23 results on '"Chi-Wang Shu"'
Search Results
2. Optimal error estimates to smooth solutions of the central discontinuous Galerkin methods for nonlinear scalar conservation laws
- Author
-
Mengjiao Jiao, Yan Jiang, Chi-Wang Shu, and Mengping Zhang
- Abstract
In this paper, we study the error estimates to sufficiently smooth solutions of the nonlinear scalar conservation laws for the semi-discrete central discontinuous Galerkin (DG) finite element methods on uniform Cartesian meshes. A general approach with an explicitly checkable condition is established for the proof of optimal L2 error estimates of the semi-discrete CDG schemes, and this condition is checked to be valid in one and two dimensions for polynomials of degree up to k = 8. Numerical experiments are given to verify the theoretical results.
- Published
- 2022
3. A local discontinuous Galerkin method for nonlinear parabolic SPDEs
- Author
-
Chi-Wang Shu, Yunzhang Li, and Shanjian Tang
- Subjects
Numerical Analysis ,Discretization ,Computer Science::Information Retrieval ,Applied Mathematics ,Degenerate energy levels ,MathematicsofComputing_NUMERICALANALYSIS ,Parabolic partial differential equation ,Stochastic partial differential equation ,Computational Mathematics ,Nonlinear system ,Discontinuous Galerkin method ,Modeling and Simulation ,Ordinary differential equation ,Applied mathematics ,Hyperbolic partial differential equation ,Analysis ,Mathematics - Abstract
In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove theL2-stability of the numerical scheme for fully nonlinear equations. Optimal error estimates (O(h(k+1))) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degreekare used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method.
- Published
- 2021
4. On a class of splines free of Gibbs phenomenon
- Author
-
Juan Ruiz, Juan Carlos Trillo, Chi-Wang Shu, Sergio Amat, Fundación Séneca, Ministerio de Economía y Competitividad, and National Science Foundation (NSF)
- Subjects
Splines ,1206 Análisis Numérico ,010103 numerical & computational mathematics ,1203.09 Diseño Con Ayuda del Ordenador ,Classification of discontinuities ,01 natural sciences ,Gibbs phenomenon ,symbols.namesake ,Applied mathematics ,Adaption to discontinuities ,0101 mathematics ,Mathematics ,Numerical Analysis ,Applied Mathematics ,Matemática Aplicada ,Interpolation ,010101 applied mathematics ,Computer aided design (modeling of curves) ,Computational Mathematics ,Discontinuity (linguistics) ,Nonlinear system ,Spline (mathematics) ,Modeling and Simulation ,Piecewise ,symbols ,Spline interpolation ,Analysis - Abstract
When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper. We would like to thank the anonymous referees for their valuable comments, which have helped to significantly improve this work. This work was funded by project 20928/PI/18 (Proyecto financiado por la Comunidad Autónoma de la Región de Murcia a través de la convocatoria de Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos, incluida en el Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2018) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia), by the national research project MTM2015- 64382-P (MINECO/FEDER) and by NSF grant DMS-1719410.
- Published
- 2021
5. Central discontinuous Galerkin methods on overlapping meshes for wave equations
- Author
-
Chi-Wang Shu, Jianfang Lu, Yong Liu, and Mengping Zhang
- Subjects
Numerical Analysis ,Applied Mathematics ,010103 numerical & computational mathematics ,Wave equation ,01 natural sciences ,Stability (probability) ,Projection (linear algebra) ,law.invention ,010101 applied mathematics ,Computational Mathematics ,Rate of convergence ,law ,Discontinuous Galerkin method ,Modeling and Simulation ,Piecewise ,Applied mathematics ,Polygon mesh ,Cartesian coordinate system ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper, we study the central discontinuous Galerkin (DG) method on overlapping meshes for second order wave equations. We consider the first order hyperbolic system, which is equivalent to the second order scalar equation, and construct the corresponding central DG scheme. We then provide the stability analysis and the optimal error estimates for the proposed central DG scheme for one- and multi-dimensional cases with piecewise Pk elements. The optimal error estimates are valid for uniform Cartesian meshes and polynomials of arbitrary degree k ≥ 0. In particular, we adopt the techniques in Liu et al. (SIAM J. Numer. Anal. 56 (2018) 520–541; ESAIM: M2AN 54 (2020) 705–726) and obtain the local projection that is crucial in deriving the optimal order of convergence. The construction of the projection here is more challenging since the unknowns are highly coupled in the proposed scheme. Dispersion analysis is performed on the proposed scheme for one dimensional problems, indicating that the numerical solution with P1 elements reaches its minimum with a suitable parameter in the dissipation term. Several numerical examples including accuracy tests and long time simulation are presented to validate the theoretical results.
- Published
- 2021
6. Analysis of optimal superconvergence of an ultraweak-local discontinuous Galerkin method for a time dependent fourth-order equation
- Author
-
Chi-Wang Shu, Yong Liu, and Qi Tao
- Subjects
Numerical Analysis ,Applied Mathematics ,Function (mathematics) ,Superconvergence ,Projection (linear algebra) ,Quadrature (mathematics) ,Computational Mathematics ,Exact solutions in general relativity ,Discontinuous Galerkin method ,Modeling and Simulation ,Piecewise ,Applied mathematics ,Order (group theory) ,Analysis ,Mathematics - Abstract
In this paper, we study superconvergence properties of the ultraweak-local discontinuous Galerkin (UWLDG) method in Tao et al. [To appear in Math. Comput. DOI: https://doi.org/10.1090/mcom/3562 (2020).] for an one-dimensional linear fourth-order equation. With special initial discretizations, we prove the numerical solution of the semi-discrete UWLDG scheme superconverges to a special projection of the exact solution. The order of this superconvergence is proved to be k + min(3, k) when piecewise ℙk polynomials with k ≥ 2 are used. We also prove a 2k-th order superconvergence rate for the cell averages and for the function values and derivatives of the UWLDG approximation at cell boundaries. Moreover, we prove superconvergence of (k + 2)-th and (k + 1)-th order of the function values and the first order derivatives of the UWLDG solution at a class of special quadrature points, respectively. Our proof is valid for arbitrary non-uniform regular meshes and for arbitrary k ≥ 2. Numerical experiments verify that all theoretical findings are sharp.
- Published
- 2020
7. Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using Pk elements
- Author
-
Mengping Zhang, Chi-Wang Shu, and Yong Liu
- Subjects
Numerical Analysis ,Constant coefficients ,Degree (graph theory) ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,law.invention ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Discontinuous Galerkin method ,law ,Modeling and Simulation ,Convergence (routing) ,Piecewise ,Applied mathematics ,Cartesian coordinate system ,0101 mathematics ,Hyperbolic partial differential equation ,Analysis ,Mathematics - Abstract
In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using Pk elements on uniform Cartesian meshes, and prove that the error in the L2 norm achieves optimal (k + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k = 0, 1, 2, 3 and k = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples.
- Published
- 2020
8. Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws
- Author
-
Lingling Zhou, Chi-Wang Shu, and Yinhua Xia
- Subjects
Numerical Analysis ,Conservation law ,Applied Mathematics ,Courant–Friedrichs–Lewy condition ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Piecewise linear function ,Computational Mathematics ,Runge–Kutta methods ,Discontinuous Galerkin method ,Modeling and Simulation ,Total variation diminishing ,Piecewise ,Applied mathematics ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.
- Published
- 2019
9. Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients
- Author
-
Waixiang Cao, Zhimin Zhang, and Chi-Wang Shu
- Subjects
Numerical Analysis ,Applied Mathematics ,Degenerate energy levels ,Mathematical analysis ,010103 numerical & computational mathematics ,Function (mathematics) ,Superconvergence ,Computer Science::Numerical Analysis ,01 natural sciences ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Function approximation ,Exact solutions in general relativity ,Discontinuous Galerkin method ,Modeling and Simulation ,Piecewise ,0101 mathematics ,Hyperbolic partial differential equation ,Analysis ,Mathematics - Abstract
In this paper, we study the superconvergence behavior of discontinuous Galerkin methods using upwind numerical fluxes for one-dimensional linear hyperbolic equations with degenerate variable coefficients. The study establishes superconvergence results for the flux function approximation as well as for the DG solution itself. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order of O (h k +2 ), when piecewise polynomials of degree k are used. We then prove that the highest superconvergence rate of the DG solution itself is O (h k +3/2 ) as the variable coefficient degenerates or achieves the value zero in the domain. As byproducts, we obtain superconvergence properties for the DG solution and the DG flux function at special points and for cell averages. All theoretical findings are confirmed by numerical experiments.
- Published
- 2017
10. Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs
- Author
-
Chi-Wang Shu, Haijin Wang, and Qiang Zhang
- Subjects
Numerical Analysis ,Partial differential equation ,Discretization ,Applied Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Derivative ,01 natural sciences ,Stability (probability) ,Projection (linear algebra) ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Third order ,Discontinuous Galerkin method ,Modeling and Simulation ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
The main purpose of this paper is to give stability analysis and error estimates of the local discontinuous Galerkin (LDG) methods coupled with three specific implicit-explicit (IMEX) Runge–Kutta time discretization methods up to third order accuracy, for solving one-dimensional time-dependent linear fourth order partial differential equations. In the time discretization, all the lower order derivative terms are treated explicitly and the fourth order derivative term is treated implicitly. By the aid of energy analysis, we show that the IMEX-LDG schemes are unconditionally energy stable, in the sense that the time step τ is only required to be upper-bounded by a constant which is independent of the mesh size h . The optimal error estimate is also derived by the aid of the elliptic projection and the adjoint argument. Numerical experiments are given to verify that the corresponding IMEX-LDG schemes can achieve optimal error accuracy.
- Published
- 2017
11. Stability analysis and error estimates of Lax–Wendroff discontinuous Galerkin methods for linear conservation laws
- Author
-
Chi-Wang Shu and Zheng Sun
- Subjects
Numerical Analysis ,Conservation law ,Lax–Wendroff theorem ,Discretization ,Lax–Wendroff method ,Applied Mathematics ,Courant–Friedrichs–Lewy condition ,Mathematical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Discontinuous Galerkin method ,Modeling and Simulation ,Time derivative ,Piecewise ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper, we analyze the Lax–Wendroff discontinuous Galerkin (LWDG) method for solving linear conservation laws. The method was originally proposed by Guo et al . in [W. Guo, J.-M. Qiu and J. Qiu, J. Sci. Comput. 65 (2015) 299–326], where they applied local discontinuous Galerkin (LDG) techniques to approximate high order spatial derivatives in the Lax–Wendroff time discretization. We show that, under the standard CFL condition τ ≤ λh (where τ and h are the time step and the maximum element length respectively and λ > 0 is a constant) and uniform or non-increasing time steps, the second order schemes with piecewise linear elements and the third order schemes with arbitrary piecewise polynomial elements are stable in the L 2 norm. The specific type of stability may differ with different choices of numerical fluxes. Our stability analysis includes multidimensional problems with divergence-free coefficients. Besides solving the equation itself, the LWDG method also gives approximations to its time derivative simultaneously. We obtain optimal error estimates for both the solution u and its first order time derivative u t in one dimension, and numerical examples are given to validate our analysis.
- Published
- 2017
12. Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems
- Author
-
Qiang Zhang, Shiping Wang, Haijin Wang, and Chi-Wang Shu
- Subjects
Numerical Analysis ,Polynomial ,Discretization ,Applied Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,Projection (linear algebra) ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Discontinuous Galerkin method ,Modeling and Simulation ,Convergence (routing) ,Piecewise ,0101 mathematics ,Convection–diffusion equation ,Analysis ,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 implicit-explicit (IMEX) time discretization schemes, for solving multi-dimensional convection-diffusion equations with nonlinear convection. By establishing the important relationship between the gradient and the interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG method, on both rectangular and triangular elements, we can obtain the same stability results as in the one-dimensional case [H.J. Wang, C.-W. Shu and Q. Zhang, SIAM J. Numer. Anal. 53 (2015) 206–227; H.J. Wang, C.-W. Shu and Q. Zhang, Appl. Math. Comput. 272 (2015) 237–258], i.e. , the IMEX LDG schemes are unconditionally stable for the multi-dimensional convection-diffusion problems, in the sense that the time-step τ is only required to be upper-bounded by a positive constant independent of the spatial mesh size h . Furthermore, by the aid of the so-called elliptic projection and the adjoint argument, we can also obtain optimal error estimates in both space and time, for the corresponding fully discrete IMEX LDG schemes, under the same condition, i.e. , if piecewise polynomial of degree k is adopted on either rectangular or triangular meshes, we can show the convergence accuracy is of order 𝒪( h k+1 + τ s ) for the s th order IMEX LDG scheme ( s = 1,2,3) under consideration. Numerical experiments are also given to verify our main results.
- Published
- 2016
13. Numerical Solution of the Viscous Surface Wave with Discontinuous Galerkin Method
- Author
-
Chi-Wang Shu and Lei Wu
- Subjects
Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,Stability (probability) ,Domain (mathematical analysis) ,Physics::Fluid Dynamics ,Surface tension ,Computational Mathematics ,Surface wave ,Discontinuous Galerkin method ,Modeling and Simulation ,Stokes wave ,Convection–diffusion equation ,Analysis ,Mathematics - Abstract
We consider an incompressible viscous flow without surface tension in a finite-depth domain of two dimensions, with free top boundary and fixed bottom boundary. This system is governed by the Navier–Stokes equations in this moving domain and the transport equation on the moving boundary. In this paper, we construct a stable numerical scheme to simulate the evolution of this system by discontinuous Galerkin method, and discuss the error analysis of the fluid under certain assumptions. Our formulation is mainly based on the geometric structure introduced in [Y. Guo and Ian Tice, Anal. PDE 6 (2013) 287–369; Y. Guo and Ian Tice, Arch. Ration. Mech. Anal. 207 (2013) 459–531; L. Wu, SIAM J. Math. Anal. 46 (2014) 2084–2135], and the natural energy estimate, which is rarely used in the numerical study of this system before.
- Published
- 2015
14. A priorierror estimates to smooth solutions of the third order Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws
- Author
-
Juan Luo, Qiang Zhang, and Chi-Wang Shu
- Subjects
Numerical Analysis ,Conservation law ,Discretization ,Applied Mathematics ,Courant–Friedrichs–Lewy condition ,Mathematical analysis ,Computer Science::Numerical Analysis ,Mathematics::Numerical Analysis ,Computational Mathematics ,Runge–Kutta methods ,Discontinuous Galerkin method ,Modeling and Simulation ,Total variation diminishing ,Piecewise ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper we present an a priori error estimate of the Runge–Kutta discontinuous Galerkin method for solving symmetrizable conservation laws, where the time is discretized with the third order explicit total variation diminishing Runge–Kutta method and the finite element space is made up of piecewise polynomials of degree k ≥ 2. Quasi-optimal error estimate is obtained by energy techniques, for the so-called generalized E-fluxes under the standard temporal-spatial CFL condition τ ≤ γh , where h is the element length and τ is time step, and γ is a positive constant independent of h and τ . Optimal estimates are also considered when the upwind numerical flux is used.
- Published
- 2015
15. Development and stability analysis of the inverse Lax−Wendroff boundary treatment for central compact schemes
- Author
-
Chi-Wang Shu and François Vilar
- Subjects
Numerical Analysis ,Discretization ,Lax–Wendroff method ,Applied Mathematics ,Mathematical analysis ,Stability (learning theory) ,Extrapolation ,Boundary (topology) ,Inverse ,Computational Mathematics ,Simple (abstract algebra) ,Modeling and Simulation ,Outflow boundary ,Analysis ,Mathematics - Abstract
In this paper, we generalize the so-called inverse Lax−Wendroff boundary treatment [S. Tan and C.-W. Shu, J. Comput. Phys. 229 (2010) 8144–8166] for the inflow boundary of a linear hyperbolic problem discretized by the recently introduced central compact schemes [X. Liu, S. Zhang, H. Zhang and C.-W. Shu, J. Comput. Phys. 248 (2013) 235–256]. The outflow boundary is treated by the classical extrapolation and a stability analysis for the resulting scheme is provided. To ensure the stability of the considered schemes provided with the chosen boundaries, the G-K-S theory [B. Gustafsson, H.-O. Kreiss and A. Sundstrom, Math. Comput. 26 (1972) 649–686] is used, first in the semidiscrete case then in the fully discrete case with the third-order TVD Runge−Kutta time discretization. Afterwards, due to the high algebraic complexity of the G-K-S theory, the stability is analyzed by visualizing the eigenspectrum of the discretized operators. We show in this paper that the results obtained with these two different approaches are perfectly consistent. We also illustrate the high accuracy of the presented schemes on simple test problems.
- Published
- 2015
16. Central local discontinuous galerkin methods on overlapping cells for diffusion equations
- Author
-
Chi-Wang Shu, Mengping Zhang, Eitan Tadmor, and Yingjie Liu
- Subjects
Numerical Analysis ,Diffusion equation ,Applied Mathematics ,Mathematical analysis ,Stability (probability) ,Mathematics::Numerical Analysis ,Computational Mathematics ,Discontinuous Galerkin method ,Modeling and Simulation ,General polynomial ,Heat equation ,Diffusion (business) ,Analysis ,Mathematics - Abstract
In this paper we present two versions of the central local discontinuous Galerkin (LDG) method on overlapping cells for solving diffusion equations, and provide their stability analysis and error estimates for the linear heat equation. A comparison between the traditional LDG method on a single mesh and the two versions of the central LDG method on overlapping cells is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis and to support conclusions for general polynomial degrees.
- Published
- 2011
17. L2stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods
- Author
-
Chi-Wang Shu, Yingjie Liu, Mengping Zhang, and Eitan Tadmor
- Subjects
Numerical Analysis ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Context (language use) ,Computer Science::Numerical Analysis ,Stability (probability) ,Mathematics::Numerical Analysis ,Computational Mathematics ,Discontinuous Galerkin method ,Modeling and Simulation ,Galerkin method ,Hyperbolic partial differential equation ,Analysis ,Linear equation ,Numerical stability ,Mathematics - Abstract
We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the central discontinuous Galerkin method and the regular discontinuous Galerkin method in this context is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis.
- Published
- 2008
18. The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws
- Author
-
Chi-Wang Shu and Bernardo Cockburn
- Subjects
Numerical Analysis ,Computational Mathematics ,Conservation law ,Runge–Kutta methods ,Discontinuous Galerkin method ,Applied Mathematics ,Modeling and Simulation ,Scalar (mathematics) ,Calculus ,Applied mathematics ,Analysis ,Finite element method ,Mathematics - Abstract
Nous introduisons et analysons le schema modele d'une nouvelle classe de methodes pour resoudre numeriquement les lois de conservation hyperboliques. La construction du schema basee sur une discretisation en espace par elements finis disconstinus et sur discretisation en temps par un schema de Runge Kutta d'ordre eleve a variation totale decroissante. Le schema obtenu satisfait un principe du maximum, est a variation totale bornee en moyenne, lineairement stable pour CFL ∈ [0,1/3], et formellement du second ordre en temps et en espace. Nous montrons numeriquement que le schema converge vers la solution entropique, et que l'ordre de convergence en dehors des singularites est optimal
- Published
- 1991
19. LOCAL DISCONTINUOUS GALERKIN METHODS WITH IMPLICIT-EXPLICIT TIME-MARCHING FOR MULTI-DIMENSIONAL CONVECTION-DIFFUSION PROBLEMS.
- Author
-
Haijin Wang, Shiping Wang, Qiang Zhang, and Chi-Wang Shu
- Abstract
The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving multi-dimensional convection-diffusion equations with nonlinear convection. By establishing the important relationship between the gradient and the interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG method, on both rectangular and triangular elements, we can obtain the same stability results as in the one-dimensional case [H.J. Wang, C.-W. Shu and Q. Zhang, SIAM J. Numer. Anal. 53 (2015) 206–227; H.J. Wang, C.-W. Shu and Q. Zhang, Appl. Math. Comput. 272 (2015) 237–258], i.e., the IMEX LDG schemes are unconditionally stable for the multi-dimensional convection-diffusion problems, in the sense that the time-step τ is only required to be upper-bounded by a positive constant independent of the spatial mesh size h. Furthermore, by the aid of the so-called elliptic projection and the adjoint argument, we can also obtain optimal error estimates in both space and time, for the corresponding fully discrete IMEX LDG schemes, under the same condition, i.e., if piecewise polynomial of degree k is adopted on either rectangular or triangular meshes, we can show the convergence accuracy is of order O(h
k+1 +τs ) for the sth order IMEX LDG scheme (s = 1, 2, 3) under consideration. Numerical experiments are also given to verify our main results. [ABSTRACT FROM AUTHOR]- Published
- 2016
- Full Text
- View/download PDF
20. A PRIORI ERROR ESTIMATES TO SMOOTH SOLUTIONS OF THE THIRD ORDER RUNGE-KUTTA DISCONTINUOUS GALERKIN METHOD FOR SYMMETRIZABLE SYSTEMS OF CONSERVATION LAWS.
- Author
-
JUAN LUO, CHI-WANG SHU, and QIANG ZHANG
- Subjects
- *
RUNGE-Kutta formulas , *GALERKIN methods , *FINITE element method , *POLYNOMIALS , *CONSERVATION laws (Mathematics) - Abstract
In this paper we present an a priori error estimate of the Runge-Kutta discontinuous Galerkin method for solving symmetrizable conservation laws, where the time is discretized with the third order explicit total variation diminishing Runge-Kutta method and the finite element space is made up of piecewise polynomials of degree k ≥ Quasi-optimal error estimate is obtained by energy techniques, for the so-called generalized E-fluxes under the standard temporal-spatial CFL condition τ ≤ γh, where h is the element length and τ is time step, and γ is a positive constant independent of h and τ. Optimal estimates are also considered when the upwind numerical flux is used. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
21. NUMERICAL SOLUTION OF THE VISCOUS SURFACE WAVE WITH DISCONTINUOUS GALERKIN METHOD.
- Author
-
LEI WU and CHI-WANG SHU
- Subjects
- *
SURFACE waves (Fluids) , *VISCOUS flow , *NAVIER-Stokes equations , *FLUID flow , *GALERKIN methods , *MATHEMATICAL models - Abstract
We consider an incompressible viscous flow without surface tension in a finite-depth domain of two dimensions, with free top boundary and fixed bottom boundary. This system is governed by the Navier-Stokes equations in this moving domain and the transport equation on the moving boundary. In this paper, we construct a stable numerical scheme to simulate the evolution of this system by discontinuous Galerkin method, and discuss the error analysis of the fluid under certain assumptions. Our formulation is mainly based on the geometric structure introduced in [Y. Guo and Ian Tice, Anal. PDE 6 (2013) 287-369; Y. Guo and Ian Tice, Arch. Ration. Mech. Anal. 207 (2013) 459-531; L. Wu, SIAM J. Math. Anal. 46 (2014) 2084-2135], and the natural energy estimate, which is rarely used in the numerical study of this system before. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
22. DEVELOPMENT AND STABILITY ANALYSIS OF THE INVERSE LAX-WENDROFF BOUNDARY TREATMENT FOR CENTRAL COMPACT SCHEMES.
- Author
-
VILAR, FRANÇOIS and CHI-WANG SHU
- Subjects
- *
NUMERICAL solutions to hyperbolic differential equations , *DISCRETIZATION methods , *EXTRAPOLATION , *STABILITY theory , *RUNGE-Kutta formulas , *OPERATOR theory - Abstract
In this paper, we generalize the so-called inverse Lax-Wendroff boundary treatment [S. Tan and C.-W. Shu, J. Comput. Phys. 229 (2010) 8144-8166] for the inflow boundary of a linear hyperbolic problem discretized by the recently introduced central compact schemes [X. Liu, S. Zhang, H. Zhang and C.-W. Shu, J. Comput. Phys. 248 (2013) 235-256]. The outflow boundary is treated by the classical extrapolation and a stability analysis for the resulting scheme is provided. To ensure the stability of the considered schemes provided with the chosen boundaries, the G-K-S theory [B. Gustafsson, H.-O. Kreiss and A. Sundström, Math. Comput. 26 (1972) 649-686] is used, first in the semidiscrete case then in the fully discrete case with the third-order TVD Runge-Kutta time discretization. Afterwards, due to the high algebraic complexity of the G-K-S theory, the stability is analyzed by visualizing the eigenspectrum of the discretized operators. We show in this paper that the results obtained with these two different approaches are perfectly consistent. We also illustrate the high accuracy of the presented schemes on simple test problems. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
23. L2stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods.
- Author
-
Yingjie Liu, Chi-Wang Shu, Eitan Tadmor, and Mengping Zhang
- Subjects
- *
GALERKIN methods , *NUMERICAL analysis , *QUANTITATIVE research , *EQUATIONS - Abstract
?We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the central discontinuous Galerkin method and the regular discontinuous Galerkin method in this context is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.