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270
ImplicitExplicit RungeKutta schemes and applications to hyperbolic systems with relaxation
 Journal of Scientific Computing
, 2000
"... We consider implicitexplicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strongstabilitypreserving (SSP) scheme, and the implicit part is treated by an Lstable diagonally implicit Runge Kutta (DIRK). The sch ..."
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Cited by 83 (14 self)
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We consider implicitexplicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strongstabilitypreserving (SSP) scheme, and the implicit part is treated by an Lstable diagonally implicit Runge Kutta (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by finite difference discretization with Weighted Essentially Non Oscillatory (WENO) reconstruction. After a brief description of the mathematical properties of the schemes, several applications will be presented. Keywords: RungeKutta methods, hyperbolic systems with relaxation, stiff systems, high order shock capturing schemes. AMS Subject Classification: 65C20, 82D25 1
Weighted Essentially NonOscillatory Schemes on Triangular Meshes
 J. Comput. Phys
, 1998
"... In this paper we construct high order weighted essentially nonoscillatory (WENO) schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. We present third order schemes using a combination of linear polynomials, and fourth order schemes using a combination of qua ..."
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Cited by 72 (12 self)
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In this paper we construct high order weighted essentially nonoscillatory (WENO) schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. We present third order schemes using a combination of linear polynomials, and fourth order schemes using a combination of quadratic polynomials. Numerical examples are shown to demonstrate the accuracies and robustness of the methods for shock calculations.
Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations
, 2006
"... An efficient, highorder, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve highcomputational efficiency and geometric flexibility; it ..."
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Cited by 44 (25 self)
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An efficient, highorder, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve highcomputational efficiency and geometric flexibility; it utilizes the concept of discontinuous and highorder local representations to achieve conservation and high accuracy; and it is based on the finitedifference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, monotonicity limiters are implemented, and tested for several problems in one and two dimensions. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids. KEY WORDS: Highorder; conservation laws; unstructured grids; spectral difference; spectral collocation method; Euler equations.
Wellbalanced finite volume schemes of arbitrary order of accuracy for shallow water flows
, 2006
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On the construction, comparison, and local characteristic decomposition for highorder central WENO schemes
 J. Comput. Phys
"... In this paper, we review and construct fifth and ninthorder central weighted essentially nonoscillatory (WENO) schemes based on a finite volume formulation, staggered mesh, and continuous extension of Runge–Kutta methods for solving nonlinear hyperbolic conservation law systems. Negative linear we ..."
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Cited by 27 (1 self)
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In this paper, we review and construct fifth and ninthorder central weighted essentially nonoscillatory (WENO) schemes based on a finite volume formulation, staggered mesh, and continuous extension of Runge–Kutta methods for solving nonlinear hyperbolic conservation law systems. Negative linear weights appear in such a formulation and they are treated using the technique recently introduced by Shi et al. (J. Comput. Phys. 175, 108 (2002)). We then perform numerical computations and comparisons with the finite difference WENO schemes of Jiang and Shu (J. Comput.
High order numerical methods for the space non homogeneous Boltzmann equation.
 J. Comput. Phys
, 2003
"... In this paper we present accurate methods for the numerical solution of the Boltzmann equation of rareed gas. The methods are based on a time splitting technique. The transport is solved by a third order accurate (in space) Positive and Flux conservative (PFC) method. The collision step is treate ..."
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Cited by 26 (7 self)
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In this paper we present accurate methods for the numerical solution of the Boltzmann equation of rareed gas. The methods are based on a time splitting technique. The transport is solved by a third order accurate (in space) Positive and Flux conservative (PFC) method. The collision step is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity, coupled with several high order integrator in time. Strang splitting is used to achieve second order accuracy in space and time. Several numerical tests illustrate the properties of the methods. Keywords: Boltzmann equation, Rareed gas dynamics, spectral methods, splitting algorithms 1
High Order Wellbalanced Finite Volume WENO Schemes for Shallow Water Equation with Moving Water
, 2007
"... A characteristic feature of hyperbolic systems of balance laws is the existence of nontrivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Recently a number of socalled wellbalanced schemes were developed which satisfy a discrete analogue of th ..."
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Cited by 26 (7 self)
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A characteristic feature of hyperbolic systems of balance laws is the existence of nontrivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Recently a number of socalled wellbalanced schemes were developed which satisfy a discrete analogue of this balance and are therefore able to maintain an equilibrium state. In most cases, applications treated equilibria at rest, where the flow velocity vanishes. Here we present a new very high order accurate, exactly wellbalanced finite volume scheme for moving flow equilibria. Numerical experiments show excellent resolution of unperturbed as well as slightly perturbed equilibria.
Level Set Methods and Their Applications in Image Science
 Comm. Math Sci
"... this article, we discuss the question "What Level Set Methods can do for image science". We examine the scope of these techniques in image science, in particular in image segmentation, and introduce some relevant level set techniques that are potentially useful for this class of applicatio ..."
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Cited by 23 (1 self)
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this article, we discuss the question "What Level Set Methods can do for image science". We examine the scope of these techniques in image science, in particular in image segmentation, and introduce some relevant level set techniques that are potentially useful for this class of applications. We will show that image science demands multidisciplinary knowledge and flexible but still robust methods. That is why the Level Set Method has become a thriving technique in this field
Level set equations on surfaces via the Closest Point Method
, 2007
"... Level set methods have been used in a great number of applications in R 2 and R 3 and it is natural to consider extending some of these methods to problems defined on surfaces embedded in R 3 or higher dimensions. In this paper we consider the treatment of level set equations on surfaces via a recen ..."
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Cited by 23 (6 self)
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Level set methods have been used in a great number of applications in R 2 and R 3 and it is natural to consider extending some of these methods to problems defined on surfaces embedded in R 3 or higher dimensions. In this paper we consider the treatment of level set equations on surfaces via a recent technique for solving partial differential equations (PDEs) on surfaces, the Closest Point Method [RM06]. Our main modification is to introduce a Weighted Essentially NonOscillatory (WENO) interpolation step into the Closest Point Method. This, in combination with standard WENO for Hamilton–Jacobi equations, gives highorder results (up to fifthorder) on a variety of smooth test problems including passive transport, normal flow and redistancing. The algorithms we propose are straightforward modifications of standard codes, are carried out in the embedding space in a welldefined band around the surface and retain the robustness of the level set method with respect to the selfintersection of interfaces. Numerous examples are provided to illustrate the flexibility of the method with respect to geometry. 1
Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information
 THE JOURNAL OF THE SOCIETY FOR THE FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
, 2001
"... We discuss the reconstruction of piecewise smooth data from its (pseudo) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious O(1) Gibbs oscillations in the neighborhood of ..."
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Cited by 22 (12 self)
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We discuss the reconstruction of piecewise smooth data from its (pseudo) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious O(1) Gibbs oscillations in the neighborhood of edges and an overall deterioration of the unacceptable firstorder convergence in rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by mollification. Here we utilize a modified version of the twoparameter family of spectral mollifiers introduced by Gottlieb and Tadmor [GoTa85]. The ubiquitous oneparameter, finiteorder mollifiers are based on dilation. In contrast, our mollifiers achieve their high resolution by an intricate process of highorder cancellation. To this end, we first implement a localization step using an edge detection procedure [GeTa00a, b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible for the high resolution. The resulting adaptive mollifier greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing the spurious oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a robust, generalpurpose “black box” procedure for accurate postprocessing of piecewise smooth data.