Adaptive mapping for high order WENO methods

Abstract

In this paper, a novel mapping approach through the use of adaptive mapping functions is introduced for high order weighted essentially non-oscillatory (WENO) methods. The new class of adaptive mapping functions are designed to adjust themselves to the solution based on a simple parameter calculated using the smoothness indicators that are readily available during computation. It is shown that this adaptive nature allows the resultant mapped WENO scheme to maintain sub-stencil weights close to the optimal weights in smooth regions without amplifying the weights of non-smooth stencils containing discontinuities. Therefore, adaptive mapping achieves enhanced accuracy in smooth regions and is more resistant against spurious oscillations near discontinuities. Taylor series analysis of the seventh order finite volume WENO scheme has been performed to demonstrate the loss of accuracy of the original WENO method near critical points. The convergence rates of the seventh order finite volume WENO scheme with adaptive mapping have been shown through a simple numerical example. Excellent results have been obtained for one-dimensional linear advection cases especially over long output times. Improved results have also been obtained for one- and two-dimensional Euler equation test cases.