Abstract
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 C 0 solution continuity at data points. It is shown that the use of these polynomials, based on extending the work of Berzins (SIAM Rev 1(4):624–627, 2007) 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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References
Berzins, M.: Adaptive polynomial interpolation on evenly spaced meshes. SIAM Rev. 1(4), 624–627 (2007)
Berzins, M.: Data bounded polynomials and preserving positivity in high order ENO and WENO methods. SCI Report UUSCI-2009-003, University of Utah, July 2009, RevisedMarch 2010 (unpublished)
Balsara, D.S., Shiu, C.W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)
Borisov, V.S., Sorek, S.: On monotonicity of difference schemes for computational physics. SIAM J. Sci. Comput. 25, 1557–1584 (2004)
Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds): Advanced numerical approximation of nonlinear hyperbolic equations. In: Lecture Notes in Mathematics 1697, pp. 325–418. Springer Berlin Heidelberg (2000)
Graillat, S., Langlois, P., Louvet, N.: Improving the compensated Horner scheme with a fused multiply and add. In: Proc. 2006 ACM Symp. on Applied Comput., pp. 1323–1327. ACM, NY, USA, ISBN:1-59593-108-2 (2006)
Higham, N.J.: Accuracy and Stability of Numerical Algorithms. Siam, Philadelphia (1996)
Greenough, J.A., Rider, W.J.: A quantitative comparison of numerical methods for the compressible Euler equations: mifth order WENO and piecewise-linear Godunov. J. Comput. Phys. 196, 259–281 (2004)
Hildebrand, F.B.: Introduction to Numerical Analysis. McGraw-Hill, New York (1956)
Hubbard, M.E., Berzins, M.: A positivity preserving finite element method for hyperbolic partial differential equations. In: Armfield, S., Morgan, P., Srinivas, K. (eds.) CFD 2002, pp. 205–210. Springer, Berlin Heidelberg New York (2003)
Shu, C.-W.: TVD properties of a class of modified ENO schemes for scalar conservation laws. IMA Preprint Series 308, University of Minnesota (1987)
Shu, C.-W.: High order WENO schemes. SIAM Rev. 51, 82–126 (2009)
Shu, C.-W.: Numerical experiments on the accuracy of ENO and modified ENO schemes. J. Sci. Comput. 5, 127–149 (1990)
Waterson, N.P., Deconinck, H.: Design principles for bounded higher-order convection schemes—a unified approach. J. Comput. Phys. 224, 182–207 (2007)
Zhang, X., Shu, C.-W.: A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws. Brown University preprint. SIAM Journal on Numerical Analysis (to appear)
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Berzins, M. Nonlinear data-bounded polynomial approximations and their applications in ENO methods. Numer Algor 55, 171–189 (2010). https://doi.org/10.1007/s11075-010-9395-8
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DOI: https://doi.org/10.1007/s11075-010-9395-8