Results 11  20
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350
An Adaptive Cartesian Grid Method For Unsteady Compressible Flow In Irregular Regions
 J. Comput. Phys
, 1993
"... In this paper we describe an adaptive Cartesian grid method for modeling timedependent inviscid compressible flow in irregular regions. In this approach a body is treated as an interface embedded in a regular Cartesian mesh. The single grid algorithm uses an unsplit secondorder Godunov algorithm fo ..."
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Cited by 48 (14 self)
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In this paper we describe an adaptive Cartesian grid method for modeling timedependent inviscid compressible flow in irregular regions. In this approach a body is treated as an interface embedded in a regular Cartesian mesh. The single grid algorithm uses an unsplit secondorder Godunov algorithm followed by a corrector applied to cells at the boundary. The discretization near the fluidbody interface is based on a volumeoffluid approach with a redistribution procedure to maintain conservation while avoiding time step restrictions arising from small cells where the boundary intersects the mesh. The single grid Cartesian mesh integration scheme is coupled to a conservative adaptive mesh refinement algorithm that selectively refines regions of the computational grid to achieve a desired level of accuracy. Examples showing the results of the combined Cartesian grid integration/adaptive mesh refinement algorithm for both two and threedimensional flows are presented. (This page intent...
Robust Numerical Methods for PDE Models of Asian Options
 Journal of Computational Finance
, 1998
"... We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this p ..."
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Cited by 46 (14 self)
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We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this problem. In particular, the usual methods generally produce solutions containing spurious oscillations. We adapt flux limiting techniques originally developed in the field of computational fluid dynamics in order to rapidly obtain accurate solutions. We show that flux limiting methods are total variation diminishing (and hence free of spurious oscillations) for nonconservative PDEs such as those typically encountered in finance, for fully explicit, and fully and partially implicit schemes. We also modify the van Leer flux limiter so that the secondorder total variation diminishing property is preserved for nonuniform grid spacing. 1 Introduction Asian options are securities with payoffs...
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
Adaptive mesh refinement using wavepropagation algorithms for hyperbolic systems
 SIAM J. Numer. Anal
, 1998
"... Dedicated to Ami Harten for his many contributions and warm sense of humor. Abstract. An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ highresolution wavepropagation algorithms in a more general framework. This allows its use on a ..."
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Cited by 43 (6 self)
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Dedicated to Ami Harten for his many contributions and warm sense of humor. Abstract. An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ highresolution wavepropagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the amrclaw package, which is freely available.
Inferring White Matter Geometry from Diffusion Tensor MRI: Application to Connectivity Mapping
 in Lecture Notes in Computer Science, T. Pajdla and
, 2004
"... We introduce a novel approach to the cerebral white matter connectivity mapping from di#usion tensor MRI. DTMRI is the unique noninvasive technique capable of probing and quantifying the anisotropic di#usion of water molecules in biological tissues. We address the problem of consistent neural ..."
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Cited by 41 (8 self)
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We introduce a novel approach to the cerebral white matter connectivity mapping from di#usion tensor MRI. DTMRI is the unique noninvasive technique capable of probing and quantifying the anisotropic di#usion of water molecules in biological tissues. We address the problem of consistent neural fibers reconstruction in areas of complex di#usion profiles with potentially multiple fibers orientations. Our method relies on a global modelization of the acquired MRI volume as a Riemannian manifold M and proceeds in 4 majors steps: First, we establish the link between Brownian motion and di#usion MRI by using the LaplaceBeltrami operator on M . We then expose how the sole knowledge of the di#usion properties of water molecules on M is su#cient to infer its geometry. There exists a direct mapping between the di#usion tensor and the metric of M . Next, having access to that metric, we propose a novel level set formulation scheme to approximate the distance function related to a radial Brownian motion on M . Finally, a rigorous numerical scheme using the exponential map is derived to estimate the geodesics of M , seen as the di#usion paths of water molecules. Numerical experimentations conducted on synthetic and real di#usion MRI datasets illustrate the potentialities of this global approach.
Equilibrium schemes for scalar conservation laws with stiff sources
 Math. Comp
, 2003
"... Abstract. We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory resu ..."
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Cited by 38 (4 self)
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Abstract. We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a socalled equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients. 1.
Relaxation Schemes For Nonlinear Kinetic Equations
 SIAM J. Numer. Anal
, 1997
"... . A class of numerical schemes for nonlinear kinetic equations of Boltzmann type is described. Following Wild's approach, the solution is represented as a power series with parameter depending exponentially on the Knudsen number. This permits us to derive accurate and stable time discretizations for ..."
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Cited by 36 (15 self)
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. A class of numerical schemes for nonlinear kinetic equations of Boltzmann type is described. Following Wild's approach, the solution is represented as a power series with parameter depending exponentially on the Knudsen number. This permits us to derive accurate and stable time discretizations for all ranges of the mean free path. These schemes preserve the main physical properties: positivity, conservation of mass, momentum, and energy. Moreover, for some particular models, the entropy property is also shown to hold. Key words. Boltzmann equation, fluid dynamic limit, Wild sum AMS subject classifications. 35L65, 65C20, 76P05, 82C40 PII. S0036142995287768 1. Introduction. Numerical resolution methods for the Boltzmann equation play an important role in practical and theoretical analysis of the time evolution of a rarefied gas. The widely used and bestknown of these methods is the direct simulation Monte Carlo method due to Bird [4]. After Bird's algorithm, more sophisticated meth...
Approximate Solutions of Nonlinear Conservation Laws and Related Equations
, 1997
"... During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical ..."
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Cited by 34 (11 self)
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During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical tools which are used in the development of convergence theories for these algorithms. These include classical compactness arguments (based on BV a priori estimates), the use of compensated compactness arguments (based on H^1compact entropy production), measure valued solutions (measured by their negative entropy production), and finally, we highlight the most recent addition to this bag of analytical tools  the use of averaging lemmas which yield new compactness and regularity results for nonlinear conservation laws and related equations. We demonstrate how these analytical tools are used in the convergence analysis of approximate solutions for hyperbolic conservation laws and related equations. Our discussion includes examples of Total Variation Diminishing (TVD) finitedifference schemes; error estimates derived from the onesided stability of Godunovtype methods for convex conservation laws (and their multidimensional analogue  viscosity solutions of demiconcave HamiltonJacobi equations); we outline, in the onedimensional case, the convergence proof of finiteelement streamlinediffusion and spectral viscosity schemes based on the divcurl lemma; we also address the questions of convergence and error estimates for multidimensional finitevolume schemes on nonrectangular grids; and finally, we indicate the convergence of approximate solutions with underlying kinetic formulation, e.g., finitevolume and relaxation schemes, once their regularizing effect is quantified in terms of the averaging lemma.
Multiphase Computations in Geometrical Optics
 J. Comp. Appl. Math
, 1996
"... In this work we propose a new set of partial differential equations (PDEs) which can be seen as a generalization of the classical eikonal and transport equations, to allow for solutions with multiple phases. The traditional geometrical optics pair of equations suffer from the fact that the class of ..."
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Cited by 32 (1 self)
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In this work we propose a new set of partial differential equations (PDEs) which can be seen as a generalization of the classical eikonal and transport equations, to allow for solutions with multiple phases. The traditional geometrical optics pair of equations suffer from the fact that the class of physically relevant solutions is limited. In particular, it does not include solutions with multiple phases, corresponding to crossing waves. Our objective has been to generalize these equations to accommodate solutions containing more than one phase. The new equations are based on the same high frequency approximation of the scalar wave equation as the eikonal and the transport equations. However, they also incorporate a finite superposition principle. The maximum allowed number of intersecting waves in the solution can be chosen arbitrarily, but a higher number means that a larger system of PDEs must be solved. The PDEs form a hyperbolic system of conservation laws with source terms. Altho...
SemiLagrangian Methods for Level Set Equations
, 1998
"... A new numerical method for solving geometric moving interface problems is presented. The method combines a level set approach and a semiLagrangian time stepping scheme which is explicit yet unconditionally stable. The combination decouples each mesh point from the others and the time step from the ..."
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Cited by 31 (6 self)
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A new numerical method for solving geometric moving interface problems is presented. The method combines a level set approach and a semiLagrangian time stepping scheme which is explicit yet unconditionally stable. The combination decouples each mesh point from the others and the time step from the CFL stability condition, permitting the construction of methods which are efficient, adaptive and modular. Analysis of a linear onedimensional model problem suggests a surprising convergence criterion which is supported by heuristic arguments and confirmed by an extensive collection of twodimensional numerical results. The new method computes correct viscosity solutions to problems involving geometry, anisotropy, curvature and complex topological events.