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Alexander Kurganov - One of the best experts on this subject based on the ideXlab platform.

  • a third order semi discrete genuinely multidimensional Central Scheme for hyperbolic conservation laws and related problems
    Numerische Mathematik, 2001
    Co-Authors: Alexander Kurganov, Guergana Petrova
    Abstract:

    We construct a new third-order semi-discrete genuinely multidimensional Central Scheme for systems of conservation laws and related convection-diffusion equations. This construction is based on a multidimensional extension of the idea, introduced in [17] – the use of more precise information about the local speeds of propagation, and integration over nonuniform control volumes, which contain Riemann fans. As in the one-dimensional case, the small numerical dissipation, which is independent of ${\cal O}(\frac{1}{\Delta t})$ , allows us to pass to a limit as $\Delta t \downarrow 0$ . This results in a particularly simple genuinely multidimensional semi-discrete Scheme. The high resolution of the proposed Scheme is ensured by the new two-dimensional piecewise quadratic non-oscillatory reconstruction. First, we introduce a less dissipative modification of the reconstruction, proposed in [29]. Then, we generalize it for the computation of the two-dimensional numerical fluxes. Our Scheme enjoys the main advantage of the Godunov-type Central Schemes –simplicity, namely it does not employ Riemann solvers and characteristic decomposition. This makes it a universal method, which can be easily implemented to a wide variety of problems. In this paper, the developed Scheme is applied to the Euler equations of gas dynamics, a convection-diffusion equation with strongly degenerate diffusion, the incompressible Euler and Navier-Stokes equations. These numerical experiments demonstrate the desired accuracy and high resolution of our Scheme.

  • semidiscrete Central upwind Schemes for hyperbolic conservation laws and hamilton jacobi equations
    SIAM Journal on Scientific Computing, 2001
    Co-Authors: Alexander Kurganov, Sebastian Noelle, Guergana Petrova
    Abstract:

    We introduce new Godunov-type semidiscrete Central Schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations. The Schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the Schemes from [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241--282; A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461--1488; A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional Central Scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720--742]. The main advantages of the proposed Central Schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications. At the same time, the developed Schemes have an upwind nature, since they respect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them Central-upwind Schemes. The constructed Schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton--Jacobi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navier--Stokes equations. The incompressibility condition in the latter equations allows us to treat them both in their conservative and transport form. We apply to these problems the Central-upwind Schemes, developed separately for each of them, and compute the corresponding numerical solutions.

  • new high resolution Central Schemes for nonlinear conservation laws and convection diffusion equations
    Journal of Computational Physics, 2000
    Co-Authors: Alexander Kurganov, Eitan Tadmor
    Abstract:

    Central Schemes may serve as universal finite-difference methods for solving nonlinear convection?diffusion equations in the sense that they are not tied to the specific eigenstructure of the problem, and hence can be implemented in a straightforward manner as black-box solvers for general conservation laws and related equations governing the spontaneous evolution of large gradient phenomena. The first-order Lax?Friedrichs Scheme (P. D. Lax, 1954) is the forerunner for such Central Schemes. The Central Nessyahu?Tadmor (NT) Scheme (H. Nessyahu and E. Tadmor, 1990) offers higher resolution while retaining the simplicity of the Riemann-solver-free approach. The numerical viscosity present in these Central Schemes is of order O((?x)2r/?t). In the convective regime where ?t~?x, the improved resolution of the NT Scheme and its generalizations is achieved by lowering the amount of numerical viscosity with increasing r. At the same time, this family of Central Schemes suffers from excessive numerical viscosity when a sufficiently small time step is enforced, e.g., due to the presence of degenerate diffusion terms.In this paper we introduce a new family of Central Schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order O(?x)2r?1)). In particular, our new Central Schemes maintain their high-resolution independent of O(1/?t), and letting ?t ? 0, they admit a particularly simple semi-discrete formulation. The main idea behind the construction of these Central Schemes is the use of more precise information of the local propagation speeds. Beyond these CFL related speeds, no characteristic information is required. As a second ingredient in their construction, these Central Schemes realize the (nonsmooth part of the) approximate solution in terms of its cell averages integrated over the Riemann fans of varying size.The semi-discrete Central Scheme is then extended to multidimensional problems, with or without degenerate diffusive terms. Fully discrete versions are obtained with Runge?Kutta solvers. We prove that a scalar version of our high-resolution Central Scheme is nonoscillatory in the sense of satisfying the total-variation diminishing property in the one-dimensional case and the maximum principle in two-space dimensions. We conclude with a series of numerical examples, considering convex and nonconvex problems with and without degenerate diffusion, and scalar and systems of equations in one- and two-space dimensions. Time evolution is carried out by the third- and fourth-order explicit embedded integration Runge?Kutta methods recently proposed by A. Medovikov (1998). These numerical studies demonstrate the remarkable resolution of our new family of Central Scheme.

  • a third order semidiscrete Central Scheme for conservation laws and convection diffusion equations
    SIAM Journal on Scientific Computing, 2000
    Co-Authors: Alexander Kurganov, Doron Levy
    Abstract:

    We present a new third-order, semidiscrete, Central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems. Our method is a high-order extension of the recently proposed second-order, semidiscrete method in [A. Kurgonov and E. Tadmor, J. Comput Phys., 160 (2000) pp. 241--282]. The method is derived independently of the specific piecewise polynomial reconstruction which is based on the previously computed cell-averages. We demonstrate our results by focusing on the new third-order Central weighted essentially nonoscillatory (CWENO) reconstruction presented in [D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 21 (1999), pp. 294--322]. The numerical results we present show the desired accuracy, high resolution, and robustness of our method.

He Huang - One of the best experts on this subject based on the ideXlab platform.

  • wnt β catenin signaling new and old players and new insights
    Current Opinion in Cell Biology, 2008
    Co-Authors: He Huang, Xi He
    Abstract:

    Wnt/β-catenin signaling has Central roles in embryogenesis and human diseases including cancer. A Central Scheme of the Wnt pathway is to stabilize the transcription coactivator β-catenin by preventing its phosphorylation-dependent degradation. Significant progress has been made toward the understanding of this crucial regulatory pathway, including the protein complex that promotes β-catenin phosphorylation–degradation, and the mechanism by which the extracellular Wnt ligand engages cell surface receptors to inhibit β-catenin phosphorylation–degradation. Here we review some recent discoveries in these two areas, and highlight some crucial questions that remain to be resolved.

  • wnt β catenin signaling new and old players and new insights
    Current Opinion in Cell Biology, 2008
    Co-Authors: He Huang
    Abstract:

    Wnt/beta-catenin signaling has Central roles in embryogenesis and human diseases including cancer. A Central Scheme of the Wnt pathway is to stabilize the transcription coactivator beta-catenin by preventing its phosphorylation-dependent degradation. Significant progress has been made toward the understanding of this crucial regulatory pathway, including the protein complex that promotes beta-catenin phosphorylation-degradation, and the mechanism by which the extracellular Wnt ligand engages cell surface receptors to inhibit beta-catenin phosphorylation-degradation. Here we review some recent discoveries in these two areas, and highlight some crucial questions that remain to be resolved.

Eitan Tadmor - One of the best experts on this subject based on the ideXlab platform.

  • new high resolution Central Schemes for nonlinear conservation laws and convection diffusion equations
    Journal of Computational Physics, 2000
    Co-Authors: Alexander Kurganov, Eitan Tadmor
    Abstract:

    Central Schemes may serve as universal finite-difference methods for solving nonlinear convection?diffusion equations in the sense that they are not tied to the specific eigenstructure of the problem, and hence can be implemented in a straightforward manner as black-box solvers for general conservation laws and related equations governing the spontaneous evolution of large gradient phenomena. The first-order Lax?Friedrichs Scheme (P. D. Lax, 1954) is the forerunner for such Central Schemes. The Central Nessyahu?Tadmor (NT) Scheme (H. Nessyahu and E. Tadmor, 1990) offers higher resolution while retaining the simplicity of the Riemann-solver-free approach. The numerical viscosity present in these Central Schemes is of order O((?x)2r/?t). In the convective regime where ?t~?x, the improved resolution of the NT Scheme and its generalizations is achieved by lowering the amount of numerical viscosity with increasing r. At the same time, this family of Central Schemes suffers from excessive numerical viscosity when a sufficiently small time step is enforced, e.g., due to the presence of degenerate diffusion terms.In this paper we introduce a new family of Central Schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order O(?x)2r?1)). In particular, our new Central Schemes maintain their high-resolution independent of O(1/?t), and letting ?t ? 0, they admit a particularly simple semi-discrete formulation. The main idea behind the construction of these Central Schemes is the use of more precise information of the local propagation speeds. Beyond these CFL related speeds, no characteristic information is required. As a second ingredient in their construction, these Central Schemes realize the (nonsmooth part of the) approximate solution in terms of its cell averages integrated over the Riemann fans of varying size.The semi-discrete Central Scheme is then extended to multidimensional problems, with or without degenerate diffusive terms. Fully discrete versions are obtained with Runge?Kutta solvers. We prove that a scalar version of our high-resolution Central Scheme is nonoscillatory in the sense of satisfying the total-variation diminishing property in the one-dimensional case and the maximum principle in two-space dimensions. We conclude with a series of numerical examples, considering convex and nonconvex problems with and without degenerate diffusion, and scalar and systems of equations in one- and two-space dimensions. Time evolution is carried out by the third- and fourth-order explicit embedded integration Runge?Kutta methods recently proposed by A. Medovikov (1998). These numerical studies demonstrate the remarkable resolution of our new family of Central Scheme.

  • nonoscillatory Central Schemes for multidimensional hyperbolic conservation laws
    SIAM Journal on Scientific Computing, 1998
    Co-Authors: Guangshan Jiang, Eitan Tadmor
    Abstract:

    We construct, analyze, and implement a new nonoscillatory high-resolution Scheme for two-dimensional hyperbolic conservation laws. The Scheme is a predictor-corrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewise-linear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a second-order, nonoscillatory Central Scheme, a natural extension of the one-dimensional second-order Central Scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408--448]. As in the one-dimensional case, the main feature of our two-dimensional Scheme is simplicity. In particular, this Central Scheme does not require the intricate and time-consuming (approximate) Riemann solvers which are essential for the high-resolution upwind Schemes; in fact, even the computation of the exact Jacobians can be avoided. Moreover, the Central Scheme is "genuinely multidimensional" in the sense that it does not necessitate dimensional splitting. We prove that the Scheme satisfies the scalar maximum principle, and in the more general context of systems, our proof indicates that the Scheme is positive (in the sense of Lax and Liu [CFD Journal, 5 (1996), pp. 1--24]). We demonstrate the application of our Central Scheme to several prototype two-dimensional Euler problems. Our numerical experiments include the resolution of shocks oblique to the computational grid; they show how our Central Scheme solves with high resolution the intricate wave interactions in the so-called double Mach reflection problem [J. Comput. Phys., 54 (1988), pp. 115--173] without following the characteristics; and finally we report on the accurate ray solutions of a weakly hyperbolic system [J. Comput. Appl. Math., 74 (1996), pp. 175--192], rays which otherwise are missed by the dimensional splitting approach. Thus, a considerable amount of simplicity and robustness is gained while achieving stability and high resolution.

Thierry Poinsot - One of the best experts on this subject based on the ideXlab platform.

  • large eddy simulation of the shock turbulence interaction
    Journal of Computational Physics, 1999
    Co-Authors: F Ducros, V Ferrand, Franck Nicoud, C Weber, D Darracq, C Gacherieu, Thierry Poinsot
    Abstract:

    The objective of this work is to derive a shock capturing tool able to treat turbulence with minimum dissipation out of the shock for a large-eddy simulation (LES) of the shock/turbulence interaction. The present numerical modeling of the shock/turbulence interaction consists of a second-order finite volume Central Scheme using a skew-symmetric form, a Jameson's type artificial dissipation, and the filtered structure function model. We focus on two areas to build simulations of increased accuracy:

Mark H. Carpenter - One of the best experts on this subject based on the ideXlab platform.

  • a systematic methodology for constructing high order energy stable weno Schemes
    Journal of Computational Physics, 2009
    Co-Authors: Nail K Yamaleev, Mark H. Carpenter
    Abstract:

    A third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference Scheme developed by the authors of this paper [N.K. Yamaleev, M.H. Carpenter, Third-order energy stable WENO Scheme, J. Comput. Phys. 228 (2009) 3025-3047] was proven to be stable in the energy norm for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, a systematic approach is presented that enables ''energy stable'' modifications for existing WENO Schemes of any order. The technique is demonstrated by developing a one-parameter family of fifth-order upwind-biased ESWENO Schemes including one sixth-order Central Scheme; ESWENO Schemes up to eighth order are presented in the Appendix. We also develop new weight functions and derive constraints on their parameters, which provide consistency, much faster convergence of the high-order ESWENO Schemes to their underlying linear Schemes for smooth solutions with arbitrary number of vanishing derivatives, and better resolution near strong discontinuities than the conventional counterparts.