321 results on '"Chi-Wang Shu"'
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2. High order numerical methods for flows with hysteretic fluxes
- Author
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Haitao Fan and Chi-Wang Shu
- Subjects
Computational Mathematics ,Numerical Analysis ,Applied Mathematics - Published
- 2023
3. L$^2$ Error Estimate to Smooth Solutions of High Order Runge--Kutta Discontinuous Galerkin Method for Scalar Nonlinear Conservation Laws with and without Sonic Points
- Author
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Jingqi Ai, Yuan Xu, Chi-Wang Shu, and Qiang Zhang
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Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2022
4. High-Resolution Viscous Terms Discretization and ILW Solid Wall Boundary Treatment for the Navier–Stokes Equations
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Francisco Augusto Aparecido Gomes, Chi-Wang Shu, Nicholas Dicati Pereira da Silva, and Rafael Brandão de Rezende Borges
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Physics ,Discretization ,business.industry ,Applied Mathematics ,Numerical analysis ,Computational fluid dynamics ,Computer Science Applications ,Euler equations ,Boundary layer ,symbols.namesake ,Flow (mathematics) ,symbols ,Applied mathematics ,Oblique shock ,Navier–Stokes equations ,business - Abstract
Robust numerical methods for CFD applications, such as WENO schemes, quickly evolved in the past few decades. Together with the Inverse Lax–Wendroff (ILW) procedure, WENO ideas were also applied in the boundary treatment. Those methods are known for their high-resolution property, i.e., good representation of nonlinear phenomena, which is an important property in solving challenging engineering problems. In light of that, the objective of this work is to present a review of well-established high-resolution numerical methods to solve the Euler equations and adapt the Navier–Stokes viscous terms discretization and boundary treatment. To test the modifications, we employed the positivity-preserving Lax–Friedrichs splitting, multi-resolution WENO scheme, third-order strong stability preserving Runge–Kutta time discretization, and ILW boundary treatment. The first problems were simple flows with analytical solutions for accuracy tests. We also tested the accuracy with nontrivial phenomena in the vortex flow. Oblique shock and complicated flow structures were captured in the Rayleigh–Taylor instability and flow past a cylinder. We showed the discretization and boundary treatment can handle non-constant viscosity, are high-order, high-resolution, and behave similarly to the well-established numerical methods. Furthermore, the methods discussed here can preserve symmetry and no approximations regarding the boundary layer were made. Therefore, the discretization and boundary treatment can be considered when solving direct numerical simulations.
- Published
- 2021
5. High-order Runge-Kutta discontinuous Galerkin methods with multi-resolution WENO limiters for solving steady-state problems
- Author
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Jianxian Qiu, Jun Zhu, and Chi-Wang Shu
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Physics::Computational Physics ,Numerical Analysis ,Sequence ,Finite volume method ,Truncation error (numerical integration) ,Applied Mathematics ,010103 numerical & computational mathematics ,Classification of discontinuities ,Residual ,01 natural sciences ,Projection (linear algebra) ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Runge–Kutta methods ,Discontinuous Galerkin method ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
Since the classical WENO schemes [27] might suffer from slight post-shock oscillations (which are responsible for the numerical residual to hang at a truncation error level) and the new high-order multi-resolution WENO schemes [59] are successful to solve for steady-state problems, we apply these high-order finite volume multi-resolution WENO techniques to serve as limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods in simulating steady-state problems. Firstly, a new troubled cell indicator is designed to precisely detect the cells which would need further limiting procedures. Then the high-order multi-resolution WENO limiting procedures are adopted on a sequence of hierarchical L 2 projection polynomials of the DG solution within the troubled cell itself. By doing so, these RKDG methods with multi-resolution WENO limiters could gradually degrade from the optimal high-order accuracy to the first-order accuracy near strong discontinuities, suppress the slight post-shock oscillations, and push the numerical residual to settle down to machine zero in steady-state simulations. These new multi-resolution WENO limiters are very simple to construct and can be easily implemented to arbitrary high-order accuracy for solving steady-state problems in multi-dimensions.
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- 2021
6. Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations
- Author
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Chi-Wang Shu and Kailiang Wu
- Subjects
Offset (computer science) ,Discretization ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,Discontinuous Galerkin method ,Robustness (computer science) ,FOS: Mathematics ,Applied mathematics ,Polygon mesh ,Mathematics - Numerical Analysis ,0101 mathematics ,Divergence (statistics) ,Instrumentation and Methods for Astrophysics (astro-ph.IM) ,Mathematics ,Applied Mathematics ,Numerical analysis ,Fluid Dynamics (physics.flu-dyn) ,Physics - Fluid Dynamics ,Numerical Analysis (math.NA) ,Computational Physics (physics.comp-ph) ,Physics - Plasma Physics ,Plasma Physics (physics.plasm-ph) ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Astrophysics - Instrumentation and Methods for Astrophysics ,Physics - Computational Physics - Abstract
We propose and analyze a class of robust, uniformly high-order accurate discontinuous Galerkin (DG) schemes for multidimensional relativistic magnetohydrodynamics (RMHD) on general meshes. A distinct feature of the schemes is their physical-constraint-preserving (PCP) property, i.e., they are proven to preserve the subluminal constraint on the fluid velocity and the positivity of density, pressure, and internal energy. This is the first time that provably PCP high-order schemes are achieved for multidimensional RMHD. Developing PCP high-order schemes for RMHD is highly desirable but remains a challenging task, especially in the multidimensional cases, due to the inherent strong nonlinearity in the constraints and the effect of the magnetic divergence-free condition. Inspired by some crucial observations at the PDE level, we construct the provably PCP schemes by using the locally divergence-free DG schemes of the recently proposed symmetrizable RMHD equations as the base schemes, a limiting technique to enforce the PCP property of the DG solutions, and the strong-stability-preserving methods for time discretization. We rigorously prove the PCP property by using a novel “quasi-linearization” approach to handle the highly nonlinear physical constraints, technical splitting to offset the influence of divergence error, and sophisticated estimates to analyze the beneficial effect of the additional source term in the symmetrizable RMHD system. Several two-dimensional numerical examples are provided to further confirm the PCP property and to demonstrate the accuracy, effectiveness and robustness of the proposed PCP schemes.
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- 2021
7. Entropy Stable Galerkin Methods with Suitable Quadrature Rules for Hyperbolic Systems with Random Inputs
- Author
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Xinghui Zhong and Chi-Wang Shu
- Subjects
Computational Mathematics ,Numerical Analysis ,Computational Theory and Mathematics ,Applied Mathematics ,General Engineering ,Software ,Theoretical Computer Science - Published
- 2022
8. An Oscillation-free Discontinuous Galerkin Method for Scalar Hyperbolic Conservation Laws
- Author
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Yong Liu, Chi-Wang Shu, and Jianfang Lu
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Numerical Analysis ,Computational Mathematics ,Conservation law ,Discontinuous Galerkin method ,Oscillation ,Applied Mathematics ,Scalar (mathematics) ,Mathematical analysis ,Superconvergence ,High order ,Spurious oscillations ,Mathematics - Abstract
In this paper, we propose a novel discontinuous Galerkin (DG) method to control the spurious oscillations when solving the scalar hyperbolic conservation laws. Usually, the high order linear numeri...
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- 2021
9. A local discontinuous Galerkin method for nonlinear parabolic SPDEs
- Author
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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
10. On a class of splines free of Gibbs phenomenon
- Author
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Juan Ruiz, Juan Carlos Trillo, Chi-Wang Shu, Sergio Amat, Fundación Séneca, Ministerio de Economía y Competitividad, and National Science Foundation (NSF)
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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
11. Central discontinuous Galerkin methods on overlapping meshes for wave equations
- Author
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Chi-Wang Shu, Jianfang Lu, Yong Liu, and Mengping Zhang
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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.
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- 2021
12. A high order positivity-preserving polynomial projection remapping method
- Author
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Nuo Lei, Juan Cheng, and Chi-Wang Shu
- Subjects
Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Computer Science Applications - Published
- 2023
13. A primal-dual approach for solving conservation laws with implicit in time approximations
- Author
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Siting Liu, Stanley Osher, Wuchen Li, and Chi-Wang Shu
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History ,Numerical Analysis ,Polymers and Plastics ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,65M06, 65K10, 49M41, 65M60 ,Numerical Analysis (math.NA) ,Industrial and Manufacturing Engineering ,Computer Science Applications ,Computational Mathematics ,Modeling and Simulation ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Business and International Management - Abstract
In this work, we propose a novel framework for the numerical solution of time-dependent conservation laws with implicit schemes via primal-dual hybrid gradient methods. We solve an initial value problem (IVP) for the partial differential equation (PDE) by casting it as a saddle point of a min-max problem and using iterative optimization methods to find the saddle point. Our approach is flexible with the choice of both time and spatial discretization schemes. It benefits from the implicit structure and gains large regions of stability, and overcomes the restriction on the mesh size in time by explicit schemes from Courant--Friedrichs--Lewy (CFL) conditions (really via von Neumann stability analysis). Nevertheless, it is highly parallelizable and easy-to-implement. In particular, no nonlinear inversions are required! Specifically, we illustrate our approach using the finite difference scheme and discontinuous Galerkin method for the spatial scheme; backward Euler and backward differentiation formulas for implicit discretization in time. Numerical experiments illustrate the effectiveness and robustness of the approach. In future work, we will demonstrate that our idea of replacing an initial-value evolution equation with this primal-dual hybrid gradient approach has great advantages in many other situations.
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- 2023
14. A high order moving boundary treatment for convection-diffusion equations
- Author
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Shihao Liu, Yan Jiang, Chi-Wang Shu, Mengping Zhang, and Shuhai Zhang
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Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Computer Science Applications - Published
- 2023
15. A high order positivity-preserving conservative WENO remapping method based on a moving mesh solver
- Author
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Xiaolu Gu, Yue Li, Juan Cheng, and Chi-Wang Shu
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Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Computer Science Applications - Published
- 2023
16. On a new centered strategy to control the accuracy of weighted essentially non oscillatory algorithm for conservation laws close to discontinuities
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Antonio Baeza, Chi-Wang Shu, Sergio Amat, and Juan Ruiz
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Computational Mathematics ,Numerical Analysis ,Conservation law ,Applied Mathematics ,Applied mathematics ,Classification of discontinuities ,Control (linguistics) ,Analysis ,Mathematics - Published
- 2020
17. Analysis of optimal superconvergence of an ultraweak-local discontinuous Galerkin method for a time dependent fourth-order equation
- Author
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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.
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- 2020
18. High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters on triangular meshes
- Author
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Jianxian Qiu, Jun Zhu, and Chi-Wang Shu
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Physics::Computational Physics ,Numerical Analysis ,Conservation law ,Finite volume method ,Applied Mathematics ,Order of accuracy ,Stencil ,Mathematics::Numerical Analysis ,Computational Mathematics ,Runge–Kutta methods ,Robustness (computer science) ,Discontinuous Galerkin method ,Applied mathematics ,Polygon mesh ,Mathematics - Abstract
In this paper, high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (WENO) limiters are designed for solving hyperbolic conservation laws on triangular meshes. These multi-resolution WENO limiters are new extensions of the associated multi-resolution WENO finite volume schemes [49] , [50] which serve as limiters for RKDG methods from structured meshes [47] to triangular meshes. Such new WENO limiters use information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF indicator [24] , to build a sequence of hierarchical L 2 projection polynomials from zeroth degree to the highest degree of the RKDG method. The second-order, third-order, and fourth-order RKDG methods with associated multi-resolution WENO limiters are developed as examples, which could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property near strong shocks or contact discontinuities by gradually degrading from the highest order to the first order. The linear weights inside the procedure of the new multi-resolution WENO limiters can be any positive numbers on the condition that their sum equals one. This is the first time that a series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell on triangular meshes. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstruction methodology improves the robustness in the simulation on the same compact spatial stencil of the original DG methods on triangular meshes. Extensive one-dimensional (run as two-dimensional problems on triangular meshes) and two-dimensional tests are performed to demonstrate the effectiveness of these RKDG methods with the new multi-resolution WENO limiters.
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- 2020
19. Essentially non-oscillatory and weighted essentially non-oscillatory schemes
- Author
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Chi-Wang Shu
- Subjects
Physics::Computational Physics ,Numerical Analysis ,Partial differential equation ,Finite volume method ,business.industry ,General Mathematics ,Finite difference ,Computational fluid dynamics ,Classification of discontinuities ,01 natural sciences ,Mathematics::Numerical Analysis ,010305 fluids & plasmas ,010101 applied mathematics ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,business ,Mathematics - Abstract
Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.
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- 2020
20. Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using Pk elements
- Author
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Mengping Zhang, Chi-Wang Shu, and Yong Liu
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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.
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- 2020
21. Error Estimate of the Fourth-Order Runge--Kutta Discontinuous Galerkin Methods for Linear Hyperbolic Equations
- Author
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Chi-Wang Shu, Yuan Xu, and Qiang Zhang
- Subjects
Physics::Computational Physics ,Numerical Analysis ,Applied Mathematics ,010103 numerical & computational mathematics ,Computer Science::Numerical Analysis ,01 natural sciences ,Energy analysis ,Mathematics::Numerical Analysis ,Computational Mathematics ,Runge–Kutta methods ,Fourth order ,Discontinuous Galerkin method ,Applied mathematics ,Condensed Matter::Strongly Correlated Electrons ,0101 mathematics ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper we consider the Runge--Kutta discontinuous Galerkin (RKDG) method to solve linear constant-coefficient hyperbolic equations, where the fourth-order explicit Runge--Kutta time-marching...
- Published
- 2020
22. 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
23. Multi-symplectic discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise
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Jiawei Sun, Chi-Wang Shu, and Yulong Xing
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Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,FOS: Mathematics ,Numerical Analysis (math.NA) ,Mathematics - Numerical Analysis ,Computer Science Applications - Abstract
One- and multi-dimensional stochastic Maxwell equations with additive noise are considered in this paper. It is known that such system can be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. High order discontinuous Galerkin methods are designed for the stochastic Maxwell equations with additive noise, and we show that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations. One- and two-dimensional numerical results are provided to demonstrate the performance of the proposed methods, and optimal error estimates and linear growth of the discrete energy can be observed for all cases.
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- 2021
24. High order conservative positivity-preserving discontinuous Galerkin method for stationary hyperbolic equations
- Author
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Ziyao Xu and Chi-Wang Shu
- Subjects
Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Computer Science Applications - Published
- 2022
25. An improved simple WENO limiter for discontinuous Galerkin methods solving hyperbolic systems on unstructured meshes
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Jie Du, Chi-Wang Shu, and Xinghui Zhong
- Subjects
Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Computer Science Applications - Published
- 2022
26. Stability of high order finite difference and local discontinuous Galerkin schemes with explicit-implicit-null time-marching for high order dissipative and dispersive equations
- Author
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Meiqi Tan, Juan Cheng, and Chi-Wang Shu
- Subjects
Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Computer Science Applications - Published
- 2022
27. A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes
- Author
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Jun Zhu and Chi-Wang Shu
- Subjects
Numerical Analysis ,Conservation law ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Series (mathematics) ,Computer science ,Applied Mathematics ,Computation ,Order of accuracy ,Mathematics::Numerical Analysis ,Computer Science Applications ,Computational Mathematics ,Robustness (computer science) ,Modeling and Simulation ,Convergence (routing) ,Applied mathematics ,Polygon mesh - Abstract
In this paper, we continue our work in [46] and propose a new type of high-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. Although termed “multi-resolution WENO schemes”, we only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. We construct new third-order, fourth-order, and fifth-order WENO schemes using three or four unequal-sized central spatial stencils, different from the classical WENO procedure using equal-sized biased/central spatial stencils for the spatial reconstruction. The new WENO schemes could obtain the optimal order of accuracy in smooth regions, and could degrade gradually to first-order of accuracy so as to suppress spurious oscillations near strong discontinuities. This is the first time that only a series of unequal-sized hierarchical central spatial stencils are used in designing arbitrary high-order finite volume WENO schemes on triangular meshes. The main advantages of these schemes are their compactness, robustness, and their ability to maintain good convergence property for steady-state computation. The linear weights of such WENO schemes can be any positive numbers on the condition that they sum to one. Extensive numerical results are provided to illustrate the good performance of these new finite volume WENO schemes.
- Published
- 2019
28. 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.
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- 2019
29. Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes
- Author
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Kailiang Wu and Chi-Wang Shu
- Subjects
Discretization ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,Discontinuous Galerkin method ,FOS: Mathematics ,Applied mathematics ,Polygon mesh ,Mathematics - Numerical Analysis ,0101 mathematics ,Divergence (statistics) ,Instrumentation and Methods for Astrophysics (astro-ph.IM) ,Mathematics ,Finite volume method ,Ideal (set theory) ,Applied Mathematics ,Numerical analysis ,Fluid Dynamics (physics.flu-dyn) ,Physics - Fluid Dynamics ,Numerical Analysis (math.NA) ,Computational Physics (physics.comp-ph) ,3. Good health ,010101 applied mathematics ,Computational Mathematics ,Magnetohydrodynamics ,Astrophysics - Instrumentation and Methods for Astrophysics ,Physics - Computational Physics - Abstract
This paper proposes and analyzes arbitrarily high-order discontinuous Galerkin (DG) and finite volume methods which provably preserve the positivity of density and pressure for the ideal MHD on general meshes. Unified auxiliary theories are built for rigorously analyzing the positivity-preserving (PP) property of MHD schemes with a HLL type flux on polytopal meshes in any space dimension. The main challenges overcome here include establishing relation between the PP property and discrete divergence of magnetic field on general meshes, and estimating proper wave speeds in the HLL flux to ensure the PP property. In 1D case, we prove that the standard DG and finite volume methods with the proposed HLL flux are PP, under condition accessible by a PP limiter. For multidimensional conservative MHD system, standard DG methods with a PP limiter are not PP in general, due to the effect of unavoidable divergence-error. We construct provably PP high-order DG and finite volume schemes by proper discretization of symmetrizable MHD system, with two divergence-controlling techniques: locally divergence-free elements and a penalty term. The former leads to zero divergence within each cell, while the latter controls the divergence error across cell interfaces. Our analysis reveals that a coupling of them is important for positivity preservation, as they exactly contribute the discrete divergence-terms absent in standard DG schemes but crucial for ensuring the PP property. Numerical tests confirm the PP property and the effectiveness of proposed PP schemes. Unlike conservative MHD system, the exact smooth solutions of symmetrizable MHD system are proved to retain the positivity even if the divergence-free condition is not satisfied. Our analysis and findings further the understanding, at both discrete and continuous levels, of the relation between the PP property and the divergence-free constraint., Comment: 49 pages, 11 figures
- Published
- 2019
30. On New Strategies to Control the Accuracy of WENO Algorithms Close to Discontinuities
- Author
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Chi-Wang Shu, Juan Ruiz, Sergio Amat, Fundación Séneca, Ministerio de Economía y Competitividad, and National Science Foundation (NSF)
- Subjects
Signal processing ,Numerical Analysis ,12 Matemáticas ,Applied Mathematics ,Order of accuracy ,Matemática Aplicada ,010103 numerical & computational mathematics ,Classification of discontinuities ,01 natural sciences ,Raising (metalworking) ,Computational Mathematics ,Nonlinear system ,Improved adaption to discontinuities ,New optimal weights ,WENO schemes ,0101 mathematics ,Control (linguistics) ,Algorithm ,Interpolation ,Mathematics - Abstract
This paper is devoted to the construction and analysis of new nonlinear optimal weights for weighted ENO (WENO) interpolation capable of raising the order of accuracy close to discontinuities. The new nonlinear optimal weights are constructed using a strategy inspired by the original WENO algorithm, and they work very well for corner or jump singularities, leading to optimal theoretical accuracy. This is the first part of a series of two papers. In this first part we analyze the performance of the new algorithms proposed for univariate function approximation in the point values (interpolation problem). In the second part, we will extend the analysis to univariate function approximation in the cell averages (reconstruction problem). Our aim is twofold: to raise the order of accuracy of the WENO type interpolation schemes both near discontinuities and in the interval which contains the singularity. The first problem can be solved using the new nonlinear optimal weights, but the second one requires a new strategy that locates the position of the singularity inside the cell in order to attain adaption. This new strategy is inspired by the ENO-SR schemes proposed by Harten [J. Comput. Phys., 83 (1989), pp. 148--184]. Thus, we will introduce two different algorithms in the point values. The first one can deal with corner singularities and jump discontinuities for intervals not containing the singularity. The second algorithm can also deal with intervals containing corner singularities, as they can be detected from the point values, but jump discontinuities cannot, as the information of their position is lost during the discretization process. As mentioned before, the second part of this work will be devoted to the cell averages and, in this context, it will be possible to work with jump discontinuities as well. The work of the authors was supported by the Programa de Apoyo a la Investigatión de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18, by the national research project MTM2015-64382-P (MINECO/FEDER), and by National Science Foundation grant DMS-1719410.
- Published
- 2019
31. Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws
- Author
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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
32. Strong Stability of Explicit Runge--Kutta Time Discretizations
- Author
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Zheng Sun and Chi-Wang Shu
- Subjects
Physics::Computational Physics ,Numerical Analysis ,Conservation law ,Applied Mathematics ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Computer Science::Numerical Analysis ,01 natural sciences ,Stability (probability) ,Mathematics::Numerical Analysis ,Computational Mathematics ,Runge–Kutta methods ,FOS: Mathematics ,Energy method ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Abstract
Motivated by studies on fully discrete numerical schemes for linear hyperbolic conservation laws, we present a framework on analyzing the strong stability of explicit Runge-Kutta (RK) time discretizations for semi-negative autonomous linear systems. The analysis is based on the energy method and can be performed with the aid of a computer. Strong stability of various RK methods, including a sixteen-stage embedded pair of order nine and eight, has been examined under this framework. Based on numerous numerical observations, we further characterize the features of strongly stable schemes. A both necessary and sufficient condition is given for the strong stability of RK methods of odd linear order.
- Published
- 2019
33. High order entropy stable and positivity-preserving discontinuous Galerkin method for the nonlocal electron heat transport model
- Author
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Nuo Lei, Juan Cheng, and Chi-Wang Shu
- Subjects
Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Computer Science Applications - Published
- 2022
34. A new type of multi-resolution WENO schemes with increasingly higher order of accuracy
- Author
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Jun Zhu and Chi-Wang Shu
- Subjects
Physics::Computational Physics ,Numerical Analysis ,Conservation law ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Computer science ,Applied Mathematics ,Finite difference ,Order of accuracy ,010103 numerical & computational mathematics ,Classification of discontinuities ,Computer Science::Numerical Analysis ,01 natural sciences ,Mathematics::Numerical Analysis ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Robustness (computer science) ,Multi resolution ,Modeling and Simulation ,Applied mathematics ,0101 mathematics ,Spurious oscillations - Abstract
In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in [25] , [45] , could obtain the optimal order of accuracy in smooth regions, and could simultaneously suppress spurious oscillations near discontinuities. The linear weights of such WENO schemes can be any positive numbers on the condition that their sum equals one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference and finite volume WENO schemes. These new WENO schemes are simple to construct and can be easily implemented to arbitrary high order of accuracy and in higher dimensions. Benchmark examples are given to demonstrate the robustness and good performance of these new WENO schemes.
- Published
- 2018
35. 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
36. A Foreword to the Special Issue in Honor of Professor Bernardo Cockburn on His 60th Birthday: A Life Time of Discontinuous Schemings
- Author
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Chi-Wang Shu, Bo Dong, and Yanlai Chen
- Subjects
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
37. 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
38. 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
39. 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
40. A new WENO-2r algorithm with progressive order of accuracy close to discontinuities
- Author
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Juan Ruiz, Chi-Wang Shu, Sergio Amat, and Dionisio F. Yáñez
- Subjects
Numerical Analysis ,Discretization ,Generalization ,Applied Mathematics ,Order of accuracy ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,Classification of discontinuities ,01 natural sciences ,Computational Mathematics ,FOS: Mathematics ,Point (geometry) ,Mathematics - Numerical Analysis ,0101 mathematics ,Algorithm ,Mathematics - Abstract
In this article we present a modification of the algorithm for data discretized in the point values introduced in [S. Amat, J. Ruiz, C.-W. Shu, On a new WENO algorithm of order 2r with improved accuracy close to discontinuities, App. Math. Lett. 105 (2020), 106-298]. In the aforementioned work, we managed to obtain an algorithm that reaches a progressive and optimal order of accuracy close to discontinuities for WENO-6. For higher orders, i.e. WENO-8, WENO-10, etc. We have found that the previous algorithm presents some shadows in the detection of discontinuities, meaning that the order of accuracy is better than the one attained by WENO of the same order, but not optimal. In this article we present a modification of the smoothness indicators used in the original algorithm, oriented to solve this problem and to attain a WENO-2r algorithm with progressive order of accuracy close to the discontinuities. We also present proofs for the accuracy and explicit formulas for all the weights used for any order 2r of the algorithm., Comment: 21 pages, 12 tables and 1 figure
- Published
- 2020
- Full Text
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41. Bounded and compact weighted essentially nonoscillatory limiters for discontinuous Galerkin schemes: Triangular elements
- Author
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Chi-Wang Shu, Vincent Perrier, Alireza Mazaheri, NASA Langley Research Center [Hampton] (LaRC), Brown University, Computational AGility for internal flows sImulations and compaRisons with Experiments (CAGIRE), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Pau et des Pays de l'Adour (UPPA), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Polynomial ,Physics and Astronomy (miscellaneous) ,Mach reflection ,Unstructured meshes ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics::Numerical Analysis ,Positivity-preserving ,symbols.namesake ,Discontinuous Galerkin method ,Inviscid flow ,Riemann problem Shu-Osher ,Applied mathematics ,[PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph] ,0101 mathematics ,Mathematics ,Numerical Analysis ,[PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph] ,Applied Mathematics ,Order of accuracy ,Computer Science Applications ,CWENO ,010101 applied mathematics ,High-order DG ,Computational Mathematics ,Riemann hypothesis ,Riemann problem ,Modeling and Simulation ,Bounded function ,symbols ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
International audience; Two new classes of compact weighted essentially nonoscillatory (WENO) polynomial limiters are presented for second-, third-, fourth-, and fifth-order discontinuous Galerkin (DG) schemes on irregular simplex elements. The presented WENO-DG procedures are extensions of the high-order WENO finite-volume and finite-difference schemes of Zhu and Shu (2017) [25], (2019) [26] to high-order unstructured DG schemes. A compact positivity preserving limiter is applied to the solutions to ensure pressure and density remain within physical ranges at all time. It is then verified that the bounded WENO-DG maintains the formal order of accuracy of the underlying DG schemes in the smooth regions. The performance of the proposed WENO-DG is also demonstrated with inviscid test cases including the classical Riemann problems, shock-turbulence interaction, scramjet, blunt body flows, and the double Mach Reflection problems.
- Published
- 2019
42. Numerical solutions of stochastic PDEs driven by arbitrary type of noise
- Author
-
Tianheng Chen, Chi-Wang Shu, and Boris Rozovskii
- Subjects
Statistics and Probability ,Partial differential equation ,Polynomial chaos ,Truncation error (numerical integration) ,Applied Mathematics ,Numerical analysis ,Noise (electronics) ,Stochastic partial differential equation ,symbols.namesake ,Rate of convergence ,Gaussian noise ,Modeling and Simulation ,symbols ,Applied mathematics ,Mathematics - Abstract
So far the theory and numerical practice of stochastic partial differential equations (SPDEs) have dealt almost exclusively with Gaussian noise or Levy noise. Recently, Mikulevicius and Rozovskii (Stoch Partial Differ Equ Anal Comput 4:319–360, 2016) proposed a distribution-free Skorokhod–Malliavin calculus framework that is based on generalized stochastic polynomial chaos expansion, and is compatible with arbitrary driving noise. In this paper, we conduct systematic investigation on numerical results of these newly developed distribution-free SPDEs, exhibiting the efficiency of truncated polynomial chaos solutions in approximating moments and distributions. We obtain an estimate for the mean square truncation error in the linear case. The theoretical convergence rate, also verified by numerical experiments, is exponential with respect to polynomial order and cubic with respect to number of random variables included.
- Published
- 2018
43. On local conservation of numerical methods for conservation laws
- Author
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Chi-Wang Shu and Cengke Shi
- Subjects
Conservation law ,Property (philosophy) ,Lax–Wendroff theorem ,General Computer Science ,Continuous galerkin ,Numerical analysis ,General Engineering ,Prove it ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Calculus ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper we introduce a definition of the local conservation property for numerical methods solving time dependent conservation laws, which generalizes the classical local conservation definition. The motivation of our definition is the Lax–Wendroff theorem, and thus we prove it for locally conservative numerical schemes per our definition in one and two space dimensions. Several numerical methods, including continuous Galerkin methods and compact schemes, which do not fit the classical local conservation definition, are given as examples of locally conservative methods under our generalized definition.
- Published
- 2018
44. Bound-preserving modified exponential Runge–Kutta discontinuous Galerkin methods for scalar hyperbolic equations with stiff source terms
- Author
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Chi-Wang Shu and Juntao Huang
- Subjects
Numerical Analysis ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Scalar (mathematics) ,Stiffness ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics::Numerical Analysis ,Computer Science Applications ,Exponential function ,010101 applied mathematics ,Computational Mathematics ,Runge–Kutta methods ,Discontinuous Galerkin method ,Modeling and Simulation ,medicine ,Applied mathematics ,Polygon mesh ,0101 mathematics ,medicine.symptom ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper, we develop bound-preserving modified exponential Runge–Kutta (RK) discontinuous Galerkin (DG) schemes to solve scalar hyperbolic equations with stiff source terms by extending the idea in Zhang and Shu [43] . Exponential strong stability preserving (SSP) high order time discretizations are constructed and then modified to overcome the stiffness and preserve the bound of the numerical solutions. It is also straightforward to extend the method to two dimensions on rectangular and triangular meshes. Even though we only discuss the bound-preserving limiter for DG schemes, it can also be applied to high order finite volume schemes, such as weighted essentially non-oscillatory (WENO) finite volume schemes as well.
- Published
- 2018
45. Conservative High Order Positivity-Preserving Discontinuous Galerkin Methods for Linear Hyperbolic and Radiative Transfer Equations
- Author
-
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
46. Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes
- Author
-
Chi-Wang Shu, Yong Liu, and Mengping Zhang
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Discretization ,Summation by parts ,Applied Mathematics ,Mathematical analysis ,Godunov's scheme ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics::Numerical Analysis ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Discontinuous Galerkin method ,Modeling and Simulation ,Total variation diminishing ,Bounded function ,Compressibility ,Dissipative system ,0101 mathematics ,Mathematics - Abstract
We present a discontinuous Galerkin (DG) scheme with suitable quadrature rules [15] for ideal compressible magnetohydrodynamic (MHD) equations on structural meshes. The semi-discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by Godunov [32] , the entropy stable DG framework with suitable quadrature rules [15] , the entropy conservative flux in [14] inside each cell and the entropy dissipative approximate Godunov type numerical flux at cell interfaces to make the scheme entropy stable. The main difficulty in the generalization of the results in [15] is the appearance of the non-conservative “source terms” added in the modified MHD model introduced by Godunov [32] , which do not exist in the general hyperbolic system studied in [15] . Special care must be taken to discretize these “source terms” adequately so that the resulting DG scheme satisfies entropy stability. Total variation diminishing / bounded (TVD/TVB) limiters and bound-preserving limiters are applied to control spurious oscillations. We demonstrate the accuracy and robustness of this new scheme on standard MHD examples.
- Published
- 2018
47. Superconvergence of Discontinuous Galerkin Method for Scalar Nonlinear Hyperbolic Equations
- Author
-
Waixiang Cao, Yang Yang, Chi-Wang Shu, and Zhimin Zhang
- Subjects
Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,Scalar (mathematics) ,010103 numerical & computational mathematics ,Superconvergence ,Computer Science::Numerical Analysis ,01 natural sciences ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Exact solutions in general relativity ,Discontinuous Galerkin method ,Bounded function ,Piecewise ,0101 mathematics ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper, we study the superconvergence behavior of the semi-discrete discontinuous Galerkin (DG) method for scalar nonlinear hyperbolic equations in one spatial dimension. Superconvergence results for problems with fixed and alternating wind directions are established. On the one hand, we prove that, if the wind direction is fixed (i.e., the derivative of the flux function is bounded away from zero), both the cell average error and numerical flux error at cell interfaces converge at a rate of $2k+1$ when upwind fluxes and piecewise polynomials of degree $k$ are used. Moreover, we also prove that the function value approximation of the DG solution is superconvergent at interior right Radau points, and the derivative value approximation is superconvergent at interior left Radau points, with an order of k+2 and k+1, respectively. As a byproduct, we show a (k+2)th order superconvergence of the DG solution towards the Gauss--Radau projection of the exact solution. On the other hand, superconvergence resu...
- Published
- 2018
48. Optimal Error Estimates of the Semidiscrete Central Discontinuous Galerkin Methods for Linear Hyperbolic Equations
- Author
-
Yong Liu, Chi-Wang Shu, and Mengping Zhang
- Subjects
Numerical Analysis ,Conservation law ,Degree (graph theory) ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics::Numerical Analysis ,law.invention ,010101 applied mathematics ,Computational Mathematics ,Dimension (vector space) ,Discontinuous Galerkin method ,law ,Piecewise ,Applied mathematics ,Polygon mesh ,Cartesian coordinate system ,0101 mathematics ,Hyperbolic partial differential equation ,Mathematics - Abstract
We analyze the central discontinuous Galerkin method for time-dependent linear conservation laws. In one dimension, optimal a priori $L^2$ error estimates of order $k+1$ are obtained for the semidiscrete scheme when piecewise polynomials of degree at most $k$ ($k\geq0$) are used on overlapping uniform meshes. We then extend the analysis to multidimensions on uniform Cartesian meshes when piecewise tensor-product polynomials are used on overlapping meshes. Numerical experiments are given to demonstrate the theoretical results.
- Published
- 2018
49. Multi-resolution HWENO schemes for hyperbolic conservation laws
- Author
-
Jianxian Qiu, Chi-Wang Shu, and Jiayin Li
- Subjects
Physics::Computational Physics ,Numerical Analysis ,Conservation law ,Hermite polynomials ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Finite difference ,Order of accuracy ,Function (mathematics) ,Computer Science Applications ,Computational Mathematics ,Robustness (computer science) ,Modeling and Simulation ,Applied mathematics ,Polygon mesh ,Mathematics - Abstract
In this paper, a new type of high-order finite volume and finite difference multi-resolution Hermite weighted essentially non-oscillatory (HWENO) schemes are designed for solving hyperbolic conservation laws on structured meshes. Here we only use the information defined on a hierarchy of nested central spatial stencils but do not introduce any equivalent multi-resolution representation, the terminology of multi-resolution HWENO follows that of the multi-resolution WENO schemes (Zhu and Shu, 2018) [29] . The main idea of our spatial reconstruction is derived from the original HWENO schemes (Qiu and Shu, 2004) [19] , in which both the function and its first-order derivative values are evolved in time and used in the reconstruction. Our HWENO schemes use the same large stencils as the classical HWENO schemes which are narrower than the stencils of the classical WENO schemes for the same order of accuracy. Only the function values need to be reconstructed by our HWENO schemes, the first-order derivative values are obtained from the high-order linear polynomials directly. Furthermore, the linear weights of such HWENO schemes can be any positive numbers as long as their sum equals one, and there is no need to do any modification or positivity-preserving flux limiting in our numerical experiments. Extensive benchmark examples are performed to illustrate the robustness and good performance of such finite volume and finite difference HWENO schemes.
- Published
- 2021
50. A high order conservative finite difference scheme for compressible two-medium flows
- Author
-
Jianxian Qiu, Chi-Wang Shu, and Feng Zheng
- Subjects
Scheme (programming language) ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Interface (Java) ,Computer science ,Applied Mathematics ,Finite difference ,Order (ring theory) ,Computer Science Applications ,Computational Mathematics ,Modeling and Simulation ,Finite difference scheme ,Compressibility ,Applied mathematics ,Algebraic function ,computer ,computer.programming_language ,Interpolation - Abstract
In this paper, a high order finite difference conservative scheme is proposed to solve two-medium flows. Our scheme has four advantages: First, our scheme is conservative, which is important to ensure the numerical solution captures the main features properly. Second, our scheme directly applies the WENO interpolation method to the primitive variables so that it can maintain the equilibrium of velocity and pressure across the interface, which is very helpful to obtain a non-oscillatory solution. Third, the usage of nodal values enables us to manipulate algebraic functions easily. Fourth, the scheme can maintain high order accuracy when the solution is smooth. Extensive numerical experiments are performed to verify the high resolution and non-oscillatory performance of this new scheme.
- Published
- 2021
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