High Accuracy Non-oscillatory Scheme with the Mean Value Theorem

Abstract

High order non-oscillatory interpolation is required for accurate computation. TVD property was introduced to limit the slope near the discontinuity and successfully suppress spurious oscillation

it also limits slope near smooth extrema and introduces unphysical clipping. ENO scheme selects the smoothest stencil in order to exclude the spurious oscillation, easily would be extended to very high order

selection of stencil would be affected by the very small error in continuous region. WENO scheme uses convex weighted sum of candidate stencils, could improve the accuracy. Weighted sum would introduce very fine oscillation in continuous region and such small error smears fine flow structures. Suggested scheme obtained fourth order accuracy interpolation at each cell interfaces. Spurious oscillation of each polynomial was tested by mean value theorem of average or derivative of average. The smallest error stencil without oscillation was selected for each cell interface. High order spatial derivatives was obtained from selected polynomials. High accuracy numerical flux was computed from Lax-Wendroff procedure. The position of discontinuity was deduced from the continuity information of each stencil. The speed of discontinuity was obtained from material derivative of discontinuity. High accuracy numerical flux at the discontinuity was computed from tracing of discontinuity. Suggested scheme showed the fourth order accuracy form the numerical test and more accurate result than WENO scheme, because the suggested scheme use central fourth order interpolation at the smooth region and also it traces the discontinuity with fourth order accuracy.