Towards a compact high-order method for non-linear hyperbolic systems. I: The Hermite Least-Square Monotone (HLSM) reconstruction

Authors:
G. Capdeville
Affiliations:
Département de Mecanique des Fluides, Ecole Centrale de Nantes, 1, rue de la Noë, B.P. 92101, 44321 Nantes cedex 3, France
Venue:
Journal of Computational Physics
Year:
2009

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Abstract

A new Hermite Least-Square Monotone (HLSM) reconstruction to calculate accurately complex flows on non-uniform meshes is presented. The coefficients defining the Hermite polynomial are calculated by using a least-square method. To introduce monotonicity conditions into the procedure, two constraints are added into the least-square system. Those constraints are derived by locally matching the high-order Hermite polynomial with a low-order TVD or ENO polynomial. To emulate these constraints only in regions of discontinuities, data-depending weights are defined; those weights are based upon normalized indicators of smoothness of the solution and are parameterized by a O(1) quantity. The reconstruction so generated is highly compact and is fifth-order accurate when the solution is smooth; this reconstruction becomes first-order in regions of discontinuities. By inserting this reconstruction into an explicit finite-volume framework, a spatially fifth-order non-oscillatory method is then generated. This method evolves in time the solution and its first derivative. In a one-dimensional context, a linear spectral analysis and extensive numerical experiments make it possible to assess the robustness and the advantages of the method in computing multi-scales problems with embedded discontinuities.