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Nonlinear DataBounded Polynomial Approximations and their Applications in ENO Methods
 NUMERICAL ALGORITHMS
, 2010
"... A class of highorder databounded polynomials on general meshes are derived and analyzed in the context of numerical solutions of hyperbolic equations. Such polynomials make it possible to circumvent the problem of Rungetype oscillations by adaptively varying the stencil and order used, but at th ..."
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A class of highorder databounded polynomials on general meshes are derived and analyzed in the context of numerical solutions of hyperbolic equations. Such polynomials make it possible to circumvent the problem of Rungetype oscillations by adaptively varying the stencil and order used, but at the cost of only enforcing C 0 solution continuity at data points. It is shown that the use of these polynomials, based on extending the work of [1] to nonuniform meshes, provides a way to develop positivity preserving polynomial approximations of potentially high order for hyperbolic equations. The central idea is to use ENO (Essentially Non Oscillatory) type approximations but to enforce additional restrictions on how the polynomial order is increased. The question of how high a polynomial order should be used will be considered, with respect to typical numerical examples. The results show that this approach is successful but that it is necessary to provide sufficient resolution inside a front if highorder methods of this type are to be used, thus emphasizing the need to consider nonuniform meshes.
Data and RangeBounded Polynomials in ENO Methods
"... Abstract Essentially NonOscillatory (ENO) methods and Weighted Essentially NonOscillatory (WENO) methods are of fundamental importance in the numerical solution of hyperbolic equations. A key property of such equations is that the solution must remain positive or lie between bounds. A modification ..."
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Abstract Essentially NonOscillatory (ENO) methods and Weighted Essentially NonOscillatory (WENO) methods are of fundamental importance in the numerical solution of hyperbolic equations. A key property of such equations is that the solution must remain positive or lie between bounds. A modification of the polynomials used in ENO methods to ensure that the modified polynomials are either bounded by adjacent values (databounded) or lie within a specified range (rangebounded) is considered. It is shown that this approach helps both in the range boundedness in the preservation of extrema in the ENO polynomial solution.