44 results on '"Chi-Wang Shu"'
Search Results
2. Implicit–Explicit Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Convection–Diffusion Problems
- Author
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Haijin Wang, Chi-Wang Shu, and Qiang Zhang
- Subjects
Numerical Analysis ,Implicit explicit ,Applied Mathematics ,Diagonal ,General Engineering ,Numerical flux ,Stability (probability) ,Projection (linear algebra) ,Theoretical Computer Science ,Computational Mathematics ,Computational Theory and Mathematics ,Discontinuous Galerkin method ,Jump ,Applied mathematics ,Convection–diffusion equation ,Software ,Mathematics - Abstract
Local discontinuous Galerkin methods with generalized alternating numerical fluxes coupled with implicit–explicit time marching for solving convection–diffusion problems is analyzed in this paper, where the explicit part is treated by a strong-stability-preserving Runge–Kutta scheme, and the implicit part is treated by an L-stable diagonally implicit Runge–Kutta method. Based on the generalized alternating numerical flux, we establish the important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient, which plays a key role in obtaining the unconditional stability of the proposed schemes. Also by the aid of the generalized Gauss–Radau projection, optimal error estimates can be shown. Numerical experiments are given to verify the stability and accuracy of the proposed schemes with different numerical fluxes.
- Published
- 2019
3. Certified Offline-Free Reduced Basis (COFRB) Methods for Stochastic Differential Equations Driven by Arbitrary Types of Noise
- Author
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Chi-Wang Shu, Tianheng Chen, Yong Liu, and Yanlai Chen
- Subjects
Numerical Analysis ,Basis (linear algebra) ,Differential equation ,Applied Mathematics ,Gaussian ,General Engineering ,Ode ,01 natural sciences ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Noise ,Stochastic differential equation ,symbols.namesake ,Computational Theory and Mathematics ,Robustness (computer science) ,Component (UML) ,symbols ,0101 mathematics ,Algorithm ,Software ,Mathematics - Abstract
In this paper, we propose, analyze, and implement a new reduced basis method (RBM) tailored for the linear (ordinary and partial) differential equations driven by arbitrary (i.e. not necessarily Gaussian) types of noise. There are four main ingredients of our algorithm. First, we propose a new space-time-like treatment of time in the numerical schemes for ODEs and PDEs. The second ingredient is an accurate yet efficient compression technique for the spatial component of the space-time snapshots that the RBM is adopting as bases. The third ingredient is a non-conventional “parameterization” of a non-parametric problem. The last is a RBM that is free of any dedicated offline procedure yet is still efficient online. The numerical experiments verify the effectiveness and robustness of our algorithms for both types of differential equations.
- Published
- 2019
4. A Third-Order Unconditionally Positivity-Preserving Scheme for Production–Destruction Equations with Applications to Non-equilibrium Flows
- Author
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Weifeng Zhao, Chi-Wang Shu, and Juntao Huang
- Subjects
Numerical Analysis ,Work (thermodynamics) ,Applied Mathematics ,General Engineering ,Finite difference ,Ode ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Third order ,Computational Theory and Mathematics ,Scheme (mathematics) ,Applied mathematics ,Production (computer science) ,0101 mathematics ,Software ,Mathematics - Abstract
In this paper, we extend our previous work in Huang and Shu (J Sci Comput, 2018. https://doi.org/10.1007/s10915-018-0852-1 ) and develop a third-order unconditionally positivity-preserving modified Patankar Runge–Kutta method for production–destruction equations. The necessary and sufficient conditions for the method to be of third-order accuracy are derived. With the same approach as Huang and Shu (2018), this time integration method is then generalized to solve a class of ODEs arising from semi-discrete schemes for PDEs and coupled with the positivity-preserving finite difference weighted essentially non-oscillatory schemes for non-equilibrium flows. Numerical experiments are provided to demonstrate the performance of our proposed scheme.
- Published
- 2018
5. A Foreword to the Special Issue in Honor of Professor Bernardo Cockburn on His 60th Birthday: A Life Time of Discontinuous Schemings
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Chi-Wang Shu, Bo Dong, and Yanlai Chen
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Numerical Analysis ,Professional career ,Applied Mathematics ,media_common.quotation_subject ,General Engineering ,Life time ,Theoretical Computer Science ,Computational Mathematics ,Presentation ,Computational Theory and Mathematics ,Discontinuous Galerkin method ,Honor ,Software ,Classics ,Mathematics ,media_common ,Theme (narrative) - Abstract
We present this special issue of the Journal of Scientific Computing to celebrate Bernardo Cockburn’s sixtieth birthday. The theme of this issue is discontinuous Galerkin methods, a hallmark of Bernardo’s distinguished professional career. This foreword provides an informal but rigorous account of what enabled Bernardo’s achievements, based on the concluding presentation he gave at the the IMA workshop “Recent Advances and Challenges in Discontinuous Galerkin Methods and Related Approaches” on July 1, 2017 which was widely deemed as the best lecture of his career so far.
- Published
- 2018
6. Positivity-Preserving Time Discretizations for Production–Destruction Equations with Applications to Non-equilibrium Flows
- Author
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Chi-Wang Shu and Juntao Huang
- Subjects
Numerical Analysis ,Applied Mathematics ,General Engineering ,Finite difference ,010103 numerical & computational mathematics ,Solver ,01 natural sciences ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Ordinary differential equation ,Applied mathematics ,Production (computer science) ,Numerical tests ,0101 mathematics ,Software ,Mathematics - Abstract
In this paper, we construct a family of modified Patankar Runge–Kutta methods, which is conservative and unconditionally positivity-preserving, for production–destruction equations, and derive necessary and sufficient conditions to obtain second-order accuracy. This ordinary differential equation solver is then extended to solve a class of semi-discrete schemes for PDEs. Combining this time integration method with the positivity-preserving finite difference weighted essentially non-oscillatory (WENO) schemes, we successfully obtain a positivity-preserving WENO scheme for non-equilibrium flows. Various numerical tests are reported to demonstrate the effectiveness of the methods.
- Published
- 2018
7. Superconvergence Analysis of the Runge–Kutta Discontinuous Galerkin Methods for a Linear Hyperbolic Equation
- Author
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Yuan Xu, Xiong Meng, Qiang Zhang, and Chi-Wang Shu
- Subjects
Physics::Computational Physics ,Numerical Analysis ,Applied Mathematics ,General Engineering ,Numerical flux ,Superconvergence ,Computer Science::Numerical Analysis ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,Computational Mathematics ,Runge–Kutta methods ,Computational Theory and Mathematics ,Discontinuous Galerkin method ,Norm (mathematics) ,Applied mathematics ,Hyperbolic partial differential equation ,Software ,Mathematics - Abstract
In this paper, we shall establish the superconvergence property of the Runge–Kutta discontinuous Galerkin (RKDG) method for solving a linear constant-coefficient hyperbolic equation. The RKDG method is made of the discontinuous Galerkin (DG) scheme with upwind-biased numerical fluxes coupled with the explicit Runge–Kutta algorithm of arbitrary orders and stages. Superconvergence results for the numerical flux, cell averages as well as the solution and derivative at some special points are shown, which are based on a systematical study of the $$\hbox {L}^2$$ -norm stability for the RKDG method and the incomplete correction techniques for the well-defined reference functions at each time stage. The result demonstrates that the superconvergence property of the semi-discrete DG method is preserved, and the optimal order in time is provided under the smoothness assumption that is independent of the number of stages. As a byproduct of the above superconvergence study, the expected order of the post-processed solution is obtained when a special initial solution is used. Some numerical experiments are also given.
- Published
- 2020
8. An Ultra-Weak Discontinuous Galerkin Method with Implicit–Explicit Time-Marching for Generalized Stochastic KdV Equations
- Author
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Chi-Wang Shu, Yunzhang Li, and Shanjian Tang
- Subjects
Numerical Analysis ,Discretization ,Applied Mathematics ,Multiplicative function ,Monte Carlo method ,General Engineering ,01 natural sciences ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Computational Theory and Mathematics ,Discontinuous Galerkin method ,Ordinary differential equation ,Applied mathematics ,0101 mathematics ,Korteweg–de Vries equation ,Hyperbolic partial differential equation ,Software ,Mathematics - Abstract
In this paper, an ultra-weak discontinuous Galerkin (DG) method is developed to solve the generalized stochastic Korteweg–de Vries (KdV) equations driven by a multiplicative temporal noise. This method is an extension of the DG method for purely hyperbolic equations and shares the advantage and flexibility of the DG method. Stability is analyzed for the general nonlinear equations. The ultra-weak DG method is shown to admit the optimal error of order $$k+1$$ in the sense of the spatial $$L^2(0,2\pi )$$-norm for semi-linear stochastic equations, when polynomials of degree $$k\ge 2$$ are used in the spatial discretization. A second order implicit–explicit derivative-free time discretization scheme is also proposed for the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples using Monte Carlo simulation are provided to illustrate the theoretical results.
- Published
- 2020
9. Conservative High Order Positivity-Preserving Discontinuous Galerkin Methods for Linear Hyperbolic and Radiative Transfer Equations
- Author
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Chi-Wang Shu, Dan Ling, and Juan Cheng
- Subjects
Numerical Analysis ,Applied Mathematics ,General Engineering ,010103 numerical & computational mathematics ,Solver ,Space (mathematics) ,01 natural sciences ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Discontinuous Galerkin method ,Convergence (routing) ,Radiative transfer ,Piecewise ,Applied mathematics ,0101 mathematics ,Scaling ,Hyperbolic partial differential equation ,Software ,Mathematics - Abstract
We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in Yuan et al. (SIAM J Sci Comput 38:A2987–A3019, 2016). The DG methods in Yuan et al. (2016) can maintain positivity and high order accuracy, but they rely both on the scaling limiter in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax–Wendroff type theorem is proved in Yuan et al. (2016), guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $$P^k$$ of piecewise polynomials of degree at most k. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu (2010) can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $$P^k$$ or $$Q^k$$ spaces (piecewise kth degree polynomials or piecewise tensor-product kth degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu (2010). Computational results are provided to demonstrate the good performance of our DG schemes.
- Published
- 2018
10. Unconditional Energy Stability Analysis of a Second Order Implicit–Explicit Local Discontinuous Galerkin Method for the Cahn–Hilliard Equation
- Author
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Huailing Song and Chi-Wang Shu
- Subjects
Numerical Analysis ,Discretization ,Continuous modelling ,Implicit explicit ,Applied Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Engineering ,010103 numerical & computational mathematics ,Time step ,Nonlinear Sciences::Cellular Automata and Lattice Gases ,01 natural sciences ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Energy stability ,Discontinuous Galerkin method ,Order (group theory) ,0101 mathematics ,Cahn–Hilliard equation ,Nonlinear Sciences::Pattern Formation and Solitons ,Software ,Mathematics - Abstract
In this article, we present a second-order in time implicit–explicit (IMEX) local discontinuous Galerkin (LDG) method for computing the Cahn–Hilliard equation, which describes the phase separation phenomenon. It is well-known that the Cahn–Hilliard equation has a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. The discretized Cahn–Hilliard system modeled by the IMEX LDG method can inherit the nonlinear stability of the continuous model. We apply a stabilization technique and prove unconditional energy stability of our scheme. Numerical experiments are performed to validate the analysis. Computational efficiency can be significantly enhanced by using this IMEX LDG method with a large time step.
- Published
- 2017
11. A Simple Bound-Preserving Sweeping Technique for Conservative Numerical Approximations
- Author
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Yuan Liu, Chi-Wang Shu, and Yingda Cheng
- Subjects
Numerical Analysis ,Conservation law ,Mathematical optimization ,Finite volume method ,Applied Mathematics ,Scalar (mathematics) ,General Engineering ,Finite difference ,010103 numerical & computational mathematics ,01 natural sciences ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Maximum principle ,Computational Theory and Mathematics ,Compressibility ,Applied mathematics ,0101 mathematics ,High order ,Spectral method ,Software ,Mathematics - Abstract
In this paper, we propose a simple bound-preserving sweeping procedure for conservative numerical approximations. Conservative schemes are of importance in many applications, yet for high order methods, the numerical solutions do not necessarily satisfy maximum principle. This paper constructs a simple sweeping algorithm to enforce the bound of the solutions. It has a very general framework acting as a postprocessing step accommodating many point-based or cell average-based discretizations. The method is proven to preserve the bounds of the numerical solution while conserving a prescribed quantity designated as a weighted average of values from all points. The technique is demonstrated to work well with a spectral method, high order finite difference and finite volume methods for scalar conservation laws and incompressible flows. Extensive numerical tests in 1D and 2D are provided to verify the accuracy of the sweeping procedure.
- Published
- 2017
12. Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model
- Author
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Yang Yang, Chi-Wang Shu, and Xingjie Helen Li
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Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Chemotaxis ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Rate of convergence ,Discontinuous Galerkin method ,Scheme (mathematics) ,Neumann boundary condition ,Limiter ,0101 mathematics ,Software ,Mathematics - Abstract
In this paper, we apply the local discontinuous Galerkin (LDG) method to 2D Keller---Segel (KS) chemotaxis model. We improve the results upon (Epshteyn and Kurganov in SIAM J Numer Anal, 47:368---408, 2008) and give optimal rate of convergence under special finite element spaces before the blow-up occurs (the exact solutions are smooth). Moreover, to construct physically relevant numerical approximations, we consider $$P^1$$P1 LDG scheme and develop a positivity-preserving limiter to the scheme, extending the idea in Zhang and Shu (J Comput Phys, 229:8918---8934, 2010). With this limiter, we can prove the $$L^1$$L1-stability of the numerical scheme. Numerical experiments are performed to demonstrate the good performance of the positivity-preserving LDG scheme. Moreover, it is known that the chemotaxis model will yield blow-up solutions under certain initial conditions. We numerically demonstrate how to find the approximate blow-up time by using the $$L^2$$L2-norm of the $$L^1$$L1-stable numerical solution.
- Published
- 2017
13. Stability Analysis of the Inverse Lax–Wendroff Boundary Treatment for High Order Central Difference Schemes for Diffusion Equations
- Author
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Chi-Wang Shu, Tingting Li, and Mengping Zhang
- Subjects
Numerical Analysis ,Lax–Wendroff theorem ,Diffusion equation ,Lax–Wendroff method ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Finite difference ,Inverse ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Boundary value problem ,0101 mathematics ,Software ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, high order central finite difference schemes in a finite interval are analyzed for the diffusion equation. Boundary conditions of the initial-boundary value problem are treated by the simplified inverse Lax–Wendroff procedure. For the fully discrete case, a third order explicit Runge–Kutta method is used as an example for the analysis. Stability is analyzed by both the Gustafsson, Kreiss and Sundstrom theory and the eigenvalue visualization method on both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate and validate the analysis results.
- Published
- 2016
14. A New Multiscale Discontinuous Galerkin Method for the One-Dimensional Stationary Schrödinger Equation
- Author
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Bo Dong, Wei Wang, and Chi-Wang Shu
- Subjects
Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Order of accuracy ,010103 numerical & computational mathematics ,01 natural sciences ,WKB approximation ,Theoretical Computer Science ,Schrödinger equation ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Rate of convergence ,Discontinuous Galerkin method ,Norm (mathematics) ,symbols ,Polygon mesh ,Boundary value problem ,0101 mathematics ,Software ,Mathematics - Abstract
In this paper, we develop and analyze a new multiscale discontinuous Galerkin (DG) method for one-dimensional stationary Schrodinger equations with open boundary conditions which have highly oscillating solutions. Our method uses a smaller finite element space than the WKB local DG method proposed in Wang and Shu (J Comput Phys 218:295---323, 2006) while achieving the same order of accuracy with no resonance errors. We prove that the DG approximation converges optimally with respect to the mesh size $$h$$h in $$L^2$$L2 norm without the typical constraint that $$h$$h has to be smaller than the wave length. Numerical experiments were carried out to verify the second order optimal convergence rate of the method and to demonstrate its ability to capture oscillating solutions on coarse meshes in the applications to Schrodinger equations.
- Published
- 2015
15. Recovering Exponential Accuracy in Fourier Spectral Methods Involving Piecewise Smooth Functions with Unbounded Derivative Singularities
- Author
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Chi-Wang Shu and Zheng Chen
- Subjects
Pointwise ,Numerical Analysis ,Partial differential equation ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Exponential polynomial ,Theoretical Computer Science ,Exponential function ,Computational Mathematics ,symbols.namesake ,Fourier transform ,Computational Theory and Mathematics ,Piecewise ,symbols ,Spectral method ,Fourier series ,Software ,Mathematics - Abstract
Fourier spectral methods achieve exponential accuracy both on the approximation level and for solving partial differential equations, if the solution is analytic. If the solution is discontinuous but piecewise analytic up to the discontinuities, Fourier spectral methods produce poor pointwise accuracy, but still maintain exponential accuracy after post-processing (Gottlieb and Shu in SIAM Rev 30:644---668, 1997) . In Chen and Shu (J Comput Appl Math 265:83---95, 2014), an extended technique is provided to recover exponential accuracy for functions which have end-point singularities, from the knowledge of point values on standard collocation points. In this paper, we develop a technique to recover exponential accuracy from the first $$N$$N Fourier coefficients of functions which are analytic in the open interval but have unbounded derivative singularities at end points. With this post-processing method, we are able to obtain exponential accuracy of spectral methods applied to linear transport equations involving such functions.
- Published
- 2015
16. Multi-scale Discontinuous Galerkin Method for Solving Elliptic Problems with Curvilinear Unidirectional Rough Coefficients
- Author
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Chi-Wang Shu, Johnny Guzmán, Yifan Zhang, and Wei Wang
- Subjects
Numerical Analysis ,Curvilinear coordinates ,Scale (ratio) ,Basis (linear algebra) ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Basis function ,Differential operator ,Space (mathematics) ,Theoretical Computer Science ,Computational Mathematics ,Computational Theory and Mathematics ,Discontinuous Galerkin method ,Algorithmic efficiency ,Software ,Mathematics - Abstract
In this paper, we propose a multi-scale discontinuous Galerkin (DG) method for second-order elliptic problems with curvilinear unidirectional rough coefficients by choosing a special non-polynomial approximation space. The key ingredient of the method lies in the incorporation of the local oscillatory features of the differential operators into the approximation space so as to capture the multi-scale solutions without having to resolve the finest scales. The unidirectional feature of the rough coefficients allows us to construct the basis functions of the DG non-polynomial approximation space explicitly, thereby greatly increasing the algorithm efficiency. Detailed error estimates for two-dimensional second-order DG methods are derived, and a general guidance on how to construct such non-polynomial basis is discussed. Numerical examples are also presented to validate and demonstrate the effectiveness of the algorithm.
- Published
- 2014
17. High Order Well-Balanced WENO Scheme for the Gas Dynamics Equations Under Gravitational Fields
- Author
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Chi-Wang Shu and Yulong Xing
- Subjects
Physics ,Numerical Analysis ,Applied Mathematics ,General Engineering ,Finite difference ,Finite difference method ,Mechanics ,Theoretical Computer Science ,Euler equations ,law.invention ,Gravitation ,Computational Mathematics ,symbols.namesake ,Classical mechanics ,Computational Theory and Mathematics ,Gravitational field ,law ,Convergence (routing) ,symbols ,Initial value problem ,Hydrostatic equilibrium ,Software - Abstract
The gas dynamics equations, coupled with a static gravitational field, admit the hydrostatic balance where the flux produced by the pressure is exactly canceled by the gravitational source term. Many astrophysical problems involve the hydrodynamical evolution in a gravitational field, therefore it is essential to correctly capture the effect of gravitational force in the simulations. Improper treatment of the gravitational force can lead to a solution which either oscillates around the equilibrium, or deviates from the equilibrium after a long time run. In this paper we design high order well-balanced finite difference WENO schemes to this system, which can preserve the hydrostatic balance state exactly and at the same time can maintain genuine high order accuracy. Numerical tests are performed to verify high order accuracy, well-balanced property, and good resolution for smooth and discontinuous solutions. The main purpose of the well-balanced schemes designed in this paper is to well resolve small perturbations of the hydrostatic balance state on coarse meshes. The more difficult issue of convergence towards such hydrostatic balance state from an arbitrary initial condition is not addressed in this paper.
- Published
- 2012
18. WENO Scheme with Subcell Resolution for Computing Nonconservative Euler Equations with Applications to One-Dimensional Compressible Two-Medium Flows
- Author
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Tao Xiong, Chi-Wang Shu, and Mengping Zhang
- Subjects
Numerical Analysis ,Finite volume method ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Classification of discontinuities ,Compressible flow ,Riemann solver ,Theoretical Computer Science ,Euler equations ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Scheme (mathematics) ,Compressibility ,symbols ,Software ,Mathematics ,Resolution (algebra) - Abstract
High order path-conservative schemes have been developed for solving nonconservative hyperbolic systems in (Pares, SIAM J. Numer. Anal. 44:300---321, 2006; Castro et al., Math. Comput. 75:1103---1134, 2006; J. Sci. Comput. 39:67---114, 2009). Recently, it has been observed in (Abgrall and Karni, J. Comput. Phys. 229:2759---2763, 2010) that this approach may have some computational issues and shortcomings. In this paper, a modification to the high order path-conservative scheme in (Castro et al., Math. Comput. 75:1103---1134, 2006) is proposed to improve its computational performance and to overcome some of the shortcomings. This modification is based on the high order finite volume WENO scheme with subcell resolution and it uses an exact Riemann solver to catch the right paths at the discontinuities. An application to one-dimensional compressible two-medium flows of nonconservative or primitive Euler equations is carried out to show the effectiveness of this new approach.
- Published
- 2012
19. Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes
- Author
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Xiangxiong Zhang, Chi-Wang Shu, and Yinhua Xia
- Subjects
Numerical Analysis ,Conservation law ,Finite volume method ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,Euler equations ,Computational Mathematics ,symbols.namesake ,Maximum principle ,Computational Theory and Mathematics ,Incompressible flow ,Discontinuous Galerkin method ,symbols ,Gaussian quadrature ,Convection–diffusion equation ,Software ,Mathematics - Abstract
In Zhang and Shu (J. Comput. Phys. 229:3091---3120, 2010), two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in Zhang and Shu (J. Comput. Phys. 229:8918---8934, 2010). The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.
- Published
- 2011
20. Improvement of Convergence to Steady State Solutions of Euler Equations with the WENO Schemes
- Author
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Chi-Wang Shu, Shuhai Zhang, and Shufen Jiang
- Subjects
Numerical Analysis ,Truncation error ,Steady state (electronics) ,Smoothness (probability theory) ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Finite difference ,Theoretical Computer Science ,Euler equations ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Convergence (routing) ,Jacobian matrix and determinant ,symbols ,Software ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The convergence to steady state solutions of the Euler equations for high order weighted essentially non-oscillatory (WENO) finite difference schemes with the Lax-Friedrichs flux splitting (Jiang and Shu, in J. Comput. Phys. 126:202---228, 1996) is investigated. Numerical evidence in Zhang and Shu (J. Sci. Comput. 31:273---305, 2007) indicates that there exist slight post-shock oscillations when we use high order WENO schemes to solve problems containing shock waves. Even though these oscillations are small in their magnitude and do not affect the "essentially non-oscillatory" property of the WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. Differently from the strategy adopted in Zhang and Shu (J. Sci. Comput. 31:273---305, 2007), in which a new smoothness indicator was introduced to facilitate convergence to steady states, in this paper we study the effect of the local characteristic decomposition on steady state convergence. Numerical tests indicate that the slight post-shock oscillation has a close relationship with the local characteristic decomposition process. When this process is based on an average Jacobian at the cell interface using the Roe average, as is the standard procedure for WENO schemes, such post-shock oscillation appears. If we instead use upwind-biased interpolation to approximate the physical variables including the velocity and enthalpy on the cell interface to compute the left and right eigenvectors of the Jacobian for the local characteristic decomposition, the slight post-shock oscillation can be removed or reduced significantly and the numerical residue settles down to lower values than other WENO schemes and can reach machine zero for many test cases. This new procedure is also effective for higher order WENO schemes and for WENO schemes with different smoothness indicators.
- Published
- 2010
21. On the Advantage of Well-Balanced Schemes for Moving-Water Equilibria of the Shallow Water Equations
- Author
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Yulong Xing, Chi-Wang Shu, and Sebastian Noelle
- Subjects
Numerical Analysis ,Mathematical optimization ,Applied Mathematics ,General Engineering ,010103 numerical & computational mathematics ,Condensed Matter::Mesoscopic Systems and Quantum Hall Effect ,01 natural sciences ,6. Clean water ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Applied mathematics ,0101 mathematics ,Spurious oscillations ,Shallow water equations ,Physics::Atmospheric and Oceanic Physics ,Software ,Mathematics - Abstract
This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-water well-balanced schemes for the shallow water equations. We concentrate on numerical examples with solutions near a moving-water equilibrium. For such examples, still-water well-balanced methods are not capable of capturing the small perturbations of the moving-water equilibrium and may generate significant spurious oscillations, unless an extremely refined mesh is used. On the other hand, moving-water well-balanced methods perform well in these tests. The numerical examples in this note clearly demonstrate the importance of utilizing moving-water well-balanced methods for solutions near a moving-water equilibrium.
- Published
- 2010
22. Fast Sweeping Fifth Order WENO Scheme for Static Hamilton-Jacobi Equations with Accurate Boundary Treatment
- Author
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Yong-Tao Zhang, Chi-Wang Shu, Tao Xiong, and Mengping Zhang
- Subjects
Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,General Engineering ,CPU time ,Hamilton–Jacobi equation ,Domain (mathematical analysis) ,Theoretical Computer Science ,law.invention ,Computational Mathematics ,Third order ,Computational Theory and Mathematics ,law ,Order (group theory) ,Polygon mesh ,Cartesian coordinate system ,Boundary value problem ,Software ,Mathematics - Abstract
A fifth order weighted essentially non-oscillatory (WENO) fast sweeping method is designed in this paper, extending the result of the third order WENO fast sweeping method in J. Sci. Comput. 29, 25---56 (2006) and utilizing the two approaches of accurate inflow boundary condition treatment in J. Comput. Math. 26, 1---11 (2008), which allows the usage of Cartesian meshes regardless of the domain boundary shape. The resulting method is tested on a variety of problems to demonstrate its good performance and CPU time efficiency when compared with lower order fast sweeping methods.
- Published
- 2010
23. Local Discontinuous Galerkin Method for Surface Diffusion and Willmore Flow of Graphs
- Author
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Yan Xu and Chi-Wang Shu
- Subjects
Surface diffusion ,Numerical Analysis ,Computer simulation ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Stability (probability) ,Finite element method ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,Computational Mathematics ,Willmore energy ,Computational Theory and Mathematics ,Flow (mathematics) ,Energy stability ,Discontinuous Galerkin method ,Software ,Mathematics - Abstract
In this paper, we develop a local discontinuous Galerkin (LDG) finite element method for surface diffusion and Willmore flow of graphs. We prove L 2 stability for the equation of surface diffusion of graphs and energy stability for the equation of Willmore flow of graphs. We provide numerical simulation results for different types of solutions of these two types of the equations to illustrate the accuracy and capability of the LDG method.
- Published
- 2008
24. High Order Strong Stability Preserving Time Discretizations
- Author
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Chi-Wang Shu, Sigal Gottlieb, and David I. Ketcheson
- Subjects
Numerical Analysis ,Partial differential equation ,Discretization ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Regular polygon ,Astrophysics::Cosmology and Extragalactic Astrophysics ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,Computational Mathematics ,Nonlinear system ,symbols.namesake ,Runge–Kutta methods ,Computational Theory and Mathematics ,Norm (mathematics) ,Euler's formula ,symbols ,High order ,Software ,Mathematics - Abstract
Strong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. SSP methods preserve the strong stability properties--in any norm, seminorm or convex functional--of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the connections between the timestep restrictions for strong stability preservation and contractivity. Numerical examples demonstrate that common linearly stable but not strong stability preserving time discretizations may lead to violation of important boundedness properties, whereas SSP methods guarantee the desired properties provided only that these properties are satisfied with forward Euler timestepping. We review optimal explicit and implicit SSP Runge---Kutta and multistep methods, for linear and nonlinear problems. We also discuss the SSP properties of spectral deferred correction methods.
- Published
- 2008
25. The WKB Local Discontinuous Galerkin Method for the Simulation of Schrödinger Equation in a Resonant Tunneling Diode
- Author
-
Chi-Wang Shu and Wei Wang
- Subjects
Numerical Analysis ,Discretization ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Finite element method ,WKB approximation ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,Schrödinger equation ,Computational Mathematics ,symbols.namesake ,Discontinuity (linguistics) ,Computational Theory and Mathematics ,Discontinuous Galerkin method ,symbols ,Reduction (mathematics) ,Galerkin method ,Software ,Mathematics - Abstract
In this paper, we develop a multiscale local discontinuous Galerkin (LDG) method to simulate the one-dimensional stationary Schrodinger-Poisson problem. The stationary Schrodinger equation is discretized by the WKB local discontinuous Galerkin (WKB-LDG) method, and the Poisson potential equation is discretized by the minimal dissipation LDG (MD-LDG) method. The WKB-LDG method we propose provides a significant reduction of both the computational cost and memory in solving the Schrodinger equation. Compared with traditional continuous finite element Galerkin methodology, the WKB-LDG method has the advantages of the DG methods including their flexibility in h-p adaptivity and allowance of complete discontinuity at element interfaces. Although not addressed in this paper, a major advantage of the WKB-LDG method is its feasibility for two-dimensional devices.
- Published
- 2008
26. A High Order WENO Scheme for a Hierarchical Size-Structured Population Model
- Author
-
Chi-Wang Shu, Jun Shen, and Mengping Zhang
- Subjects
Scheme (programming language) ,Numerical Analysis ,education.field_of_study ,Mathematical optimization ,Applied Mathematics ,Population ,General Engineering ,Grid ,Theoretical Computer Science ,Computational Mathematics ,Nonlinear system ,Monotone polygon ,Computational Theory and Mathematics ,Population model ,Bounded function ,Boundary value problem ,education ,computer ,Software ,Mathematics ,computer.programming_language - Abstract
In this paper we develop a high order explicit finite difference weighted essentially non-oscillatory (WENO) scheme for solving a hierarchical size-structured population model with nonlinear growth, mortality and reproduction rates. The main technical complication is the existence of global terms in the coefficient and boundary condition for this model. We carefully design approximations to these global terms and boundary conditions to ensure high order accuracy. Comparing with the first order monotone and second order total variation bounded schemes for the same model, the high order WENO scheme is more efficient and can produce accurate results with far fewer grid points. Numerical examples including one in computational biology for the evolution of the population of Gambussia affinis, are presented to illustrate the good performance of the high order WENO scheme.
- Published
- 2007
27. A New Smoothness Indicator for the WENO Schemes and Its Effect on the Convergence to Steady State Solutions
- Author
-
Chi-Wang Shu and Shuhai Zhang
- Subjects
Shock wave ,Numerical Analysis ,Truncation error ,Steady state (electronics) ,Smoothness (probability theory) ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Domain (mathematical analysis) ,Theoretical Computer Science ,Euler equations ,Shock (mechanics) ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Convergence (routing) ,symbols ,Software ,Mathematics - Abstract
The convergence to steady state solutions of the Euler equations for the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme with the Lax---Friedrichs flux splitting [7, (1996) J. Comput. Phys. 126, 202---228.] is studied through systematic numerical tests. Numerical evidence indicates that this type of WENO scheme suffers from slight post-shock oscillations. Even though these oscillations are small in magnitude and do not affect the "essentially non-oscillatory" property of WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. We propose a new smoothness indicator for the WENO schemes in steady state calculations, which performs better near the steady shock region than the original smoothness indicator in [7, (1996) J. Comput. Phys. 126, 202---228.]. With our new smoothness indicator, the slight post-shock oscillations are either removed or significantly reduced and convergence is improved significantly. Numerical experiments show that the residue for the WENO scheme with this new smoothness indicator can converge to machine zero for one and two dimensional (2D) steady problems with strong shock waves when there are no shocks passing through the domain boundaries.
- Published
- 2006
28. Recovering High-Order Accuracy in WENO Computations of Steady-State Hyperbolic Systems
- Author
-
Sigal Gottlieb, Chi-Wang Shu, and David Gottlieb
- Subjects
Numerical Analysis ,Steady state (electronics) ,Applied Mathematics ,Computation ,Mathematical analysis ,Nozzle ,General Engineering ,Order of accuracy ,Hyperbolic systems ,Theoretical Computer Science ,Euler equations ,Shock (mechanics) ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Flow (mathematics) ,symbols ,Software ,Mathematics - Abstract
In this note we consider the application of the WENO scheme to simulations of steady-state flow in a converging diverging nozzle. We demonstrate the recovery of design accuracy through Gegenbauer postprocessing, despite the degradation of the order of accuracy for the numerical solution of the Euler equations to first-order in regions where the characteristics passed through the shock. We have shown a case in which the Gegenbauer postprocessing can recover the order of accuracy right up to the shock location. This suggests that high-order accurate information which crosses through the shock may not be irretrievably lost, and we can strive to recover it through various types of postprocessing.
- Published
- 2006
29. High-Order Well-Balanced Finite Difference WENO Schemes for a Class of Hyperbolic Systems with Source Terms
- Author
-
Yulong Xing and Chi-Wang Shu
- Subjects
Numerical Analysis ,Conservation law ,Steady state ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Finite difference ,Classification of discontinuities ,Wave equation ,Theoretical Computer Science ,Separable space ,Computational Mathematics ,Computational Theory and Mathematics ,Flow (mathematics) ,Shallow water equations ,Software ,Mathematics - Abstract
In this paper, we generalize the high order well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, designed earlier by us in Xing and Shu (2005, J. Comput. phys. 208, 206---227) for the shallow water equations, to solve a wider class of hyperbolic systems with separable source terms including the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. Properties of the scheme for the shallow water equations (Xing and Shu 2005, J. Comput. phys. 208, 206---227), such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions, are maintained for the scheme when applied to this general class of hyperbolic systems
- Published
- 2005
30. Numerical Convergence Study of Nearly Incompressible, Inviscid Taylor–Green Vortex Flow
- Author
-
David Gottlieb, Wai Sun Don, Chi-Wang Shu, Oleg Schilling, and Leland Jameson
- Subjects
Numerical Analysis ,Conservation law ,Computer simulation ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,General Engineering ,Enstrophy ,Theoretical Computer Science ,Vortex ,Computational Mathematics ,Computational Theory and Mathematics ,Inviscid flow ,Taylor–Green vortex ,Spectral method ,Software ,Mathematics - Abstract
A spectral method and a fifth-order weighted essentially non-oscillatory method were used to examine the consequences of filtering in the numerical simulation of the three-dimensional evolution of nearly-incompressible, inviscid Taylor--Green vortex flow. It was found that numerical filtering using the high-order exponential filter and low-pass filter with sharp high mode cutoff applied in the spectral simulations significantly affects the convergence of the numerical solution. While the conservation property of the spectral method is highly desirable for fluid flows described by a system of hyperbolic conservation laws, spectral methods can yield erroneous results and conclusions at late evolution times when the flow eventually becomes under-resolved. In particular, it is demonstrated that the enstrophy and kinetic energy, which are two integral quantities often used to evaluate the quality of numerical schemes, can be misleading and should not be used unless one can assure that the solution is sufficiently well-resolved. In addition, it is shown that for the Taylor--Green vortex (for example) it is useful to compare the predictions of at least two numerical methods with different algorithmic foundations (such as a spectral and finite-difference method) in order to corroborate the conclusions from the numerical solutions when the analytical solution is not known.
- Published
- 2005
31. Numerical Simulation of High Mach Number Astrophysical Jets with Radiative Cooling
- Author
-
Carl L. Gardner, Chi-Wang Shu, Youngsoo Ha, and Anne Gelb
- Subjects
Physics ,Numerical Analysis ,Computer simulation ,Radiative cooling ,business.industry ,Applied Mathematics ,Numerical analysis ,General Engineering ,Computational fluid dynamics ,Theoretical Computer Science ,Computational physics ,Physics::Fluid Dynamics ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Mach number ,symbols ,business ,Software - Abstract
Computational fluid dynamics simulations using the WENO-LF method are applied to high Mach number nonrelativistic astrophysical jets, including the effects of radiative cooling. Our numerical methods have allowed us to simulate astrophysical jets at much higher Mach numbers than have been attained (Mach 20) in the literature. Our simulations of the HH 1-2 astrophysical jets are at Mach 80. Simulations at high Mach numbers and with radiative cooling are essential for achieving detailed agreement with the astrophysical images.
- Published
- 2005
32. Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations
- Author
-
Chi-Wang Shu and Fengyan Li
- Subjects
Numerical Analysis ,Polynomial ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Classification of discontinuities ,Theoretical Computer Science ,Computational Mathematics ,symbols.namesake ,Nonlinear system ,Computational Theory and Mathematics ,Maxwell's equations ,Discontinuous Galerkin method ,symbols ,Piecewise ,Galerkin method ,Software ,Mathematics ,Numerical stability - Abstract
In this paper, we continue our investigation of the locally divergence-free discontinuous Galerkin method, originally developed for the linear Maxwell equations (J. Comput. Phys. 194 588-610 (2004)), to solve the nonlinear ideal magnetohydrodynamics (MHD) equations. The distinctive feature of such method is the use of approximate solutions that are exactly divergence-free inside each element for the magnetic field. As a consequence, this method has a smaller computational cost than the traditional discontinuous Galerkin method with standard piecewise polynomial spaces. We formulate the locally divergence-free discontinuous Galerkin method for the MHD equations and perform extensive one and two-dimensional numerical experiments for both smooth solutions and solutions with discontinuities. Our computational results demonstrate that the locally divergence-free discontinuous Galerkin method, with a reduced cost comparing to the traditional discontinuous Galerkin method, can maintain the same accuracy for smooth solutions and can enhance the numerical stability of the scheme and reduce certain nonphysical features in some of the test cases.
- Published
- 2005
33. [Untitled]
- Author
-
Kurt Sebastian and Chi-Wang Shu
- Subjects
Numerical Analysis ,Finite volume method ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Finite difference method ,Lagrange polynomial ,Finite difference ,Order of accuracy ,Finite difference coefficient ,Theoretical Computer Science ,Computational Mathematics ,symbols.namesake ,Complex geometry ,Computational Theory and Mathematics ,Robustness (computer science) ,symbols ,Applied mathematics ,Software ,Mathematics - Abstract
High order finite difference WENO methods have the advantage of simpler coding and smaller computational cost for multi-dimensional problems, compared with finite volume WENO methods of the same order of accuracy. However a main restriction is that conservative finite difference methods of third and higher order of accuracy can only be used on uniform rectangular or smooth curvilinear meshes. In order to overcome this difficulty, in this paper we develop a multidomain high order WENO finite difference method which uses an interpolation procedure at the subdomain interfaces. A simple Lagrange interpolation procedure is implemented and compared to a WENO interpolation procedure. Extensive numerical examples are shown to indicate the effectiveness of each procedure, including the measurement of conservation errors, orders of accuracy, essentially non-oscillatory properties at the domain interfaces, and robustness for problems containing strong shocks and complex geometry. Our numerical experiments have shown that the simple and efficient Lagrange interpolation suffices for the subdomain interface treatment in the multidomain WENO finite difference method, to retain essential conservation, full high order of accuracy, essentially non-oscillatory properties at the domain interfaces even for strong shocks, and robustness for problems containing strong shocks and complex geometry. The method developed in this paper can be used to solve problems in relatively complex geometry at a much smaller CPU cost than the finite volume version of the same method for the same accuracy. The method can also be used for high order finite difference ENO schemes and an example is given to demonstrate a similar result as that for the WENO schemes.
- Published
- 2003
34. [Untitled]
- Author
-
Chi-Wang Shu, Yinfan Li, and Tie Zhou
- Subjects
Numerical Analysis ,Finite volume method ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Lagrange polynomial ,Finite difference method ,Theoretical Computer Science ,Computational Mathematics ,symbols.namesake ,Runge–Kutta methods ,Discontinuity (linguistics) ,Computational Theory and Mathematics ,Discontinuous Galerkin method ,symbols ,Galerkin method ,Convection–diffusion equation ,Software ,Mathematics - Abstract
High order WENO (weighted essentially non-oscillatory) schemes and discontinuous Galerkin methods are two classes of high order, high resolution methods suitable for convection dominated simulations with possible discontinuous or sharp gradient solutions. In this paper we first review these two classes of methods, pointing out their similarities and differences in algorithm formulation, theoretical properties, implementation issues, applicability, and relative advantages. We then present some quantitative comparisons of the third order finite volume WENO methods and discontinuous Galerkin methods for a series of test problems to assess their relative merits in accuracy and CPU timing.
- Published
- 2001
35. Foreword
- Author
-
Chi-Wang Shu and Bernardo Cockburn
- Subjects
Computational Mathematics ,Numerical Analysis ,Computational Theory and Mathematics ,Applied Mathematics ,General Engineering ,Engineering ethics ,Software ,Theoretical Computer Science ,Mathematics - Published
- 2009
36. Foreword
- Author
-
David Gottlieb, Jan S. Hesthaven, George Karniadakis, and Chi-Wang Shu
- Subjects
Computational Mathematics ,Numerical Analysis ,Computational Theory and Mathematics ,Applied Mathematics ,General Engineering ,Software ,Theoretical Computer Science - Published
- 2006
37. A note on the accuracy of spectral method applied to nonlinear conservation laws
- Author
-
Peter S. Wong and Chi-Wang Shu
- Subjects
Numerical Analysis ,Partial differential equation ,Gegenbauer polynomials ,Applied Mathematics ,Mathematical analysis ,Spectral element method ,General Engineering ,Order of accuracy ,Theoretical Computer Science ,Computational Mathematics ,Nonlinear system ,symbols.namesake ,Computational Theory and Mathematics ,Fourier analysis ,symbols ,Spectral method ,Software ,Mathematics ,Analytic function - Abstract
Fourier spectral method can achieve exponential accuracy both on the approximation level and for solving partial differential equations if the solutions are analytic. For a linear partial differential equation with a discontinuous solution, Fourier spectral method produces poor point-wise accuracy without post-processing, but still maintains exponential accuracy for all moments against analytic functions. In this note we assess the accuracy of Fourier spectral method applied to nonlinear conservation laws through a numerical case study. We find that the moments with respect to analytic functions are no longer very accurate. However the numerical solution does contain accurate information which can be extracted by a post-processing based on Gegenbauer polynomials.
- Published
- 1995
38. Numerical experiments on the accuracy of ENO and modified ENO schemes
- Author
-
Chi-Wang Shu
- Subjects
Numerical Analysis ,Conservation law ,Partial differential equation ,business.industry ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Finite difference method ,Order of accuracy ,Computational fluid dynamics ,Theoretical Computer Science ,Computational Mathematics ,Runge–Kutta methods ,Computational Theory and Mathematics ,Applied mathematics ,Boundary value problem ,business ,Degeneracy (mathematics) ,Software ,Mathematics - Abstract
Further numerical experiments are made assessing an accuracy degeneracy phenomena. A modified essentially non-oscillatory (ENO) scheme is proposed, which recovers the correct order of accuracy for all the test problems with smooth initial conditions and gives comparable results with the original ENO schemes for discontinuous problems.
- Published
- 1990
39. Preface
- Author
-
Chi-Wang Shu
- Subjects
Computational Mathematics ,Numerical Analysis ,Computational Theory and Mathematics ,Applied Mathematics ,General Engineering ,Software ,Theoretical Computer Science - Published
- 2012
40. Foreword
- Author
-
Sigal Gottlieb and Chi-Wang Shu
- Subjects
Computational Mathematics ,Numerical Analysis ,Computational Theory and Mathematics ,Applied Mathematics ,General Engineering ,Software ,Theoretical Computer Science - Published
- 2010
41. Foreword
- Author
-
Mark H. Carpenter, David I. Gottlieb, Jan S. Hesthaven, and Chi-Wang Shu
- Subjects
Computational Mathematics ,Numerical Analysis ,Computational Theory and Mathematics ,Applied Mathematics ,General Engineering ,Software ,Theoretical Computer Science - Published
- 2005
42. Multidomain WENO Finite Difference Method with Interpolation at Subdomain Interfaces.
- Author
-
Sebastian, Kurt and Chi-Wang Shu
- Subjects
FINITE differences ,INTERPOLATION ,NUMERICAL analysis ,COST ,ERRORS ,SCIENTIFIC experimentation ,GEOMETRY - Abstract
High order finite difference WENO methods have the advantage of simpler coding and smaller computational cost for multi-dimensional problems, compared with finite volume WENO methods of the same order of accuracy. However a main restriction is that conservative finite difference methods of third and higher order of accuracy can only be used on uniform rectangular or smooth curvilinear meshes. In order to overcome this difficulty, in this paper we develop a multidomain high order WENO finite difference method which uses an interpolation procedure at the subdomain interfaces. A simple Lagrange interpolation procedure is implemented and compared to a WENO interpolation procedure. Extensive numerical examples are shown to indicate the effectiveness of each procedure, including the measurement of conservation errors, orders of accuracy, essentially non-oscillatory properties at the domain interfaces, and robustness for problems containing strong shocks and complex geometry. Our numerical experiments have shown that the simple and efficient Lagrange interpolation suffices for the subdomain interface treatment in the multidomain WENO finite difference method, to retain essential conservation, full high order of accuracy, essentially non-oscillatory properties at the domain interfaces even for strong shocks, and robustness for problems containing strong shocks and complex geometry. The method developed in this paper can be used to solve problems in relatively complex geometry at a much smaller CPU cost than the finite volume version of the same method for the same accuracy. The method can also be used for high order finite difference ENO schemes and an example is given to demonstrate a similar result as that for the WENO schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2003
- Full Text
- View/download PDF
43. On the Conservation and Convergence to Weak Solutions of Global Schemes.
- Author
-
Mark H. Carpenter, David Gottlieb, and Chi-Wang Shu
- Published
- 2003
44. Foreword.
- Author
-
Chi-Wang Shu
- Subjects
ALGORITHMS ,HYPERBOLIC differential equations ,SCIENCE ,ENGINEERING - Abstract
The article presents an introduction to various articles published in this issue of the periodical. The issue is dedicated to scholar Stanley Osher on the occasion of his sixtieth birthday. It consists of papers covering a wide spectrum in algorithm design, analysis, implementation and application in the sciences and engineering. A level set algorithm is used in the paper by Tariq D. Aslam to track discontinuities in hyperbolic systems of conservation laws. Jean-David Benamou gives an extensive overview of the development over the past ten years in Eulerian geometric optics.
- Published
- 2003
- Full Text
- View/download PDF
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