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276
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 70 (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.
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 29 (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 27 (8 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
Adaptive Mollifiers  High Resolution Recovery of Piecewise Smooth Data from . . .
, 2001
"... this paper we use the G # regular cuto c (x)=exp(cx # =(x # # # )) which led to the fractional power 1=2. Similar results holds in the discrete case. Indeed, in this case, one can bypass the discrete Fourier coe cients: expressed in terms of the given equidistant discrete values, #f(y ..."
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Cited by 23 (13 self)
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this paper we use the G # regular cuto c (x)=exp(cx # =(x # # # )) which led to the fractional power 1=2. Similar results holds in the discrete case. Indeed, in this case, one can bypass the discrete Fourier coe cients: expressed in terms of the given equidistant discrete values, #f(y )#, of piecewise analytic f,wehave { consult Theorem 3.2 below, # N #N## # ## p; (x # y )f(y ) # f(x)##Const # (d(x)N) # # e # # d#x#N : Thus, the discrete convolution # p; (x#y )f(y ) forms an exponentially accurate nearbyinterpolant
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
High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallowwater systems
 Math. Comp
"... Abstract. This paper is concerned with the development of high order methods for the numerical approximation of onedimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO r ..."
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Cited by 22 (4 self)
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Abstract. This paper is concerned with the development of high order methods for the numerical approximation of onedimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO reconstruction of states. We also investigate the wellbalanced properties of the resulting schemes. Finally, we will focus on applications to shallowwater systems. 1.
Nonoscillatory central schemes for one and twodimensional MHD equations: I
, 2004
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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 21 (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