A class of ENO schemes with adaptive order for solving hyperbolic conservation laws

Abstract

When solving hyperbolic conservation laws, in order to avoid spurious oscillations at discontinuities, namely the Gibbs phenomenon, essentially non-oscillatory (ENO) schemes select the smoothest stencil from several candidates of an equal size to approximate the fluxes. However, when all candidate stencils are smooth, we can use the union of the candidate stencils to construct a polynomial with higher-order accuracy. In extreme situations, all of the candidate stencils of the same size are possibly non-smooth. In view of this problem, we proposed ENO schemes with adaptive order (ENO-AO) which select a polynomial from several candidates that are reconstructed on stencils of unequal sizes. The selected polynomial is the optimal one in the sense that it predicts the cell average of the neighboring cell on either side of the target stencil with the smallest error. In this way, all the stencils containing discontinuities are automatically excluded, and the selected polynomial approximates the flux to adaptive order, ranging from the first-order all the way up to the designed high-order. When applied to piecewise smooth data, the proposed ENO-AO schemes give high-order accuracy whenever the data is smooth but avoid the Gibbs phenomenon at discontinuities. Several numerical examples of scalar conservation laws and the compressible Euler equations are carried out to test the accuracy, robustness, and efficiency of the proposed schemes.