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112
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 43 (6 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
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 35 (24 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.
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 32 (7 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.
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 20 (14 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
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 17 (6 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
Finitedifference quasiP traveltimes for anisotropic media
 Geophysics
, 2002
"... The firstarrival quasiP wave traveltime field in an anisotropic elastic solid solves a firstorder nonlinear partial differential equation, the qP eikonal equation. The difficulty in solving this eikonal equation by a finitedifference method is that for anisotropic media the ray (group) velocity ..."
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Cited by 13 (2 self)
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The firstarrival quasiP wave traveltime field in an anisotropic elastic solid solves a firstorder nonlinear partial differential equation, the qP eikonal equation. The difficulty in solving this eikonal equation by a finitedifference method is that for anisotropic media the ray (group) velocity direction is not the same as the direction of traveltime gradient, so that the traveltime gradient can no longer serve as an indicator of the group velocity direction in extrapolating the traveltime field. However, establishing an explicit relation between the ray velocity vector and the phase velocity vector overcomes this difficulty. Furthermore, the solution of the paraxial qP eikonal equation, an evolution equation in depth, gives the firstarrival traveltime along downward propagating rays. A secondorder upwind finitedifference scheme solves this paraxial eikonal equation in O(N) floating point operations, where N is 1 the number of grid points. Numerical experiments using 2D and 3D transversely isotropic models demonstrate the accuracy of the scheme.
A Variational Technique for Time Consistent Tracking of Curves and Motion
 J MATH IMAGING VIS
"... In this paper, a new framework for the tracking of closed curves and their associated motion fields is described. The proposed method enables a continuous tracking along an image sequence of both a deformable curve and its velocity field. Such an approach is formalized through the minimization of a ..."
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Cited by 13 (6 self)
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In this paper, a new framework for the tracking of closed curves and their associated motion fields is described. The proposed method enables a continuous tracking along an image sequence of both a deformable curve and its velocity field. Such an approach is formalized through the minimization of a global spatiotemporal continuous cost functional, w.r.t a set of variables representing the curve and its related motion field. The resulting minimization process relies on optimal control approach and consists in a forward integration of an evolution law followed by a backward integration of an adjoint evolution model. This latter pde includes a term related to the discrepancy between the current estimation of the state variable and discrete noisy measurements of the system. The closed curves are represented through implicit surface modeling, whereas the motion is described either by a vector field or through vorticity and divergence maps depending on the kind of targeted applications. The efficiency of the approach is demonstrated on two types of image sequences showing deformable objects and fluid motions.
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 applications. We wil ..."
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Cited by 13 (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
Axisymmetric Numerical Relativity
, 2005
"... Chapters 2, 3 and 6 contain work done in collaboration with my supervisor and published in a joint paper [119]. The dynamical shift conditions in chapter 6 are a later addition by myself. The remaining chapters are my own work. All computer programmes were written by myself unless otherwise stated. ..."
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Cited by 12 (6 self)
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Chapters 2, 3 and 6 contain work done in collaboration with my supervisor and published in a joint paper [119]. The dynamical shift conditions in chapter 6 are a later addition by myself. The remaining chapters are my own work. All computer programmes were written by myself unless otherwise stated. c○Oliver Rinne, 2005 This thesis is concerned with formulations of the Einstein equations in axisymmetric spacetimes which are suitable for numerical evolutions. The common basis for our formulations is provided by the (2+1)+1 formalism. General matter sources and rotational degrees of freedom are included. A first evolution system adopts elliptic gauge conditions arising from maximal slicing and conformal flatness. The numerical implementation is based on the finitedifference approach, using a Multigrid algorithm for the elliptic equations and the method of lines for the hyperbolic evolution equations.
Shock Capturing, Level Sets and PDE Based Methods in Computer Vision and Image Processing: A Review of Osher's Contributions
 J. Comput. Phys
, 2001
"... In this paper we review the algorithm development and applications in high resolution shock capturing methods, level set methods and PDE based methods in computer vision and image processing. The emphasis is on Stanley Osher's contribution in these areas and the impact of his work. We will start wit ..."
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Cited by 12 (0 self)
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In this paper we review the algorithm development and applications in high resolution shock capturing methods, level set methods and PDE based methods in computer vision and image processing. The emphasis is on Stanley Osher's contribution in these areas and the impact of his work. We will start with shock capturing methods and will review the EngquistOsher scheme, TVD schemes, entropy conditions, ENO and WENO schemes and numerical schemes for HamiltonJacobi type equations. Among level set methods we will review level set calculus, numerical techniques, fluids and materials, variational approach, high codimension motion, geometric optics, and the computation of discontinuous solutions to HamiltonJacobi equations. Among computer vision and image processing we will review the total variation model for image denoising, images on implicit surfaces, and the level set method in image processing and computer vision.