373 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
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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. Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation
- Author
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Haijin Wang, Fengyan Li, Chi-Wang Shu, and Qiang Zhang
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Computational Mathematics ,Algebra and Number Theory ,Applied Mathematics - Abstract
In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the L 2 L^2 norm of the numerical solution does not increase in time, under the time step condition τ ≤ F ( h / c , d / c 2 ) \tau \le \mathcal {F}(h/c, d/c^2) , with the convection coefficient c c , the diffusion coefficient d d , and the mesh size h h . The function F \mathcal {F} depends on the specific IMEX temporal method, the polynomial degree k k of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes τ ≲ h / c \tau \lesssim h/c in the convection-dominated regime and it becomes τ ≲ d / c 2 \tau \lesssim d/c^2 in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.
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- 2023
5. RKDG Methods with Multi-resolution WENO Limiters for Solving Steady-State Problems on Triangular Meshes
- Author
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Jun Zhu, Chi-Wang Shu, and Jianxian Qiu
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Computational Mathematics ,Applied Mathematics - Published
- 2023
6. A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws
- Author
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Liang Li, Jun Zhu, Chi-Wang Shu, and Yong-Tao Zhang
- Subjects
Computational Mathematics ,Applied Mathematics ,MathematicsofComputing_NUMERICALANALYSIS - Abstract
Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifth-order fixed-point fast sweeping WENO method with an ILW procedure to solve steady-state solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show high-order accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.
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- 2022
7. An Essentially Oscillation-Free Discontinuous Galerkin Method for Hyperbolic Systems
- Author
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Yong Liu, Jianfang Lu, and Chi-Wang Shu
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Computational Mathematics ,Applied Mathematics - Published
- 2022
8. Multi-layer Perceptron Estimator for the Total Variation Bounded Constant in Limiters for Discontinuous Galerkin Methods
- Author
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Xinyue Yu and Chi-Wang Shu
- Subjects
Maxima and minima ,Nonlinear system ,Discontinuous Galerkin method ,Multilayer perceptron ,Piecewise ,Applied mathematics ,Constant (mathematics) ,Hyperbolic partial differential equation ,Mathematics ,Numerical partial differential equations - Abstract
The discontinuous Galerkin (DG) method is widely used in numerical solution of partial differential equations, especially for hyperbolic equations. However, for problems containing strong shocks, the DG method often needs to be supplemented by a limiter to control spurious oscillations and to ensure nonlinear stability. The total variation bounded (TVB) limiter is a popular choice and can maintain the original high order accuracy of the DG scheme in smooth regions and keep a sharp and non-oscillatory discontinuity transition, when a certain TVB constant M is chosen adequately. For scalar conservation laws, suitable choice of this constant M can be based on solid mathematical analysis. However, for nonlinear hyperbolic systems, there is no rigorous mathematical guiding principle for the determination of this constant, and numerical experiments often use ad hoc choices based on experience and through trial and error. In this paper, we develop a TVB constant artificial neural network (ANN) based estimator by constructing a multi-layer perceptron (MLP) model. We generate the training data set by constructing piecewise smooth functions containing local maxima, local minima, and discontinuities. By using the supervised learning strategy, the MLP model is trained offline. The proposed method gives the TVB constant M with robust performance to capture sharp and non-oscillatory shock transitions while maintaining the original high order accuracy in smooth regions. Numerical results using this new estimator in the TVB limiter for DG methods in one and two dimensions are given, and its performance is compared with the classical ad hoc choices of this TVB constant.
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- 2021
9. High-Resolution Viscous Terms Discretization and ILW Solid Wall Boundary Treatment for the Navier–Stokes Equations
- Author
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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.
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- 2021
10. Preface to the Focused Issue in Honor of Professor Tong Zhang on the Occasion of His 90th Birthday
- Author
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Jiequan Li, Wancheng Sheng, Chi-Wang Shu, Ping Zhang, and Yuxi Zheng
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Computational Mathematics ,Applied Mathematics - Published
- 2022
11. 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
12. On the approximation of derivative values using a WENO algorithm with progressive order of accuracy close to discontinuities
- Author
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Sergio Amat, Juan Ruiz-Álvarez, Chi-Wang Shu, and Dionisio F. Yáñez
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Computational Mathematics ,Applied Mathematics - Abstract
In this article, we introduce a new WENO algorithm that aims to calculate an approximation to derivative values of a function in a non-regular grid. We adapt the ideas presented in [Amat et al., SIAM J. Numer. Anal. (2020)] to design the nonlinear weights in a manner such that the order of accuracy is maximum in the intervals close to the discontinuities. Some proofs, remarks on the choice of the stencils and explicit formulas for the weights and smoothness indicators are given. We also present some numerical experiments to confirm the theoretical results.
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- 2022
13. 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.
- Published
- 2021
14. 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
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Computational Mathematics ,Numerical Analysis ,Computational Theory and Mathematics ,Applied Mathematics ,General Engineering ,Software ,Theoretical Computer Science - Published
- 2022
15. Development and analysis of two new finite element schemes for a time-domain carpet cloak model
- Author
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Jichun Li, Chi-Wang Shu, and Wei Yang
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Computational Mathematics ,Applied Mathematics - Published
- 2022
16. 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...
- Published
- 2021
17. 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.
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- 2021
18. 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)
- 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
19. 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.
- Published
- 2021
20. A high order positivity-preserving polynomial projection remapping method
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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
21. Local discontinuous Galerkin methods for diffusive–viscous wave equations
- Author
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Dan Ling, Chi-Wang Shu, and Wenjing Yan
- Subjects
Computational Mathematics ,Applied Mathematics - Published
- 2023
22. 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
- Subjects
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.
- Published
- 2023
23. 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
- Subjects
Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Computer Science Applications - Published
- 2023
24. 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
- Subjects
Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Computer Science Applications - Published
- 2023
25. 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
26. 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
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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
27. A Sequel of Inverse Lax–Wendroff High Order Wall Boundary Treatment for Conservation Laws
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Sirui Tan, Nicholas Dicati Pereira da Silva, Francisco Augusto Aparecido Gomes, Chi-Wang Shu, and Rafael Brandão de Rezende Borges
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Discretization ,Computer science ,business.industry ,Lax–Wendroff method ,Applied Mathematics ,Finite difference method ,Boundary (topology) ,02 engineering and technology ,Computational fluid dynamics ,Solver ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Inviscid flow ,0202 electrical engineering, electronic engineering, information engineering ,Oblique shock ,Applied mathematics ,020201 artificial intelligence & image processing ,0101 mathematics ,business - Abstract
When solving CFD problems, the solver, or the numerical code, plays an important role. Depending on the phenomena and problem domain, designing such numerical codes can be hard work. One strategy is to start with simple problems and construct the code as building blocks. The purpose of this work is to provide a detailed review of the theory to compute analytical and exact solutions, and recent numerical methods to construct a code to solve compressible and inviscid fluid flows with high-resolution, arbitrary domains, non-linear phenomena, and on rectangular meshes. We also propose a modification to the inverse Lax–Wendroff procedure solid wall treatment and two-dimensional WENO-type extrapolation stencil selection and weights to handle more generic situations. To test our modifications, we use the finite difference method, Lax–Friedrichs splitting, WENO-Z+ scheme, and third-order strong stability preserving Runge-Kutta time discretization. Our first problem is a simple one-dimensional transient problem with periodic boundary conditions, which is useful for constructing the core solver. Then, we move to the one-dimensional Rayleigh flow, which can handle flows with heat exchange and requires more detailed boundary treatment. The next problem is the quasi-one-dimensional nozzle flow with and without shock, where the boundary treatment needs a few adjustments. The first two-dimensional problem is the Ringleb flow, and despite being smooth, it has a curved wall as the left boundary. Finally, the last problem is a two-dimensional conical flow, which presents an oblique shock and an inclined straight line wall being the cone surface. We show that the designed accuracy is being reached for smooth problems, that high-resolution is being attained for non-smooth problems, and that our modifications produce similar results while providing a more generic way to treat solid walls.
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- 2020
28. High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters on triangular meshes
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Jianxian Qiu, Jun Zhu, and Chi-Wang Shu
- Subjects
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.
- Published
- 2020
29. Review Article:Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes
- Author
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Tianheng Chen Chi-Wang Shu
- Subjects
Conservation law ,Simplex ,Discontinuous Galerkin method ,Applied mathematics ,Polygon mesh ,General Medicine ,Mathematics - Published
- 2020
30. 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
31. 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
- 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
32. A Discontinuous Galerkin Method for Stochastic Conservation Laws
- Author
-
Chi-Wang Shu, Shanjian Tang, and Yunzhang Li
- Subjects
Computational Mathematics ,Conservation law ,Discontinuous Galerkin method ,Applied Mathematics ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Itō's lemma ,01 natural sciences ,Stability (probability) ,Mathematics - Abstract
In this paper we present a discontinuous Galerkin (DG) method to approximate stochastic conservation laws, which is an efficient high-order scheme. We study the stability for the semidiscrete DG me...
- Published
- 2020
33. 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
34. Existence and Computation of Solutions of a Model of Traffic Involving Hysteresis
- Author
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Chi-Wang Shu and Haitao Fan
- Subjects
Hysteresis ,Applied Mathematics ,Computation ,Total variation diminishing ,Applied mathematics ,Upwind scheme ,Two-phase flow ,Traffic flow ,Borel measure ,Hyperbolic systems ,Mathematics - Abstract
The meaning of weak solutions of a nonconservative hyperbolic system with discontinuous coefficients modeling traffic flows involving hysteresis is defined. An upwinding approximation scheme for th...
- Published
- 2020
35. 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
36. Multi-symplectic discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise
- Author
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Jiawei Sun, Chi-Wang Shu, and Yulong Xing
- Subjects
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.
- Published
- 2021
37. 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
38. An improved simple WENO limiter for discontinuous Galerkin methods solving hyperbolic systems on unstructured meshes
- Author
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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
39. Bound-Preserving High Order Schemes for Hyperbolic Equations: Survey and Recent Developments
- Author
-
Chi-Wang Shu
- Subjects
symbols.namesake ,Conservation law ,Maximum principle ,Finite volume method ,Discontinuous Galerkin method ,MathematicsofComputing_NUMERICALANALYSIS ,symbols ,Finite difference ,Applied mathematics ,Flux limiter ,Hyperbolic partial differential equation ,Euler equations ,Mathematics - Abstract
Solutions to many hyperbolic equations have convex invariant regions, for example, solutions to scalar conservation laws satisfy the maximum principle, solutions to compressible Euler equations satisfy the positivity-preserving property for density and internal energy. It is, however, a challenge to design schemes whose solutions also honor such invariant regions. This is especially the case for high-order accurate schemes. In this contribution, we survey strategies in the recent literature to design high-order bound-preserving schemes, including a general framework in constructing high-order bound-preserving finite volume and discontinuous Galerkin schemes for scalar and systems of hyperbolic equations through a simple scaling limiter and a convex combination argument based on first-order bound-preserving building blocks, and various flux limiters to design high-order bound-preserving finite difference schemes. We also discuss a few recent developments, including high-order bound-preserving schemes for relativistic hydrodynamics, high-order discontinuous Galerkin Lagrangian schemes, and high-order discontinuous Galerkin methods for radiative transfer equations.
- Published
- 2021
40. 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
41. Preface to Focused Issue on Discontinuous Galerkin Methods
- Author
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Chi-Wang Shu, Yan Xu, Jennifer K. Ryan, Zhimin Zhang, Jan S. Hesthaven, Jaap van der Vegt, Qiang Zhang, Mathematics of Computational Science, and MESA+ Institute
- Subjects
Computer science ,Discontinuous Galerkin method ,General Earth and Planetary Sciences ,Computational Science and Engineering ,Applied mathematics ,n/a OA procedure ,General Environmental Science - Abstract
The discontinuous Galerkin (DG) method is a class of finite element methods using completely discontinuous piecewise smooth functions (typically polynomials) as basis and test functions. Since its inception in 1973 [10], it has seen a sustained development, both in the computational mathematics community and in many scientific and engineering application communities. The DG methods have several advantages, such as its extreme flexibility in dealing with complex geometry and adaptive computation (both h- and p-adaptivities are easy to implement), extremely high parallel efficiency, good stability properties (energy and entropy stability has been established for DG methods in many situations), nice convergence and superconvergence properties, and capability to solve hyperbolic and convection- dominated problems effectively.
- Published
- 2022
42. 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
43. A brief review on the convergence to steady state solutions of Euler equations with high-order WENO schemes
- Author
-
Shuhai Zhang, Jun Zhu, and Chi-Wang Shu
- Subjects
Steady state (electronics) ,Computer science ,lcsh:Motor vehicles. Aeronautics. Astronautics ,WENO compact scheme ,01 natural sciences ,Projection (linear algebra) ,010305 fluids & plasmas ,symbols.namesake ,WENO scheme ,Smoothness indicator ,0103 physical sciences ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Steady state solution ,Order of accuracy ,General Medicine ,Shock (mechanics) ,Euler equations ,010101 applied mathematics ,Flow (mathematics) ,lcsh:TA1-2040 ,symbols ,lcsh:TL1-4050 ,Convergence ,lcsh:Engineering (General). Civil engineering (General) ,Interpolation - Abstract
Weighted essentially non-oscillatory (WENO) schemes are a class of high-order shock capturing schemes which have been designed and applied to solve many fluid dynamics problems to study the detailed flow structures and their evolutions. However, like many other high-order shock capturing schemes, WENO schemes also suffer from the problem that it can not easily converge to a steady state solution if there is a strong shock wave. This is a long-standing difficulty for high-order shock capturing schemes. In recent years, this non-convergence problem has been studied extensively for WENO schemes. Numerical tests show that the key reason of the non-convergence to steady state is the slight post shock oscillations, which are at the small local truncation error level but prevent the residue to settle down to machine zero. Several strategies have been proposed to reduce these slight post shock oscillations, including the design of new smoothness indicators for the fifth-order WENO scheme, the development of a high-order weighted interpolation in the procedure of the local characteristic projection for WENO schemes of higher order of accuracy, and the design of a new type of WENO schemes. With these strategies, the convergence to steady states is improved significantly. Moreover, the strategies are applicable to other types of weighted schemes. In this paper, we give a brief review on the topic of convergence to steady state solutions for WENO schemes applied to Euler equations.
- Published
- 2019
44. Certified Offline-Free Reduced Basis (COFRB) Methods for Stochastic Differential Equations Driven by Arbitrary Types of Noise
- Author
-
Chi-Wang Shu, Tianheng Chen, Yong Liu, and Yanlai Chen
- Subjects
Numerical Analysis ,Basis (linear algebra) ,Differential equation ,Applied Mathematics ,Gaussian ,General Engineering ,Ode ,01 natural sciences ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Noise ,Stochastic differential equation ,symbols.namesake ,Computational Theory and Mathematics ,Robustness (computer science) ,Component (UML) ,symbols ,0101 mathematics ,Algorithm ,Software ,Mathematics - Abstract
In this paper, we propose, analyze, and implement a new reduced basis method (RBM) tailored for the linear (ordinary and partial) differential equations driven by arbitrary (i.e. not necessarily Gaussian) types of noise. There are four main ingredients of our algorithm. First, we propose a new space-time-like treatment of time in the numerical schemes for ODEs and PDEs. The second ingredient is an accurate yet efficient compression technique for the spatial component of the space-time snapshots that the RBM is adopting as bases. The third ingredient is a non-conventional “parameterization” of a non-parametric problem. The last is a RBM that is free of any dedicated offline procedure yet is still efficient online. The numerical experiments verify the effectiveness and robustness of our algorithms for both types of differential equations.
- Published
- 2019
45. Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes
- Author
-
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
46. High order finite difference hermite WENO schemes for the Hamilton–Jacobi equations on unstructured meshes
- Author
-
Chi-Wang Shu, Feng Zheng, and Jianxian Qiu
- Subjects
Polynomial ,Hermite polynomials ,General Computer Science ,Compact stencil ,General Engineering ,Finite difference ,01 natural sciences ,Hamilton–Jacobi equation ,Stability (probability) ,010305 fluids & plasmas ,010101 applied mathematics ,0103 physical sciences ,Applied mathematics ,Node (circuits) ,Polygon mesh ,0101 mathematics ,Mathematics - Abstract
In this paper, a new type of high order Hermite weighted essentially non-oscillatory (HWENO) methods is proposed to solve the Hamilton–Jacobi (HJ) equations on unstructured meshes. We use a fourth order accurate scheme to demonstrate our procedure. Both the solution and its spatial derivatives are evolved in time. Our schemes have three advantages. First, they are more compact than the one in [38] as more information is used at each node which allows us to achieve the same high order accuracy with a more compact stencil. Second, the new HWENO approximation on the unstructured mesh allows arbitrary positive linear weights, which enhances the stability of our scheme. Third, the new HWENO procedure produces an approximation polynomial on each triangle, which allows us to compute all the spatial derivatives at the three nodes of each triangle based on this single polynomial, instead of computing each derivative individually with different linear weights in the classical HWENO framework, which improves the efficiency of our scheme. Extensive numerical experiments are performed to verify the accuracy, high resolution and efficiency of this new scheme.
- Published
- 2019
47. Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations
- Author
-
Yong Liu, Chi-Wang Shu, and Mengping Zhang
- Subjects
Exact solutions in general relativity ,Discretization ,Discontinuous Galerkin method ,Piecewise ,General Earth and Planetary Sciences ,Applied mathematics ,Order (ring theory) ,Superconvergence ,Hyperbolic partial differential equation ,Projection (linear algebra) ,Mathematics::Numerical Analysis ,General Environmental Science ,Mathematics - Abstract
In this paper, we study the superconvergence properties of the energy-conserving discontinuous Galerkin (DG) method in [18] for one-dimensional linear hyperbolic equations. We prove the approximate solution superconverges to a particular projection of the exact solution. The order of this superconvergence is proved to be $$k+2$$ when piecewise $$\mathbb {P}^k$$ polynomials with $$k \ge 1$$ are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise $$\mathbb {P}^k$$ polynomials with arbitrary $$k \ge 1$$ . Furthermore, we find that the derivative and function value approximations of the DG solution are superconvergent at a class of special points, with an order of $$k+1$$ and $$k+2$$ , respectively. We also prove, under suitable choice of initial discretization, a ( $$2k+1$$ )-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages. Numerical experiments are given to demonstrate these theoretical results.
- Published
- 2019
48. An energy-conserving ultra-weak discontinuous Galerkin method for the generalized Korteweg–de Vries equation
- Author
-
Guosheng Fu and Chi-Wang Shu
- Subjects
Convection ,Degree (graph theory) ,Applied Mathematics ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Dimension (vector space) ,Discontinuous Galerkin method ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,Order (group theory) ,Polygon mesh ,Mathematics - Numerical Analysis ,0101 mathematics ,Korteweg–de Vries equation ,Mathematics - Abstract
We propose an energy-conserving ultra-weak discontinuous Galerkin (DG) method for the generalized Korteweg-De Vries(KdV) equation in one dimension. Optimal a priori error estimate of order $k + 1$ is obtained for the semi-discrete scheme for the KdV equation without convection term on general nonuniform meshes when polynomials of degree $k\ge 2$ is used. We also numerically observed optimal convergence of the method for the KdV equation with linear or nonlinear convection terms. It is numerically observed for the new method to have a superior performance for long-time simulations over existing DG methods., Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1804.10307
- Published
- 2019
49. On New Strategies to Control the Accuracy of WENO Algorithms Close to Discontinuities
- Author
-
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
50. 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
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