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Nonlinear Data-Bounded Polynomial Approximations and their Applications in ENO Methods
M. Berzins. In Numerical Algorithms, Vol. 55, No. 2, pp. 171. 2010.
A class of high-order data-bounded 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 Runge-type oscillations by adaptively varying the stencil and order used, but at the cost of only enforcing C0 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 high-order methods of this type are to be used, thus emphasizing the need to consider nonuniform meshes.
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