107 results on '"Chi-Wang Shu"'
Search Results
2. 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
3. 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
4. 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
5. 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
6. 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
7. 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
8. 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
9. 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
10. 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
11. 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
12. Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes
- Author
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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
13. Multi-resolution HWENO schemes for hyperbolic conservation laws
- Author
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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
14. A high order conservative finite difference scheme for compressible two-medium flows
- Author
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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
15. Numerical study on the convergence to steady state solutions of a new class of high order WENO schemes
- Author
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Chi-Wang Shu and Jun Zhu
- Subjects
Numerical Analysis ,Steady state (electronics) ,Physics and Astronomy (miscellaneous) ,Truncation error (numerical integration) ,Applied Mathematics ,Mathematical analysis ,Order of accuracy ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics::Numerical Analysis ,Computer Science Applications ,Euler equations ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Dimension (vector space) ,Modeling and Simulation ,Quartic function ,Convergence (routing) ,symbols ,Convex combination ,0101 mathematics ,Mathematics - Abstract
A new class of high order weighted essentially non-oscillatory (WENO) schemes (Zhu and Qiu, 2016, [50] ) is applied to solve Euler equations with steady state solutions. It is known that the classical WENO schemes (Jiang and Shu, 1996, [23] ) might suffer from slight post-shock oscillations. Even though such post-shock oscillations are small enough in magnitude and do not visually affect the essentially non-oscillatory property, they are truly responsible for the residue to hang at a truncation error level instead of converging to machine zero. With the application of this new class of WENO schemes, such slight post-shock oscillations are essentially removed and the residue can settle down to machine zero in steady state simulations. This new class of WENO schemes uses a convex combination of a quartic polynomial with two linear polynomials on unequal size spatial stencils in one dimension and is extended to two dimensions in a dimension-by-dimension fashion. By doing so, such WENO schemes use the same information as the classical WENO schemes in Jiang and Shu (1996) [23] and yield the same formal order of accuracy in smooth regions, yet they could converge to steady state solutions with very tiny residue close to machine zero for our extensive list of test problems including shocks, contact discontinuities, rarefaction waves or their interactions, and with these complex waves passing through the boundaries of the computational domain.
- Published
- 2017
16. A new troubled-cell indicator for discontinuous Galerkin methods for hyperbolic conservation laws
- Author
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Chi-Wang Shu and Guosheng Fu
- Subjects
Numerical Analysis ,Conservation law ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Hyperbolic systems ,Mathematics::Numerical Analysis ,Computer Science Applications ,Euler equations ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Discontinuous Galerkin method ,Simple (abstract algebra) ,Modeling and Simulation ,symbols ,Polygon mesh ,0101 mathematics ,High order ,Mathematics - Abstract
We introduce a new troubled-cell indicator for the discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. This indicator can be defined on unstructured meshes for high order DG methods and depends only on data from the target cell and its immediate neighbors. It is able to identify shocks without PDE sensitive parameters to tune. Extensive one- and two-dimensional simulations on the hyperbolic systems of Euler equations indicate the good performance of this new troubled-cell indicator coupled with a simple minmod-type TVD limiter for the Runge–Kutta DG (RKDG) methods.
- Published
- 2017
17. Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws
- Author
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Chi-Wang Shu and Tianheng Chen
- Subjects
Numerical Analysis ,Conservation law ,Physics and Astronomy (miscellaneous) ,Summation by parts ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics::Numerical Analysis ,Computer Science Applications ,Numerical integration ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Discontinuous Galerkin method ,Modeling and Simulation ,symbols ,Gaussian quadrature ,0101 mathematics ,Entropy (arrow of time) ,Legendre polynomials ,Mathematics - Abstract
It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws (Jiang and Shu (1994) [39] ) and symmetric hyperbolic systems (Hou and Liu (2007) [36] ), in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and/or when the integration is approximated by a numerical quadrature. In this paper, we develop a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in Carpenter et al. (2014) [5] and Gassner (2013) [19] . The main ingredients are summation-by-parts (SBP) operators derived from Legendre Gauss–Lobatto quadrature, the entropy conservative flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection–diffusion equations. Extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.
- Published
- 2017
18. Finite difference Hermite WENO schemes for the Hamilton–Jacobi equations
- Author
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Chi-Wang Shu, Jianxian Qiu, and Feng Zheng
- Subjects
Numerical Analysis ,Hermite polynomials ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Finite difference ,Field (mathematics) ,010103 numerical & computational mathematics ,01 natural sciences ,Hamilton–Jacobi equation ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Compact space ,Simple (abstract algebra) ,Modeling and Simulation ,Convergence (routing) ,Point (geometry) ,0101 mathematics ,Mathematics - Abstract
In this paper, a new type of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are constructed for solving Hamilton–Jacobi (HJ) equations. Point values of both the solution and its first derivatives are used in the HWENO reconstruction and evolved via time advancing. While the evolution of the solution is still through the classical numerical fluxes to ensure convergence to weak solutions, the evolution of the first derivatives of the solution is through a simple dimension-by-dimension non-conservative procedure to gain efficiency. The main advantages of this new scheme include its compactness in the spatial field and its simplicity in the reconstructions. Extensive numerical experiments in one and two dimensional cases are performed to verify the accuracy, high resolution and efficiency of this new scheme.
- Published
- 2017
19. Maximum-principle-satisfying space-time conservation element and solution element scheme applied to compressible multifluids
- Author
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Chi-Wang Shu, Chih-yung Wen, Matteo Parsani, and Hua Shen
- Subjects
Numerical Analysis ,Conservation law ,Equation of state ,Mathematical optimization ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Space time ,Upwind scheme ,01 natural sciences ,010305 fluids & plasmas ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Maximum principle ,Modeling and Simulation ,0103 physical sciences ,Compressibility ,Applied mathematics ,Flux limiter ,0101 mathematics ,Element (category theory) ,Mathematics - Abstract
A maximum-principle-satisfying space-time conservation element and solution element (CE/SE) scheme is constructed to solve a reduced five-equation model coupled with the stiffened equation of state for compressible multifluids. We first derive a sufficient condition for CE/SE schemes to satisfy maximum-principle when solving a general conservation law. And then we introduce a slope limiter to ensure the sufficient condition which is applicative for both central and upwind CE/SE schemes. Finally, we implement the upwind maximum-principle-satisfying CE/SE scheme to solve the volume-fraction-based five-equation model for compressible multifluids. Several numerical examples are carried out to carefully examine the accuracy, efficiency, conservativeness and maximum-principle-satisfying property of the proposed approach.
- Published
- 2017
20. An efficient class of WENO schemes with adaptive order
- Author
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Sudip K. Garain, Dinshaw S. Balsara, and Chi-Wang Shu
- Subjects
Numerical Analysis ,Mathematical optimization ,Conservation law ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Finite difference ,Order of accuracy ,Lower order ,01 natural sciences ,Stencil ,010305 fluids & plasmas ,Computer Science Applications ,010101 applied mathematics ,Maxima and minima ,Computational Mathematics ,Third order ,Modeling and Simulation ,0103 physical sciences ,0101 mathematics ,Legendre polynomials ,Algorithm ,Mathematics - Abstract
Finite difference WENO schemes have established themselves as very worthy performers for entire classes of applications that involve hyperbolic conservation laws. In this paper we report on two major advances that make finite difference WENO schemes more efficient.The first advance consists of realizing that WENO schemes require us to carry out stencil operations very efficiently. In this paper we show that the reconstructed polynomials for any one-dimensional stencil can be expressed most efficiently and economically in Legendre polynomials. By using Legendre basis, we show that the reconstruction polynomials and their corresponding smoothness indicators can be written very compactly. The smoothness indicators are written as a sum of perfect squares. Since this is a computationally expensive step, the efficiency of finite difference WENO schemes is enhanced by the innovation which is reported here.The second advance consists of realizing that one can make a non-linear hybridization between a large, centered, very high accuracy stencil and a lower order WENO scheme that is nevertheless very stable and capable of capturing physically meaningful extrema. This yields a class of adaptive order WENO schemes, which we call WENO-AO (for adaptive order). Thus we arrive at a WENO-AO(5,3) scheme that is at best fifth order accurate by virtue of its centered stencil with five zones and at worst third order accurate by virtue of being non-linearly hybridized with an r=3 CWENO scheme. The process can be extended to arrive at a WENO-AO(7,3) scheme that is at best seventh order accurate by virtue of its centered stencil with seven zones and at worst third order accurate. We then recursively combine the above two schemes to arrive at a WENO-AO(7,5,3) scheme which can achieve seventh order accuracy when that is possible; graciously drop down to fifth order accuracy when that is the best one can do; and also operate stably with an r=3 CWENO scheme when that is the only thing that one can do. Schemes with ninth order of accuracy are also presented.Several accuracy tests and several stringent test problems are presented to demonstrate that the method works very well.
- Published
- 2016
21. Weighted ghost fluid discontinuous Galerkin method for two-medium problems
- Author
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Yun-Long Liu, Chi-Wang Shu, and A-Man Zhang
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Discretization ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Compressible flow ,Computer Science Applications ,Euler equations ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Riemann problem ,Discontinuous Galerkin method ,Modeling and Simulation ,Compressibility ,symbols ,Applied mathematics ,Convex combination ,0101 mathematics ,Normal ,Mathematics - Abstract
A new interface treating method is proposed to simulate compressible two-medium problems with the Runge-Kutta discontinuous Galerkin (RKDG) method. In the present work, both the Euler equation and the level-set equation are discretized with the RKDG method which is compact and of high-order accuracy. The linearized interface inside an interface cell is recovered by the level-set function. The new solution of this cell is taken as a convex combination of two auxiliary solutions. One is the solution obtained by the RKDG method for a single-medium cell with proper numerical fluxes, and the other one is the intermediate state of the two-medium Riemann problem constructed in the normal direction. The weights of the two auxiliary solutions are carefully chosen according to the location of the interface inside the cell. Thus, it ensures a smooth transition when the interface leaves one cell and enters a neighboring cell. The entropy-fix technique is adopted to minimize the overshoots or undershoots in problems with large entropy ratio across the interface. The scheme is justified in a 1-dimensional situation and extended to 2-dimensional problems. Several 1-dimensional two-medium problems, including both smooth and discontinuous examples, are simulated and compared with exact solutions. Also, three 2-dimensional benchmark problems are simulated to validate the present method in two-medium problems.
- Published
- 2021
22. An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary
- Author
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Chi-Wang Shu, Sirui Tan, Mengping Zhang, and Jianfang Lu
- Subjects
Numerical Analysis ,Conservation law ,Partial differential equation ,Physics and Astronomy (miscellaneous) ,Lax–Wendroff method ,Applied Mathematics ,Extrapolation ,Boundary (topology) ,010103 numerical & computational mathematics ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Modeling and Simulation ,Jacobian matrix and determinant ,symbols ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we reconsider the inverse Lax-Wendroff (ILW) procedure, which is a numerical boundary treatment for solving hyperbolic conservation laws, and propose a new approach to evaluate the values on the ghost points. The ILW procedure was firstly proposed to deal with the “cut cell” problems, when the physical boundary intersects with the Cartesian mesh in an arbitrary fashion. The key idea of the ILW procedure is repeatedly utilizing the partial differential equations (PDEs) and inflow boundary conditions to obtain the normal derivatives of each order on the boundary. A simplified ILW procedure was proposed in [28] and used the ILW procedure for the evaluation of the first order normal derivatives only. The main difference between the simplified ILW procedure and the proposed ILW procedure here is that we define the unknown u and the flux f ( u ) on the ghost points separately. One advantage of this treatment is that it allows the eigenvalues of the Jacobian f ′ ( u ) to be close to zero on the boundary, which may appear in many physical problems. We also propose a new weighted essentially non-oscillatory (WENO) type extrapolation at the outflow boundaries, whose idea comes from the multi-resolution WENO schemes in [32] . The WENO type extrapolation maintains high order accuracy if the solution is smooth near the boundary and it becomes a low order extrapolation automatically if a shock is close to the boundary. This WENO type extrapolation preserves the property of self-similarity, thus it is more preferable in computing the hyperbolic conservation laws. We provide extensive numerical examples to demonstrate that our method is stable, high order accurate and has good performance for various problems with different kinds of boundary conditions including the solid wall boundary condition, when the physical boundary is not aligned with the grids.
- Published
- 2021
23. High order conservative Lagrangian schemes for one-dimensional radiation hydrodynamics equations in the equilibrium-diffusion limit
- Author
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Peng Song, Juan Cheng, and Chi-Wang Shu
- Subjects
Physics ,Coupling ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Spacetime ,Advection ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Flow (mathematics) ,Modeling and Simulation ,symbols ,Applied mathematics ,Limit (mathematics) ,0101 mathematics ,Newton's method ,Interpolation - Abstract
Radiation hydrodynamics (RH) describes the interaction between matter and radiation which affects the thermodynamic states and the dynamic flow characteristics of the matter-radiation system. Its application areas are mainly in high-temperature hydrodynamics, including gaseous stars in astrophysics, combustion phenomena, reentry vehicles fusion physics and inertial confinement fusion (ICF). Solving the radiation hydrodynamics equations (RHE), even in the equilibrium-diffusion limit, is a difficult task. In this paper, we will discuss the methodology to construct fully explicit and implicit-explicit (IMEX) high order Lagrangian schemes solving one dimensional RHE in the equilibrium-diffusion limit respectively, which can be used to simulate multi-material problems with the coupling of radiation and hydrodynamics. The schemes are based on the HLLC numerical flux, the essentially non-oscillatory (ENO) reconstruction for the advection term, ENO reconstruction or high order central reconstruction and interpolation for the radiation diffusion term, the Newton iteration method (for the IMEX scheme), and the strong stability preserving (SSP) high order time discretizations. The schemes can maintain conservation and uniformly high order accuracy both in space and time. The issue of positivity-preserving for the high order explicit Lagrangian scheme is also discussed. Various numerical tests for the high order Lagrangian schemes are provided to demonstrate the desired properties of the schemes such as high order accuracy, non-oscillation, and positivity-preserving.
- Published
- 2020
24. Inverse Lax–Wendroff procedure for numerical boundary conditions of convection–diffusion equations
- Author
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Sirui Tan, Chi-Wang Shu, Jianfang Lu, Jinwei Fang, and Mengping Zhang
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Mixed boundary condition ,Singular boundary method ,Boundary knot method ,01 natural sciences ,Robin boundary condition ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Boundary conditions in CFD ,Modeling and Simulation ,Free boundary problem ,Boundary value problem ,0101 mathematics ,Mathematics ,Numerical partial differential equations - Abstract
We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure 28, in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, has been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.
- Published
- 2016
25. High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments
- Author
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Chi-Wang Shu
- Subjects
Numerical Analysis ,Partial differential equation ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Finite difference ,010103 numerical & computational mathematics ,Mixed finite element method ,01 natural sciences ,Finite element method ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Discontinuous Galerkin method ,Modeling and Simulation ,0101 mathematics ,Spectral method ,Extended finite element method ,Mathematics - Abstract
For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, namely the weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and discontinuous Galerkin (DG) finite element methods, emphasizing several of their recent developments: bound-preserving limiters for DG, finite volume and finite difference schemes, which address issues in robustness and accuracy; WENO limiters for DG methods, which address issues in non-oscillatory performance when there are strong shocks, and inverse Lax-Wendroff type boundary treatments for finite difference schemes, which address issues in solving complex geometry problems using Cartesian meshes.
- Published
- 2016
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26. Bound-preserving discontinuous Galerkin methods for relativistic hydrodynamics
- Author
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Chi-Wang Shu, Yang Yang, and Tong Qin
- Subjects
Numerical Analysis ,Ideal (set theory) ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Dissipation ,01 natural sciences ,Stability (probability) ,Speed of light (cellular automaton) ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Discontinuous Galerkin method ,Modeling and Simulation ,Bounded function ,Limiter ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we develop a discontinuous Galerkin (DG) method to solve the ideal special relativistic hydrodynamics (RHD) and design a bound-preserving (BP) limiter for this scheme by extending the idea in X. Zhang and C.-W. Shu, (2010) 56. For RHD, the density and pressure are positive and the velocity is bounded by the speed of light. One difficulty in numerically solving the RHD in its conservative form is that the failure of preserving these physical bounds will result in ill-posedness of the problem and blowup of the code, especially in extreme relativistic cases. The standard way in dealing with this difficulty is to add extra numerical dissipation, while in doing so there is no guarantee of maintaining the high order of accuracy. Our BP limiter has the following features. It can theoretically guarantee to preserve the physical bounds for the numerical solution and maintain its designed high order accuracy. The limiter is local to the cell and hence is very easy to implement. Moreover, it renders L 1 -stability to the numerical scheme. Numerical experiments are performed to demonstrate the good performance of this bound-preserving DG scheme. Even though we only discuss the BP limiter for DG schemes, it can be applied to high order finite volume schemes, such as weighted essentially non-oscillatory (WENO) finite volume schemes as well.
- Published
- 2016
- Full Text
- View/download PDF
27. Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case
- Author
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François Vilar, Chi-Wang Shu, Pierre-Henri Maire, Department of Applied Mathematics, The division of Applied Mathematics [Providence], Brown University-Brown University, Centre d'études scientifiques et techniques d'Aquitaine (CESTA), Direction des Applications Militaires (DAM), and Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)
- Subjects
Numerical Analysis ,positivity-preserving high-order methods ,equations of state ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,updated and total Lagrangian formulations ,Godunov-type method ,010103 numerical & computational mathematics ,multi-material compressible flows ,01 natural sciences ,Computer Science Applications ,cell-centered Lagrangian schemes ,010101 applied mathematics ,Computational Mathematics ,Modeling and Simulation ,Riemann solver ,0101 mathematics ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
International audience; One of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in [89, 90, 22], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in [85], and proves to fit a wide range of numerical schemes in the literature, such as those presented in [19, 64, 15, 82, 84].
- Published
- 2016
28. Parallel adaptive mesh refinement method based on WENO finite difference scheme for the simulation of multi-dimensional detonation
- Author
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Chi-Wang Shu, XinZhuang Dong, and Cheng Wang
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Computer simulation ,Adaptive mesh refinement ,Applied Mathematics ,Detonation ,Message Passing Interface ,Finite difference ,Data_CODINGANDINFORMATIONTHEORY ,Data structure ,Computer Science Applications ,Computational science ,Computational Mathematics ,Modeling and Simulation ,Code (cryptography) ,Polygon mesh ,Mathematics - Abstract
For numerical simulation of detonation, computational cost using uniform meshes is large due to the vast separation in both time and space scales. Adaptive mesh refinement (AMR) is advantageous for problems with vastly different scales. This paper aims to propose an AMR method with high order accuracy for numerical investigation of multi-dimensional detonation. A well-designed AMR method based on finite difference weighted essentially non-oscillatory (WENO) scheme, named as AMR&WENO is proposed. A new cell-based data structure is used to organize the adaptive meshes. The new data structure makes it possible for cells to communicate with each other quickly and easily. In order to develop an AMR method with high order accuracy, high order prolongations in both space and time are utilized in the data prolongation procedure. Based on the message passing interface (MPI) platform, we have developed a workload balancing parallel AMR&WENO code using the Hilbert space-filling curve algorithm. Our numerical experiments with detonation simulations indicate that the AMR&WENO is accurate and has a high resolution. Moreover, we evaluate and compare the performance of the uniform mesh WENO scheme and the parallel AMR&WENO method. The comparison results provide us further insight into the high performance of the parallel AMR&WENO method.
- Published
- 2015
29. On the conservation of finite difference WENO schemes in non-rectangular domains using the inverse Lax-Wendroff boundary treatments
- Author
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Chi-Wang Shu, Mengping Zhang, and Shengrong Ding
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Numerical Analysis ,Conservation law ,Physics and Astronomy (miscellaneous) ,Lax–Wendroff method ,Applied Mathematics ,Scalar (mathematics) ,Finite difference ,Inverse ,Classification of discontinuities ,Grid ,Computer Science Applications ,Computational Mathematics ,Modeling and Simulation ,Applied mathematics ,High order ,Mathematics - Abstract
We discuss the issue of conservation of the total mass for finite difference WENO schemes solving hyperbolic conservation laws on a Cartesian mesh using the inverse Lax-Wendroff boundary treatments in arbitrary physical domains whose boundaries do not coincide with grid lines. The numerical fluxes near the boundary are suitably modified so that strict conservation of the total mass is achieved and the high order accuracy and non-oscillatory performance are not compromised. The key point is a suitable definition of the total mass, which is consistent with the high order accuracy finite difference framework over an arbitrary domain with a boundary not necessarily coinciding with grid lines. Extensive numerical examples are provided to demonstrate that our modified method is strictly conservative, and is high order accurate and has as good performance as the original high order WENO schemes with the Lax-Wendroff boundary treatments, for both smooth problems and problems with discontinuities, in both one- and two-dimensional problems involving both scalar equations and systems.
- Published
- 2020
30. A new type of third-order finite volume multi-resolution WENO schemes on tetrahedral meshes
- Author
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Chi-Wang Shu and Jun Zhu
- Subjects
Numerical Analysis ,Conservation law ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Series (mathematics) ,Discretization ,Applied Mathematics ,Order of accuracy ,Mathematics::Numerical Analysis ,Computer Science Applications ,Computational Mathematics ,Compact space ,Robustness (computer science) ,Modeling and Simulation ,Convergence (routing) ,Applied mathematics ,Mathematics - Abstract
In this continuing paper of [J. Comput. Phys., 375 (2018), 659-683; J. Comput. Phys., 392 (2019), 19-33], we design a new third-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws on tetrahedral meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation. Comparing with classical third-order finite volume WENO schemes [Commun. Comput. Phys., 5 (2009), 836-848] on tetrahedral meshes, the crucial advantages of such new multi-resolution WENO schemes are their simplicity and compactness with the application of only three unequal-sized central stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, their easy choice of arbitrary positive linear weights without considering the topology of the tetrahedral meshes, their optimal order of accuracy in smooth regions, and their suppression of spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO scheme can be any positive numbers on the condition that their sum is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite volume WENO scheme on tetrahedral meshes. By performing such new spatial reconstruction procedures and adopting a third-order TVD Runge-Kutta method for time discretization, the occupied memory is decreased and the computing efficiency is increased. This new third-order finite volume multi-resolution WENO scheme is suitable for large scale engineering applications and could maintain good convergence property for steady-state problems on tetrahedral meshes. Benchmark examples are computed to demonstrate the robustness and good performance of these new finite volume WENO schemes.
- Published
- 2020
31. High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters
- Author
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Jun Zhu, Chi-Wang Shu, and Jianxian Qiu
- Subjects
Physics::Computational Physics ,Numerical Analysis ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Series (mathematics) ,Applied Mathematics ,Finite difference ,Order of accuracy ,Stencil ,Projection (linear algebra) ,Mathematics::Numerical Analysis ,Computer Science Applications ,Computational Mathematics ,Runge–Kutta methods ,Discontinuous Galerkin method ,Modeling and Simulation ,Applied mathematics ,Mathematics - Abstract
In this paper, a new type of multi-resolution weighted essentially non-oscillatory (WENO) limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods is designed. This type of multi-resolution WENO limiters is an extension of the multi-resolution WENO finite volume and finite difference schemes developed in [43] . Such new limiters use information of the DG solution essentially only within the troubled cell itself, 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, fourth-order, and fifth-order RKDG methods with these multi-resolution WENO limiters have been developed as examples, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong discontinuities. The linear weights of such 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. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original DG methods. Benchmark examples are given to demonstrate the good performance of these RKDG methods with the associated multi-resolution WENO limiters.
- Published
- 2020
32. Assessment of aeroacoustic resolution properties of DG schemes and comparison with DRP schemes
- Author
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Ziqiang Cheng, Jinwei Fang, Chi-Wang Shu, and Mengping Zhang
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Computer science ,Applied Mathematics ,Finite difference ,010103 numerical & computational mathematics ,Dissipation ,Topology ,01 natural sciences ,Stencil ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Complex geometry ,Discontinuous Galerkin method ,Modeling and Simulation ,Polygon mesh ,Computational aeroacoustics ,0101 mathematics ,Block (data storage) - Abstract
We discuss the aeroacoustic resolution properties of the discontinuous Galerkin (DG) schemes in detail and compare their performance with the state of the art finite difference (FD) schemes, including the classical dispersion-relation-preserving (DRP) schemes. Analysis shows that, even though the DG schemes are slightly dissipative, their overall dispersion and dissipation properties are comparable with the corresponding DRP schemes on the same stencil. For the convenience of a direct comparison with FD schemes, we write the DG schemes in the form of block finite difference schemes. Ample numerical tests, including the tests on nozzle flow problems, are performed and we observe that the DG schemes and DRP schemes with the same stencil produce comparable numerical results. Since the DG schemes are flexible on non-uniform meshes and general unstructured meshes, they should be good candidates for computational aeroacoustics, especially those on complex geometry.
- Published
- 2019
33. A new class of central compact schemes with spectral-like resolution II: Hybrid weighted nonlinear schemes
- Author
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Xuliang Liu, Hanxin Zhang, Chi-Wang Shu, and Shuhai Zhang
- Subjects
Shock wave ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Resolution (electron density) ,Mathematical analysis ,Classification of discontinuities ,Computer Science Applications ,Computational Mathematics ,Nonlinear system ,Modeling and Simulation ,Scheme (mathematics) ,Applied mathematics ,Flux limiter ,Low dissipation ,Interpolation ,Mathematics - Abstract
In this paper, we develop a class of nonlinear compact schemes based on our previous linear central compact schemes with spectral-like resolution (X. Liu et al., 2013 20). In our approach, we compute the flux derivatives on the cell-nodes by the physical fluxes on the cell nodes and numerical fluxes on the cell centers. To acquire the numerical fluxes on the cell centers, we perform a weighted hybrid interpolation of an upwind interpolation and a central interpolation. Through systematic analysis and numerical tests, we show that our nonlinear compact scheme has high order, high resolution and low dissipation, and has the same ability to capture strong discontinuities as regular weighted essentially non-oscillatory (WENO) schemes. It is a good choice for the simulation of multiscale problems with shock waves.
- Published
- 2015
34. Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media
- Author
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Yulong Xing, Chi-Wang Shu, and Ching-Shan Chou
- Subjects
Numerical Analysis ,Mathematical optimization ,Physics and Astronomy (miscellaneous) ,Wave propagation ,Applied Mathematics ,Numerical analysis ,Wave equation ,Computer Science Applications ,Energy conservation ,Computational Mathematics ,Discontinuous Galerkin method ,Modeling and Simulation ,Integrator ,Convergence (routing) ,Energy (signal processing) ,Mathematics - Abstract
Solving wave propagation problems within heterogeneous media has been of great interest and has a wide range of applications in physics and engineering. The design of numerical methods for such general wave propagation problems is challenging because the energy conserving property has to be incorporated in the numerical algorithms in order to minimize the phase or shape errors after long time integration. In this paper, we focus on multi-dimensional wave problems and consider linear second-order wave equation in heterogeneous media. We develop and analyze an LDG method, in which numerical fluxes are carefully designed to maintain the energy conserving property and accuracy. Compatible high order energy conserving time integrators are also proposed. The optimal error estimates and the energy conserving property are proved for the semi-discrete methods. Our numerical experiments demonstrate optimal rates of convergence, and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property.
- Published
- 2014
35. Second order symmetry-preserving conservative Lagrangian scheme for compressible Euler equations in two-dimensional cylindrical coordinates
- Author
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Chi-Wang Shu and Juan Cheng
- Subjects
Numerical Analysis ,Conservation law ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Symmetry (physics) ,Control volume ,Computer Science Applications ,Euler equations ,Momentum ,Computational Mathematics ,symbols.namesake ,Classical mechanics ,Parabolic cylindrical coordinates ,Modeling and Simulation ,symbols ,Circular symmetry ,Cylindrical coordinate system ,Mathematics - Abstract
In applications such as astrophysics and inertial confinement fusion, there are many three-dimensional cylindrical-symmetric multi-material problems which are usually simulated by Lagrangian schemes in the two-dimensional cylindrical coordinates. For this type of simulation, a critical issue for the schemes is to keep spherical symmetry in the cylindrical coordinate system if the original physical problem has this symmetry. In the past decades, several Lagrangian schemes with such symmetry property have been developed, but all of them are only first order accurate. In this paper, we develop a second order cell-centered Lagrangian scheme for solving compressible Euler equations in cylindrical coordinates, based on the control volume discretizations, which is designed to have uniformly second order accuracy and capability to preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid. The scheme maintains several good properties such as conservation for mass, momentum and total energy, and the geometric conservation law. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of accuracy, symmetry, non-oscillation and robustness. The advantage of higher order accuracy is demonstrated in these examples.
- Published
- 2014
36. Positivity-preserving Lagrangian scheme for multi-material compressible flow
- Author
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Juan Cheng and Chi-Wang Shu
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,business.industry ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Computational fluid dynamics ,Compressible flow ,Riemann solver ,Computer Science Applications ,Euler equations ,Computational Mathematics ,symbols.namesake ,Robustness (computer science) ,Modeling and Simulation ,symbols ,Compressibility ,Polygon mesh ,business ,Mathematics - Abstract
Robustness of numerical methods has attracted an increasing interest in the community of computational fluid dynamics. One mathematical aspect of robustness for numerical methods is the positivity-preserving property. At high Mach numbers or for flows near vacuum, solving the conservative Euler equations may generate negative density or internal energy numerically, which may lead to nonlinear instability and crash of the code. This difficulty is particularly profound for high order methods, for multi-material flows and for problems with moving meshes, such as the Lagrangian methods. In this paper, we construct both first order and uniformly high order accurate conservative Lagrangian schemes which preserve positivity of physically positive variables such as density and internal energy in the simulation of compressible multi-material flows with general equations of state (EOS). We first develop a positivity-preserving approximate Riemann solver for the Lagrangian scheme solving compressible Euler equations with both ideal and non-ideal EOS. Then we design a class of high order positivity-preserving and conservative Lagrangian schemes by using the essentially non-oscillatory (ENO) reconstruction, the strong stability preserving (SSP) high order time discretizations and the positivity-preserving scaling limiter which can be proven to maintain conservation and uniformly high order accuracy and is easy to implement. One-dimensional and two-dimensional numerical tests for the positivity-preserving Lagrangian schemes are provided to demonstrate the effectiveness of these methods.
- Published
- 2014
37. Discontinuous Galerkin method for Krauseʼs consensus models and pressureless Euler equations
- Author
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Yang Yang, Chi-Wang Shu, and Dongming Wei
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Probability density function ,Computer Science Applications ,Euler equations ,Computational Mathematics ,symbols.namesake ,Singularity ,Maximum principle ,Discontinuous Galerkin method ,Modeling and Simulation ,Scheme (mathematics) ,symbols ,Limiter ,Single point ,Mathematics - Abstract
In this paper, we apply discontinuous Galerkin (DG) methods to solve two model equations: [email protected]?s consensus models and pressureless Euler equations. These two models are used to describe the collisions of particles, and the distributions can be identified as density functions. If the particles are placed at a single point, then the density function turns out to be a @d-function and is difficult to be well approximated numerically. In this paper, we use DG method to approximate such a singularity and demonstrate the good performance of the scheme. Since the density functions are always positive, we apply a positivity-preserving limiter to them. Moreover, for pressureless Euler equations, the velocity satisfies the maximum principle. We also construct special limiters to fulfill this requirement.
- Published
- 2013
38. A homotopy method based on WENO schemes for solving steady state problems of hyperbolic conservation laws
- Author
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Chi-Wang Shu, Yong-Tao Zhang, Andrew J. Sommese, Wenrui Hao, Zhiliang Xu, and Jonathan D. Hauenstein
- Subjects
Numerical Analysis ,Conservation law ,Physics and Astronomy (miscellaneous) ,Differential equation ,Applied Mathematics ,Courant–Friedrichs–Lewy condition ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Finite difference ,Adaptive stepsize ,Computer Science Applications ,Computational Mathematics ,Robustness (computer science) ,Modeling and Simulation ,Adaptive system ,Homotopy analysis method ,Mathematics - Abstract
Homotopy continuation is an efficient tool for solving polynomial systems. Its efficiency relies on utilizing adaptive stepsize and adaptive precision path tracking, and endgames. In this article, we apply homotopy continuation to solve steady state problems of hyperbolic conservation laws. A third-order accurate finite difference weighted essentially non-oscillatory (WENO) scheme with Lax-Friedrichs flux splitting is utilized to derive the difference equation. This new approach is free of the CFL condition constraint. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency and robustness of the new method.
- Published
- 2013
39. Corrigendum to 'Spurious behavior of shock-capturing methods by the fractional step approach: Problems containing stiff source terms and discontinuities' [J. Comput. Phys. 241 (2013) 266–291]
- Author
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D. V. Kotov, Chi-Wang Shu, Wei Wang, and Helen C. Yee
- Subjects
Physics ,Computational Mathematics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Shock capturing method ,Applied mathematics ,Classification of discontinuities ,Spurious relationship ,Algorithm ,Computer Science Applications - Published
- 2013
40. Runge–Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes
- Author
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Jianxian Qiu, Xinghui Zhong, Chi-Wang Shu, and Jun Zhu
- Subjects
Numerical Analysis ,Conservation law ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Finite element method ,Mathematics::Numerical Analysis ,Computer Science Applications ,Computational Mathematics ,symbols.namesake ,Runge–Kutta methods ,Nonlinear system ,Discontinuous Galerkin method ,Modeling and Simulation ,Euler's formula ,symbols ,Polygon mesh ,Mathematics - Abstract
In this paper we generalize a new type of limiters based on the weighted essentially non-oscillatory (WENO) finite volume methodology for the Runge–Kutta discontinuous Galerkin (RKDG) methods solving nonlinear hyperbolic conservation laws, which were recently developed in [32] for structured meshes, to two-dimensional unstructured triangular meshes. The key idea of such limiters is to use the entire polynomials of the DG solutions from the troubled cell and its immediate neighboring cells, and then apply the classical WENO procedure to form a convex combination of these polynomials based on smoothness indicators and nonlinear weights, with suitable adjustments to guarantee conservation. The main advantage of this new limiter is its simplicity in implementation, especially for the unstructured meshes considered in this paper, as only information from immediate neighbors is needed and the usage of complicated geometric information of the meshes is largely avoided. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good performance of this procedure.
- Published
- 2013
41. A new class of central compact schemes with spectral-like resolution I: Linear schemes
- Author
-
Xuliang Liu, Chi-Wang Shu, Hanxin Zhang, and Shuhai Zhang
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Direct numerical simulation ,Compact finite difference ,Function (mathematics) ,Grid ,Topology ,Computer Science Applications ,Computational Mathematics ,symbols.namesake ,Fourier analysis ,Modeling and Simulation ,symbols ,Computational aeroacoustics ,Flux limiter ,Interpolation ,Mathematics - Abstract
In this paper, we design a new class of central compact schemes based on the cell-centered compact schemes of Lele [S.K. Lele, Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics 103 (1992) 16–42]. These schemes equate a weighted sum of the nodal derivatives of a smooth function to a weighted sum of the function on both the grid points (cell boundaries) and the cell-centers. In our approach, instead of using a compact interpolation to compute the values on cell-centers, the physical values on these half grid points are stored as independent variables and updated using the same scheme as the physical values on the grid points. This approach increases the memory requirement but not the computational costs. Through systematic Fourier analysis and numerical tests, we observe that the schemes have excellent properties of high order, high resolution and low dissipation. It is an ideal class of schemes for the simulation of multi-scale problems such as aeroacoustics and turbulence.
- Published
- 2013
42. Positivity-preserving method for high-order conservative schemes solving compressible Euler equations
- Author
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Chi-Wang Shu, Xiangyu Hu, and Nikolaus A. Adams
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Order of accuracy ,010103 numerical & computational mathematics ,Classification of discontinuities ,01 natural sciences ,Compressible flow ,Computer Science Applications ,Euler equations ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Flow (mathematics) ,Simple (abstract algebra) ,Modeling and Simulation ,symbols ,Flux limiter ,0101 mathematics ,Mathematics - Abstract
In this work a simple method to enforce the positivity-preserving property for general high-order conservative schemes is proposed for solving compressible Euler equations. The method detects critical numerical fluxes which may lead to negative density and pressure, and for such critical fluxes imposes a simple flux limiter by combining the high-order numerical flux with the first-order Lax-Friedrichs flux to satisfy a sufficient condition for preserving positivity. Though an extra time-step size condition is required to maintain the formal order of accuracy, it is less restrictive than those in previous works. A number of numerical examples suggest that this method, when applied on an essentially non-oscillatory scheme, can be used to prevent positivity failure when the flow involves vacuum or near vacuum and very strong discontinuities.
- Published
- 2013
43. Spurious behavior of shock-capturing methods by the fractional step approach: Problems containing stiff source terms and discontinuities
- Author
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Chi-Wang Shu, Helen C. Yee, D. V. Kotov, and Wei Wang
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Discretization ,Truncation error (numerical integration) ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Classification of discontinuities ,Computer Science Applications ,Computational Mathematics ,Discontinuity (linguistics) ,Nonlinear system ,Classical mechanics ,Strang splitting ,Modeling and Simulation ,Shock capturing method ,Mathematics - Abstract
The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The present study focuses only on solving the reactive system by the fractional step method using the Strang splitting. Studies shows that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general.
- Published
- 2013
44. Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes
- Author
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Chi-Wang Shu, Xiangxiong Zhang, and Yifan Zhang
- Subjects
Numerical Analysis ,Conservation law ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Computer Science Applications ,Computational Mathematics ,Nonlinear system ,symbols.namesake ,Maximum principle ,Discontinuous Galerkin method ,Modeling and Simulation ,Euler's formula ,symbols ,Initial value problem ,Convection–diffusion equation ,Mathematics - Abstract
We propose second order accurate discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle for general nonlinear convection-diffusion equations on unstructured triangular meshes. Motivated by genuinely high order maximum-principle-satisfying DG schemes for hyperbolic conservation laws (Perthame, 1996) and (Zhang, 2010) [14,26], we prove that under suitable time step restriction for forward Euler time stepping, for general nonlinear convection-diffusion equations, the same scaling limiter coupled with second order DG methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy. Similar to the purely convection cases, the limiters are mass conservative and easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. Following the idea in Zhang (2012) [30], we extend the schemes to two-dimensional convection-diffusion equations on triangular meshes. There are no geometric constraints on the mesh such as angle acuteness. Numerical results including incompressible Navier-Stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods.
- Published
- 2013
45. A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods
- Author
-
Chi-Wang Shu and Xinghui Zhong
- Subjects
Physics::Computational Physics ,Numerical Analysis ,Conservation law ,Polynomial ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Mathematics::Numerical Analysis ,Computer Science Applications ,Computational Mathematics ,Nonlinear system ,Runge–Kutta methods ,Discontinuous Galerkin method ,Modeling and Simulation ,Limiter ,Polygon mesh ,Convex combination ,Mathematics - Abstract
In this paper, we investigate a simple limiter using weighted essentially non-oscillatory (WENO) methodology for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving conservation laws, with the goal of obtaining a robust and high order limiting procedure to simultaneously achieve uniform high order accuracy and sharp, non-oscillatory shock transitions. The idea of this limiter is to reconstruct the entire polynomial, instead of reconstructing point values or moments in the classical WENO reconstructions. That is, the reconstruction polynomial on the target cell is a convex combination of polynomials on this cell and its neighboring cells and the nonlinear weights of the convex combination follow the classical WENO procedure. The main advantage of this limiter is its simplicity in implementation, especially for multi-dimensional meshes. Numerical results in one and two dimensions are provided to illustrate the behavior of this procedure.
- Published
- 2013
46. Positivity-preserving high order finite difference WENO schemes for compressible Euler equations
- Author
-
Xiangxiong Zhang and Chi-Wang Shu
- Subjects
Numerical Analysis ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Finite difference ,Finite difference coefficient ,Compressible flow ,Mathematics::Numerical Analysis ,Computer Science Applications ,Euler equations ,Computational Mathematics ,symbols.namesake ,Discontinuous Galerkin method ,Modeling and Simulation ,symbols ,Order (group theory) ,Mathematics - Abstract
In Zhang and Shu (2010) [20], Zhang and Shu (2011) [21] and Zhang et al. (in press) [23], we constructed uniformly high order accurate discontinuous Galerkin (DG) and finite volume schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this framework to construct positivity-preserving high order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) finite difference schemes for compressible Euler equations. General equations of state and source terms are also discussed. Numerical tests of the fifth order finite difference WENO scheme are reported to demonstrate the good behavior of such schemes.
- Published
- 2012
47. Efficient implementation of high order inverse Lax–Wendroff boundary treatment for conservation laws
- Author
-
Sirui Tan, Chi-Wang Shu, Cheng Wang, and Jianguo Ning
- Subjects
Numerical Analysis ,Conservation law ,Physics and Astronomy (miscellaneous) ,Lax–Wendroff method ,Applied Mathematics ,Mathematical analysis ,Finite difference ,Order of accuracy ,Computer Science Applications ,Euler equations ,Computational Mathematics ,symbols.namesake ,Inviscid flow ,Modeling and Simulation ,symbols ,Boundary value problem ,Outflow boundary ,Mathematics - Abstract
In [20], two of the authors developed a high order accurate numerical boundary condition procedure for hyperbolic conservation laws, which allows the computation using high order finite difference schemes on Cartesian meshes to solve problems in arbitrary physical domains whose boundaries do not coincide with grid lines. This procedure is based on the so-called inverse Lax-Wendroff (ILW) procedure for inflow boundary conditions and high order extrapolation for outflow boundary conditions. However, the algebra of the ILW procedure is quite heavy for two dimensional (2D) hyperbolic systems, which makes it difficult to implement the procedure for order of accuracy higher than three. In this paper, we first discuss a simplified and improved implementation for this procedure, which uses the relatively complicated ILW procedure only for the evaluation of the first order normal derivatives. Fifth order WENO type extrapolation is used for all other derivatives, regardless of the direction of the local characteristics and the smoothness of the solution. This makes the implementation of a fifth order boundary treatment practical for 2D systems with source terms. For no-penetration boundary condition of compressible inviscid flows, a further simplification is discussed, in which the evaluation of the tangential derivatives involved in the ILW procedure is avoided. We test our simplified and improved boundary treatment for Euler equations with or without source terms representing chemical reactions in detonations. The results demonstrate the designed fifth order accuracy, stability, and good performance for problems involving complicated interactions between detonation/shock waves and solid boundaries.
- Published
- 2012
48. Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations
- Author
-
Xiangxiong Zhang, Chi-Wang Shu, Cheng Wang, and Jianguo Ning
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Time evolution ,Detonation ,Computer Science Applications ,Euler equations ,Computational Mathematics ,symbols.namesake ,Third order ,Simple (abstract algebra) ,Discontinuous Galerkin method ,Modeling and Simulation ,Limiter ,symbols ,Order (group theory) ,Mathematics - Abstract
One of the main challenges in computational simulations of gas detonation propagation is that negative density or negative pressure may emerge during the time evolution, which will cause blow-ups. Therefore, schemes with provable positivity-preserving of density and pressure are desired. First order and second order positivity-preserving schemes were well studied, e.g., [6,10]. For high order discontinuous Galerkin (DG) method, even though the characteristicwise TVB limiter in [1,2] can kill oscillations, it is not sufficient to maintain the positivity. A simple solution for arbitrarily high order positivity-preserving schemes solving Euler equations was proposed recently in [22]. In this paper, we first discuss an extension of the technique in [22-24] to design arbitrarily high order positivity-preserving DG schemes for reactive Euler equations. We then present a simpler and more robust implementation of the positivity-preserving limiter than the one in [22]. Numerical tests, including very demanding examples in gaseous detonations, indicate that the third order DG scheme with the new positivity-preserving limiter produces satisfying results even without the TVB limiter.
- Published
- 2012
49. High order finite difference methods with subcell resolution for advection equations with stiff source terms
- Author
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Helen C. Yee, Chi-Wang Shu, Wei Wang, and Björn Sjögreen
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Finite difference method ,Scalar (physics) ,Euler system ,Classification of discontinuities ,Computer Science Applications ,Computational Mathematics ,Discontinuity (linguistics) ,Orders of magnitude (time) ,Modeling and Simulation ,Fluid dynamics ,Mathematics - Abstract
A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with the computed flow variables in the shock region modified by the Harten subcell resolution idea. For numerical experiments, a fifth-order finite-difference WENO scheme and its anti-diffusion WENO variant are considered. A wide range of 1D and 2D scalar and Euler system test cases are investigated. Studies indicate that for the considered test cases, the new method maintains high order accuracy in space for smooth flows, and for stiff source terms with discontinuities, it can capture the correct propagation speed of discontinuities in very coarse meshes with reasonable CFL numbers.
- Published
- 2012
50. Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov–Poisson system
- Author
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Chi-Wang Shu and Jing-Mei Qiu
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Vlasov equation ,Weak formulation ,Computer Science Applications ,Computational Mathematics ,Nonlinear system ,Strang splitting ,Discontinuous Galerkin method ,Modeling and Simulation ,Boundary value problem ,Galerkin method ,Linear equation ,Mathematics - Abstract
Semi-Lagrangian (SL) methods have been very popular in the Vlasov simulation community [29,3,4,18,32,5,24,26]. In this paper, we propose a new Strang split SL discontinuous Galerkin (DG) method for solving the Vlasov equation. Specifically, we apply the Strang splitting for the Vlasov equation [6], as a way to decouple the nonlinear Vlasov system into a sequence of 1-D advection equations, each of which has an advection velocity that only depends on coordinates that are transverse to the direction of propagation. To evolve the decoupled linear equations, we propose to couple the SL framework with the semi-discrete DG formulation. The proposed SL DG method is free of time step restriction compared with the Runge-Kutta DG method, which is known to suffer from numerical time step limitation with relatively small CFL numbers according to linear stability analysis. We apply the recently developed positivity preserving (PP) limiter [37], which is a low-cost black box procedure, to our scheme to ensure the positivity of the unknown probability density function without affecting the high order accuracy of the base SL DG scheme. We analyze the stability and accuracy properties of the SL DG scheme by establishing its connection with the direct and weak formulations of the characteristics/Lagrangian Galerkin method [23]. The quality of the proposed method is demonstrated via basic test problems, such as linear advection and rigid body rotation, and via classical plasma problems, such as Landau damping and the two stream instability.
- Published
- 2011
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