A wavelet-based adaptive WENO algorithm for Euler equations

• WENO scheme is a well known high order scheme but costly.
• An effective multi-resolution approximation is used to detect singularities.
• We apply WENO scheme at only the singularities.
• Our adaptive scheme gives a large decrease of CPU time with accurate results.

In this paper, we present an efficient and adaptive weighted essentially non-oscillatory (WENO) scheme, for Euler equations arising from conservation laws. WENO schemes are popular high order oscillation-control schemes for application to hyperbolic conservation laws. WENO schemes are computationally expensive, especially when solving non-linear system arising from hyperbolic conservation laws, since they compute non-linear weights involving smoothness indicators. We propose an algorithm which applies WENO schemes only at necessary points, in order to reduce computational costs. An effective multi-resolution approximation (MRA) is used to detect singularities. We adaptively apply WENO schemes at those singularities and skip other points by performing fixed stencil differentiations which are relatively faster than WENO. As expected, large decreases in computational costs are observed, while obtaining accurate results. In other words, the proposed scheme has still high order accuracy. We use several well known numerical examples to display the computational efficiency, the order of accuracy, conservative error of our approach.