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

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.