uniformly Accurate Discontinuous Galerkin Fast Sweeping Methods for Eikonal Equations
- Author
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Shanqin Chen, Chi-Wang Shu, Yong-Tao Zhang, Fengyan Li, and Hongkai Zhao
- Subjects
Computational Mathematics ,Computational complexity theory ,Discontinuous Galerkin method ,Eikonal equation ,Applied Mathematics ,Norm (mathematics) ,Mathematical analysis ,Finite element solver ,Finite difference ,Solver ,First order ,Mathematics - Abstract
In [F. Li, C.-W. Shu, Y.-T. Zhang, H. Zhao, J. Comput. Phys., 227 (2008) pp. 8191-8208], we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous Galerkin (DG) finite element solver and a first order finite difference solver for Eikonal equations. The method has second order accuracy in the $L^1$ norm and a very fast convergence speed, but only first order accuracy in the $L^\infty$ norm for the general cases. This is an obstacle to the design of higher order DG fast sweeping methods. In this paper, we overcome this problem by developing uniformly accurate DG fast sweeping methods for solving Eikonal equations. We design novel causality indicators which guide the information flow directions for the DG local solver. The values of these indicators are initially provided by the first order finite difference fast sweeping method, and they are updated during iterations along with the solution. We observe both a uniform second order accuracy in the $L^\infty$ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples.
- Published
- 2011
23. High Order Finite Difference WENO Schemes for Nonlinear Degenerate Parabolic Equations
- Author
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Chi-Wang Shu, Mengping Zhang, and Yuanyuan Liu
- Subjects
Computational Mathematics ,Nonlinear system ,Conservation law ,Partial differential equation ,Discretization ,Applied Mathematics ,Mathematical analysis ,Finite difference ,Finite difference coefficient ,Parabolic partial differential equation ,Second derivative ,Mathematics - Abstract
High order accurate weighted essentially nonoscillatory (WENO) schemes are usually designed to solve hyperbolic conservation laws or to discretize the first derivative convection terms in convection dominated partial differential equations. In this paper we discuss a high order WENO finite difference discretization for nonlinear degenerate parabolic equations which may contain discontinuous solutions. A porous medium equation (PME) is used as an example to demonstrate the algorithm structure and performance. By directly approximating the second derivative term using a conservative flux difference, the sixth order and eighth order finite difference WENO schemes are constructed. Numerical examples are provided to demonstrate the accuracy and nonoscillatory performance of these schemes.
- Published
- 2011
24. Superconvergence of Discontinuous Galerkin and Local Discontinuous Galerkin Schemes for Linear Hyperbolic and Convection-Diffusion Equations in One Space Dimension
- Author
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Chi-Wang Shu and Yingda Cheng
- Subjects
Numerical Analysis ,Conservation law ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Superconvergence ,Mathematics::Numerical Analysis ,Computational Mathematics ,Discontinuous Galerkin method ,Piecewise ,Galerkin method ,Hyperbolic partial differential equation ,Linear equation ,Mathematics - Abstract
In this paper, we study the superconvergence property for the discontinuous Galerkin (DG) and the local discontinuous Galerkin (LDG) methods for solving one-dimensional time dependent linear conservation laws and convection-diffusion equations. We prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection-diffusion equations. The order of superconvergence for both cases is proved to be $k+\frac{3}{2}$ when piecewise $P^k$ polynomials with $k\geq1$ are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise $P^k$ polynomials with arbitrary $k\geq1$, improving upon the results in [Y. Cheng and C.-W. Shu, J. Comput. Phys., 227 (2008), pp. 9612-9627], [Y. Cheng and C.-W. Shu, Computers and Structures, 87 (2009), pp. 630-641] in which the proof based on Fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise $P^1$ polynomials.
- Published
- 2010
25. Stability Analysis and A Priori Error Estimates of the Third Order Explicit Runge–Kutta Discontinuous Galerkin Method for Scalar Conservation Laws
- Author
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Chi-Wang Shu and Qiang Zhang
- Subjects
Numerical Analysis ,Conservation law ,Discretization ,Applied Mathematics ,Courant–Friedrichs–Lewy condition ,Mathematical analysis ,Scalar (mathematics) ,Computer Science::Numerical Analysis ,Mathematics::Numerical Analysis ,Computational Mathematics ,Runge–Kutta methods ,Discontinuous Galerkin method ,Mathematics ,Numerical stability ,Linear stability - Abstract
In this paper we present an analysis of the Runge-Kutta discontinuous Galerkin method for solving scalar conservation laws, where the time discretization is the third order explicit total variation diminishing Runge-Kutta method. We use an energy technique to prove the $\mathrm{L}^2$-norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws. Quasi-optimal order is obtained for general numerical fluxes, and optimal order is given for upwind fluxes. The theoretical results are obtained for piecewise polynomials with any degree $k\geq1$ under the standard temporal-spatial CFL condition $\tau\leq\gamma h$, where $h$ and $\tau$ are the element length and time step, respectively, and the positive constant $\gamma$ is independent of $h$ and $\tau$.
- Published
- 2010
26. A Genuinely High Order Total Variation Diminishing Scheme for One-Dimensional Scalar Conservation Laws
- Author
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Chi-Wang Shu and Xiangxiong Zhang
- Subjects
Numerical Analysis ,Conservation law ,Finite volume method ,Applied Mathematics ,Numerical analysis ,Finite difference ,Computational Mathematics ,Third order ,Norm (mathematics) ,Total variation diminishing ,Calculus ,Applied mathematics ,Flux limiter ,Mathematics - Abstract
It is well known that finite difference or finite volume total variation diminishing (TVD) schemes solving one-dimensional scalar conservation laws degenerate to first order accuracy at smooth extrema [S. Osher and S. Chakravarthy, SIAM J. Numer. Anal., 21 (1984), pp. 955-984], thus TVD schemes are at most second order accurate in the $L^1$ norm for general smooth and nonmonotone solutions. However, Sanders [Math. Comp., 51 (1988), pp. 535-558] introduced a third order accurate finite volume scheme which is TVD, where the total variation is defined by measuring the variation of the reconstructed polynomials rather than the traditional way of measuring the variation of the grid values. By adopting the definition of the total variation for the numerical solutions as in [R. Sanders, Math. Comp., 51 (1988), pp. 535-558], it is possible to design genuinely high order accurate TVD schemes. In this paper, we construct a finite volume scheme which is TVD in this sense with high order accuracy (up to sixth order) in the $L^1$ norm. Numerical tests for a fifth order accurate TVD scheme will be reported, which include test cases from traffic flow models.
- Published
- 2010
27. High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems
- Author
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Chi-Wang Shu
- Subjects
Partial differential equation ,Differential equation ,business.industry ,Applied Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,Computational fluid dynamics ,Classification of discontinuities ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,Computational Mathematics ,Discontinuity (linguistics) ,Microscopic traffic flow model ,Flow (mathematics) ,Calculus ,Applied mathematics ,business ,Hyperbolic partial differential equation ,Mathematics - Abstract
High order accurate weighted essentially nonoscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations (PDEs) and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, nonoscillatory, and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. At the heart of the WENO schemes is actually an approximation procedure not directly related to PDEs, hence the WENO procedure can also be used in many non-PDE applications. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic PDEs and other convection dominated problems, and present a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications. Finally, we mention a few topics concerning WENO schemes that are currently under investigation.
- Published
- 2009
28. Local Discontinuous Galerkin Method for the Hunter–Saxton Equation and Its Zero-Viscosity and Zero-Dispersion Limits
- Author
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Yan Xu and Chi-Wang Shu
- Subjects
Computational Mathematics ,Nonlinear system ,Viscosity ,Computer simulation ,Discontinuous Galerkin method ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Hunter–Saxton equation ,Stability (probability) ,Mathematics::Numerical Analysis ,Numerical stability ,Mathematics - Abstract
In this paper, we develop, analyze, and test a local discontinuous Galerkin (LDG) method for solving the Hunter-Saxton (HS) equation and its zero-viscosity and zero-dispersion limits. The energy stability for general solutions are proved, and numerical simulation results for different types of solutions of the nonlinear HS equation are provided to illustrate the accuracy and capability of the LDG method. The zero-viscosity and zero-dispersion properties of the HS equation are studied in a numerical simulation.
- Published
- 2009
29. Postprocessing for the Discontinuous Galerkin Method over Nonuniform Meshes
- Author
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Robert M. Kirby, Chi-Wang Shu, Jennifer K. Ryan, and Sean Curtis
- Subjects
Mathematical optimization ,Numerical linear algebra ,Applied Mathematics ,Order of accuracy ,computer.software_genre ,Computational Mathematics ,Kernel (image processing) ,Discontinuous Galerkin method ,Projection method ,Applied mathematics ,Polygon mesh ,Galerkin method ,computer ,Condition number ,Mathematics - Abstract
A postprocessing technique based on negative order norm estimates for the discontinuous Galerkin methods was previously introduced by Cockburn, Luskin, Shu, and Suli [Proceedings of the International Symposium on Discontinuous Galerkin Methods, Springer, New York, pp. 291-300; Math. Comput., 72 (2003), pp. 577-606]. The postprocessor allows improvement in accuracy of the discontinuous Galerkin method for time-dependent linear hyperbolic equations from order $k$+1 to order 2$k$+1 over a uniform mesh. Assumptions on the convolution kernel along with uniformity in mesh size give a local translation invariant postprocessor that allows for simple implementation using small matrix-vector multiplications. In this paper, we present two alternatives for extending this postprocessing technique to include smoothly varying meshes. The first method uses a simple local $L^2$-projection of the smoothly varying mesh to a locally uniform mesh and uses this projected solution to compute the postprocessed solution. By using this local $L^2$-projection, recalculating the convolution kernel for every element can be avoided, and 2$k$+1 order accuracy of the postprocessed solution can be achieved. The second method uses the idea of characteristic length based upon the largest element size for the scaling of the postprocessing kernel. These two methods, local projection and characteristic length, are also applied to approximations over a mesh with elements that vary in size randomly. We discuss the computational issues in using these two techniques and demonstrate numerically that we obtain the 2$k$+1 order of accuracy for the smoothly varying meshes, and that although the 2$k$+1 order of accuracy is not fully realized for random meshes, there is significant improvement in the $L^2$-errors.
- Published
- 2008
30. The Discontinuous Galerkin Method for the Multiscale Modeling of Dynamics of Crystalline Solids
- Author
-
Xiantao Li, Wei Wang, and Chi-Wang Shu
- Subjects
Computer simulation ,Ecological Modeling ,Computation ,Numerical analysis ,General Physics and Astronomy ,General Chemistry ,Multiscale modeling ,Computer Science Applications ,Computational physics ,Nonlinear system ,Discontinuous Galerkin method ,Modeling and Simulation ,Displacement field ,Statistical physics ,Galerkin method ,Mathematics - Abstract
We present a multiscale model for numerical simulation of dynamics of crystalline solids. The method couples nonlinear elastodynamics as the continuum description and molecular dynamics as another component at the atomic scale. The governing equations on the macroscale are solved by the discontinuous Galerkin method, which is built up with an appropriate local curl-free space to produce a coherent displacement field. The constitutive data are based on the underlying atomistic model: it is either calibrated prior to the computation or obtained from molecular dynamics as the computation proceeds. The decision to use either the former or the latter is made locally for each cell based on suitable criteria.
- Published
- 2008
31. Convergence of High Order Finite Volume Weighted Essentially Nonoscillatory Scheme and Discontinuous Galerkin Method for Nonconvex Conservation Laws
- Author
-
Jing-Mei Qiu and Chi-Wang Shu
- Subjects
Computational Mathematics ,Conservation law ,Monotone polygon ,Finite volume method ,Discontinuous Galerkin method ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Entropy (information theory) ,Galerkin method ,Finite element method ,Mathematics - Abstract
In this paper, we consider the issue of convergence toward entropy solutions for the high order finite volume weighted essentially nonoscillatory (WENO) scheme and the discontinuous Galerkin (DG) finite element method approximating scalar nonconvex conservation laws. Although such high order nonlinearly stable schemes can usually converge to entropy solutions of convex conservation laws, convergence may fail for certain nonconvex conservation laws. We perform a detailed study to demonstrate such convergence issues for a few representative examples and suggest a modification of the high order schemes based on either first order monotone schemes or a second order entropic projection [Bouchut, Bourdarias, and Perthame, Math. Comp., 65 (1996), pp. 1438-1461] to achieve convergence toward entropy solutions while maintaining high order accuracy in smooth regions.
- Published
- 2008
32. A Local Discontinuous Galerkin Method for the Camassa–Holm Equation
- Author
-
Yan Xu and Chi-Wang Shu
- Subjects
Numerical Analysis ,Camassa–Holm equation ,Computer simulation ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Stability (probability) ,Mathematics::Numerical Analysis ,Computational Mathematics ,Nonlinear system ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Discontinuous Galerkin method ,Galerkin method ,Mathematics ,Numerical stability - Abstract
In this paper, we develop, analyze, and test a local discontinuous Galerkin (LDG) method for solving the Camassa–Holm equation which contains nonlinear high-order derivatives. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the $L^2$ stability for general solutions and give a detailed error estimate for smooth solutions, and provide numerical simulation results for different types of solutions of the nonlinear Camassa–Holm equation to illustrate the accuracy and capability of the LDG method.
- Published
- 2008
33. Central Discontinuous Galerkin Methods on Overlapping Cells with a Nonoscillatory Hierarchical Reconstruction
- Author
-
Eitan Tadmor, Chi-Wang Shu, Mengping Zhang, and Yingjie Liu
- Subjects
Numerical Analysis ,Conservation law ,Finite volume method ,Applied Mathematics ,Mathematical analysis ,Godunov's scheme ,Order of accuracy ,Computational Mathematics ,symbols.namesake ,Riemann problem ,Discontinuous Galerkin method ,symbols ,Applied mathematics ,MUSCL scheme ,Galerkin method ,Mathematics - Abstract
The central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408-463] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys., 160 (2000), pp. 241-282] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu, J. Comput. Phys., 209 (2005), pp. 82-104]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunov-type finite volume schemes. Overlapping cells lend themselves to the development of a central-type discontinuous Galerkin (DG) method, following the series of works by Cockburn and Shu [J. Comput. Phys., 141 (1998), pp. 199-224] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities beyond those of Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which removes spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine, and Flaherty [Appl. Numer. Math., 14 (1994), pp. 255-283] but is otherwise different in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multilayer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes, and retains the overall preprocessed order of accuracy while effectively removing spurious oscillations around shocks.
- Published
- 2007
34. High Resolution Schemes for a Hierarchical Size‐Structured Model
- Author
-
Jun Shen, Chi-Wang Shu, and Mengping Zhang
- Subjects
Numerical Analysis ,Applied Mathematics ,Numerical analysis ,Finite difference method ,Stability (learning theory) ,Upwind scheme ,Computational Mathematics ,Convergence (routing) ,Calculus ,Applied mathematics ,Flux limiter ,High-resolution scheme ,Mathematics ,Numerical stability - Abstract
In this paper we discuss two explicit finite difference schemes, namely a first order upwind scheme and a second order high resolution scheme, for solving a hierarchical size-structured population model with nonlinear growth, mortality, and reproduction rates. We prove stability and convergence for both schemes and provide numerical examples to demonstrate their capability in solving smooth and discontinuous solutions.
- Published
- 2007
35. Central WENO Schemes for Hamilton–Jacobi Equations on Triangular Meshes
- Author
-
Yong-Tao Zhang, Doron Levy, Chi-Wang Shu, and Suhas Nayak
- Subjects
Computational Mathematics ,Monotone polygon ,Finite volume method ,Applied Mathematics ,Scheme (mathematics) ,Numerical analysis ,Applied mathematics ,Polygon mesh ,Variety (universal algebra) ,Topology ,Hamilton–Jacobi equation ,Mathematics::Numerical Analysis ,Mathematics - Abstract
We derive Godunov-type semidiscrete central schemes for Hamilton-Jacobi equations on triangular meshes. High-order schemes are then obtained by combining our new numerical fluxes with high-order WENO reconstructions on triangular meshes. The numerical fluxes are shown to be monotone in certain cases. The accuracy and high-resolution properties of our scheme are demonstrated in a variety of numerical examples.
- Published
- 2006
36. Error Estimates to Smooth Solutions of Runge–Kutta Discontinuous Galerkin Method for Symmetrizable Systems of Conservation Laws
- Author
-
Chi-Wang Shu and Qiang Zhang
- Subjects
Numerical Analysis ,Discretization ,Applied Mathematics ,Courant–Friedrichs–Lewy condition ,Mathematical analysis ,Computer Science::Numerical Analysis ,Finite element method ,Mathematics::Numerical Analysis ,Piecewise linear function ,Computational Mathematics ,Runge–Kutta methods ,Discontinuous Galerkin method ,Piecewise ,Galerkin method ,Mathematics - Abstract
In this paper we study the error estimates to sufficiently smooth solutions of symmetrizable systems of conservation laws for the Runge–Kutta discontinuous Galerkin (RKDG) method. Time discretization is the second‐order explicit TVD (total variation diminishing) Runge–Kutta method, and the $\mathbb{P}^k$ (piecewise polynomial) finite element is used. When $k=1$ (piecewise linear finite element), the error estimate is obtained under the usual CFL condition $\dt\leq \beta h$ for nonlinear systems in one dimension and for linear systems in multiple space dimensions. Here, h is the maximum element length, τ is the time step, and β is a positive constant independent of h and τ. Error estimates for $\mathbb{P}^k$ finite elements with $k>1$ are obtained under a more restrictive CFL condition.
- Published
- 2006
37. The Heterogeneous Multiscale Method Based on the Discontinuous Galerkin Method for Hyperbolic and Parabolic Problems
- Author
-
Shanqin Chen, Chi-Wang Shu, and Weinan E
- Subjects
Nonlinear system ,Discontinuous Galerkin method ,Ecological Modeling ,Modeling and Simulation ,Mathematical analysis ,Scalar (mathematics) ,General Physics and Astronomy ,General Chemistry ,Hidden Markov model ,Homogenization (chemistry) ,Linear equation ,Computer Science Applications ,Mathematics - Abstract
In this paper we develop a discontinuous Galerkin (DG) method, within the framework of the heterogeneous multiscale method (HMM), for solving hyperbolic and parabolic multiscale problems. Hyperbolic scalar equations and systems, as well as parabolic scalar problems, are considered. Error estimates are given for the linear equations, and numerical results are provided for the linear and nonlinear problems to demonstrate the capability of the method.
- Published
- 2005
38. Approximation of Hyperbolic Models for Chemosensitive Movement
- Author
-
Chi-Wang Shu and Francis Filbet
- Subjects
Computational Mathematics ,Steady state (electronics) ,Finite volume method ,Computer simulation ,Property (programming) ,Applied Mathematics ,Numerical analysis ,Scheme (mathematics) ,Mathematical analysis ,Finite difference ,Finite difference method ,Mathematics - Abstract
Numerical methods with different orders of accuracy are proposed to approximate hyperbolic models for chemosensitive movements. On the one hand, first- and second-order well-balanced finite volume schemes are presented. This approach provides exact conservation of the steady state solutions. On the other hand, a high-order finite difference weighted essentially nonoscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme to exactly preserve steady states and to retain high-order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments.
- Published
- 2005
39. A Comparison of Troubled-Cell Indicators for Runge--Kutta Discontinuous Galerkin Methods Using Weighted Essentially Nonoscillatory Limiters
- Author
-
Jianxian Qiu and Chi-Wang Shu
- Subjects
Computational Mathematics ,Runge–Kutta methods ,Finite volume method ,Hermite polynomials ,Discontinuous Galerkin method ,Applied Mathematics ,Numerical analysis ,Bounded function ,Mathematical analysis ,Limiter ,Galerkin method ,Mathematics - Abstract
In [SIAM J. Sci. Comput., 26 (2005), pp. 907--929], we initiated the study of using WENO (weighted essentially nonoscillatory) methodology as limiters for the RKDG (Runge--Kutta discontinuous Galerkin) methods. The idea is to first identify "troubled cells," namely, those cells where limiting might be needed, then to abandon all moments in those cells except the cell averages and reconstruct those moments from the information of neighboring cells using a WENO methodology. This technique works quite well in our one- and two-dimensional test problems [SIAM J. Sci. Comput., 26 (2005), pp. 907--929] and in the follow-up work where more compact Hermite WENO methodology is used in the troubled cells. In these works we used the classical minmod-type TVB (total variation bounded) limiters to identify the troubled cells; that is, whenever the minmod limiter attempts to change the slope, the cell is declared to be a troubled cell. This troubled-cell indicator has a TVB parameter $M$ to tune and may identify more cells than necessary as troubled cells when $M$ is not chosen adequately, making the method costlier than necessary. In this paper we systematically investigate and compare a few different limiter strategies as troubled-cell indicators with an objective of obtaining the most efficient and reliable troubled-cell indicators to save computational cost.
- Published
- 2005
40. Extension of a Post Processing Technique for the Discontinuous Galerkin Method for Hyperbolic Equations with Application to an Aeroacoustic Problem
- Author
-
Chi-Wang Shu, Harold L. Atkins, and Jennifer K. Ryan
- Subjects
Partial differential equation ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Superconvergence ,Finite element method ,Mathematics::Numerical Analysis ,Euler equations ,Computational Mathematics ,symbols.namesake ,Tensor product ,Discontinuous Galerkin method ,symbols ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper we further explore a local postprocessing technique, originally developed by Bramble and Schatz [Math. Comp., 31 (1977), pp. 94--111] using continuous finite element methods for elliptic problems and later by Cockburn et al. [Math. Comp., 72 (2003), pp. 577--606] using discontinuous Galerkin methods for hyperbolic equations. We investigate the technique in the context of superconvergence of the derivatives of the numerical solution, two space dimensions for both tensor product local basis and the usual kth degree polynomials basis, multidomain problems with different mesh sizes, variable coefficient linear problems including those with discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. We demonstrate through extensive numerical examples that the technique is very effective in all these situations in enhancing the accuracy of the discontinuous Galerkin solutions.
- Published
- 2005
41. Runge--Kutta Discontinuous Galerkin Method Using WENO Limiters
- Author
-
Jianxian Qiu. and Chi-Wang Shu
- Subjects
Physics::Computational Physics ,Conservation law ,Finite volume method ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Finite difference method ,Computer Science::Numerical Analysis ,Finite element method ,Mathematics::Numerical Analysis ,Computational Mathematics ,Runge–Kutta methods ,Discontinuous Galerkin method ,Hyperbolic partial differential equation ,Mathematics - Abstract
The Runge--Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws. It uses ideas from high resolution finite volume schemes, such ...
- Published
- 2005
42. The Dynamics of a Plane Diode
- Author
-
Yan Guo, Chi-Wang Shu, and Tie Zhou
- Subjects
Computational Mathematics ,Partial differential equation ,Plane (geometry) ,Applied Mathematics ,Bounded function ,Mathematical analysis ,Boundary problem ,Boundary (topology) ,Uniqueness ,Inflow ,Boundary value problem ,Analysis ,Mathematics - Abstract
The dynamics of a plane diode is described by the Vlasov--Poisson system over an interval with inflow boundary conditions at two ends. In this article, the uniqueness and regularity of such dynamics are investigated. It is shown that a rather general initial and boundary datum leads to a unique solution with bounded variations (BV). Moreover, such a solution becomes discontinuous if the external voltage is large enough, while it can remain C1 if the external voltage is sufficiently small or absent.
- Published
- 2004
43. Error Estimates to Smooth Solutions of Runge--Kutta Discontinuous Galerkin Methods for Scalar Conservation Laws
- Author
-
Qiang Zhang and Chi-Wang Shu
- Subjects
Numerical Analysis ,Conservation law ,Discretization ,Applied Mathematics ,Courant–Friedrichs–Lewy condition ,Mathematical analysis ,Scalar (mathematics) ,Mathematics::Numerical Analysis ,Piecewise linear function ,Computational Mathematics ,Runge–Kutta methods ,Monotone polygon ,Discontinuous Galerkin method ,Mathematics - Abstract
In this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for Runge--Kutta discontinuous Galerkin (RKDG) methods, where the time discretization is the second order explicit total variation diminishing (TVD) Runge--Kutta method. Error estimates for the $\mathbb{P}^1$ (piecewise linear) elements are obtained under the usual CFL condition $\tau\leq \gamma h$ for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and $\tau$ are the maximum element lengths and time steps, respectively, and the positive constant $\gamma$ is independent of $h$ and $\tau$. However, error estimates for higher order $\mathbb{P}^k(k\geq 2)$ elements need a more restrictive time step $\tau\leq \gamma h^{4/3}$. We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition $\tau\leq\gamma h$ for the $\mathbb{P}^k$ elements of degree $k\geq 2$. Error estimates of $O(h^{k+1/2}+\tau^2)$ are obtained for general monotone numerical fluxes, and optimal error estimates of $O(h^{k+1}+\tau^2)$ are obtained for upwind numerical fluxes.
- Published
- 2004
44. High-Order WENO Schemes for Hamilton--Jacobi Equations on Triangular Meshes
- Author
-
Chi-Wang Shu and Yong-Tao Zhang
- Subjects
Computational Mathematics ,Nonlinear system ,Partial differential equation ,Monotone polygon ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Viscosity solution ,Hamilton–Jacobi equation ,Stencil ,Mathematics::Numerical Analysis ,Numerical stability ,Mathematics - Abstract
In this paper we construct high-order weighted essentially nonoscillatory (WENO) schemes for solving the nonlinear Hamilton--Jacobi equations on two-dimensional unstructured meshes. The main ideas are nodal based approximations, the usage of monotone Hamiltonians as building blocks on unstructured meshes, nonlinear weights using smooth indicators of second and higher derivatives, and a strategy to choose diversified smaller stencils to make up the bigger stencil in the WENO procedure. Both third-order and fourth-order WENO schemes using combinations of second-order approximations with nonlinear weights are constructed. Extensive numerical experiments are performed to demonstrate the stability and accuracy of the methods. High-order accuracy in smooth regions, good resolution of derivative singularities, and convergence to viscosity solutions are observed.
- Published
- 2003
45. Finite Difference WENO Schemes with Lax--Wendroff-Type Time Discretizations
- Author
-
Jianxian Qiu and Chi-Wang Shu
- Subjects
Partial differential equation ,Discretization ,Lax–Wendroff method ,Applied Mathematics ,Mathematical analysis ,Finite difference ,Finite difference method ,Mathematics::Numerical Analysis ,Euler equations ,Computational Mathematics ,symbols.namesake ,symbols ,Euler's formula ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper we develop a Lax--Wendroff time discretization procedure for high order finite difference weighted essentially nonoscillatory schemes to solve hyperbolic conservation laws. This is an alternative method for time discretization to the popular TVD Runge--Kutta time discretizations. We explore the possibility in avoiding the local characteristic decompositions or even the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining nonoscillatory properties for problems with strong shocks. As a result, the Lax--Wendroff time discretization procedure is more cost effective than the Runge--Kutta time discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics.
- Published
- 2003
46. A Local Discontinuous Galerkin Method for KdV Type Equations
- Author
-
Jue Yan and Chi-Wang Shu
- Subjects
010101 applied mathematics ,Numerical Analysis ,Computational Mathematics ,Applied Mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences - Published
- 2002
47. Strong Stability-Preserving High-Order Time Discretization Methods
- Author
-
Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor
- Subjects
Discretization ,Differential equation ,Applied Mathematics ,Courant–Friedrichs–Lewy condition ,Method of lines ,Mathematical analysis ,Numerical methods for ordinary differential equations ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,Computational Mathematics ,Runge–Kutta methods ,Total variation diminishing ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.
- Published
- 2001
48. A Discontinuous Galerkin Finite Element Method for Hamilton--Jacobi Equations
- Author
-
Chi-Wang Shu and Changqing Hu
- Subjects
Computational Mathematics ,Discontinuous Galerkin method ,Compact stencil ,Applied Mathematics ,Spectral element method ,Mathematical analysis ,Mixed finite element method ,Galerkin method ,Discontinuous Deformation Analysis ,Finite element method ,Mathematics::Numerical Analysis ,Mathematics ,Extended finite element method - Abstract
In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton--Jacobi equations. This method is based on the Runge--Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high-order accuracy with a local, compact stencil, and is suited for efficient parallel implementation. One- and two-dimensional numerical examples are given to illustrate the capability of the method. At least kth order of accuracy is observed for smooth problems when kth degree polynomials are used, and derivative singularities are resolved well without oscillations, even without limiters.
- Published
- 1999
49. The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
- Author
-
Chi-Wang Shu and Bernardo Cockburn
- Subjects
Numerical Analysis ,Partial differential equation ,Applied Mathematics ,Mathematical analysis ,Triangulation (social science) ,Mathematics::Numerical Analysis ,Computational Mathematics ,Nonlinear system ,Runge–Kutta methods ,Rate of convergence ,Discontinuous Galerkin method ,Convection–diffusion equation ,Galerkin method ,Mathematics - Abstract
In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.
- Published
- 1998
50. On the Gibbs Phenomenon and Its Resolution
- Author
-
David Gottlieb and Chi-Wang Shu
- Subjects
Sigma approximation ,Series (mathematics) ,Applied Mathematics ,Mathematical analysis ,Order of accuracy ,Function (mathematics) ,Theoretical Computer Science ,Gibbs phenomenon ,Computational Mathematics ,symbols.namesake ,symbols ,Runge's phenomenon ,Statistical physics ,Series expansion ,Fourier series ,Mathematics - Abstract
The nonuniform convergence of the Fourier series for discontinuous functions, and in particular the oscillatory behavior of the finite sum, was already analyzed by Wilbraham in 1848. This was later named the Gibbs phenomenon. This article is a review of the Gibbs phenomenon from a different perspective. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a function from its expansion coefficients. Alternatively it can be viewed as the possibility of the recovery of local information from global information. The main theme here is not the structure of the Gibbs oscillations but the understanding and resolution of the phenomenon in a general setting. The purpose of this article is to review the Gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case. This is done by using the finite expansion series to construct a different, rapidly convergent, approximation.
- Published
- 1997
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