Mathematics Research Works

Permanent URI for this collectionhttp://hdl.handle.net/1903/1595

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Start date: 1990End date: 1999

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Now showing 1 - 10 of 12
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    SPECTRAL METHODS FOR HYPERBOLIC PROBLEMS
    (1994-01) Tadmor, Eitan
    We review several topics concerning spectral approximations of time-dependent problems, primarily | the accuracy and stability of Fourier and Chebyshev methods for the approximate solutions of hyperbolic systems. To make these notes self contained, we begin with a very brief overview of Cauchy problems. Thus, the main focus of the first part is on hyperbolic systems which are dealt with two (related) tools: the energy method and Fourier analysis. The second part deals with spectral approximations. Here we introduce the main ingredients of spectral accuracy, Fourier and Chebyshev interpolants, aliasing, differentiation matrices ... The third part is devoted to Fourier method for the approximate solution of periodic systems. The questions of stability and convergence are answered by combining ideas from the first two sections. In this context we highlight the role of aliasing and smoothing; in particular, we explain how the lack of resolution might excite small scales weak instability, which is avoided by high modes smoothing. The forth and final part deals with non-periodic problems. We study the stability of the Chebyshev method, paying special attention to the intricate issue of the CFL stability restriction on the permitted time-step.
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    High-resolution non-oscillatory central schemes with non-staggered grids for hyperbolic conservation laws
    (Copyright: Society for Industrial and Applied Mathematics, 1998-12) JIANG, G.-S.; LEVY, D.; LIN, C.-T.; OSHER, S.; TADMOR, E.
    We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408{463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397{425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892{1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.
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    FROM SEMIDISCRETE TO FULLY DISCRETE: STABILITY OF RUNGE-KUTTA SCHEMES BY THE ENERGY METHOD
    (Copyright: Society for Industrial and Applied Mathematics, 1998-03) LEVY, DORON; TADMO, EITAN
    The integration of semidiscrete approximations for time-dependent problems is encountered in a variety of applications. The Runge{Kutta (RK) methods are widely used to integrate the ODE systems which arise in this context, resulting in large ODE systems called methods of lines. These methods of lines are governed by possibly ill-conditioned systems with a growing dimension; consequently, the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading. Instead, we present here a stability analysis of RK methods for well-posed semidiscrete approximations, based on a general energy method. We review the stability question for such RK approximations, and highlight its intricate dependence on the growing dimension of the problem. In particular, we prove the strong stability of general fully discrete RK methods governed by coercive approximations. We conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection-diffusion equations with variable coeficients. A straightforward implementation of our results verify the strong stability of RK methods for local finite-difference schemes as well as global spectral approximations. Since our approach is based on the energy method (which is carried in the physical space), and since it avoids the von Neumann analysis (which is carried in the dual Fourier space), we are able to easily adapt additional extensions due to nonperiodic boundary conditions, general geometries, etc.
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    The convergence rate of Godunov type schemes
    (Copyright: Society for Industrial and Applied Mathematics, 1994-02) Nessyahu, Haim; Tadmor, Eitan; Tassa, Tamir
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    Legendre pseudospectral viscosity method for nonlinear conservation laws
    (Copyright: Society for Industrial and Applied Mathematics, 1993-04) Maday, Yvon; Kaber, Sidi M. Ould; Tadmor, Eitan
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    The convergence rate of approximate solutions for nonlinear scalar conservation laws
    (Copyright: Society for Industrial and Applied Mathematics, 1992-12) Nessyahu, Haim; Tadmor, Eitan
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    Local error estimates for discontinuous solutions of nonlinear hyperbolic equations
    (Copyright: Society for Industrial and Applied Mathematics, 1991-08) Tadmor, Eitan
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    An O(N2) method for computing the eigensystem of N x N symmetric tri-diagonal matrices by the divide and conquer approach
    (Copyright: Society for Industrial and Applied Mathematics, 1990-01) Gill, Doron; Tadmor, Eitan
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    A kinetic formulation of multidimensional scalar conservation laws and related equations
    (American Mathematical Society, 1994-01) Lions, P. L.; Perthame, B.; Tadmor, E.
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    Spectral viscosity approximations to multidimensional scalar conservation laws
    (American Mathematical Society, 1993-10) Chen, Gui-Qiang; Du, Qiang; Tadmor, Eitan