Multidimensional WENO-AO Reconstructions Using a Simplified Smoothness Indicator and Applications to Conservation Laws

Abstract Finite volume, weighted essentially non-oscillatory (WENO) schemes require the computation of a smoothness indicator. This can be expensive, especially in multiple space dimensions. We consider use simple indicator $$\sigma ^{\textrm{S}}= \frac{1}{N_{\textrm{S}}-1}\sum _{j} ({\bar{u}}_{j} - {\bar{u}}_{m})^2$$ σ S = 1 N - ∑ j ( u ¯ m ) 2 , where $$N_{\textrm{S}}$$ is number mesh elements stencil, $${\bar{u}}_j$$ local function average over element j and index m gives target element. Reconstructions utilizing standard WENO weighting fail with this develop modification WENO-Z that reliable accurate reconstruction adaptive order, which we denote as SWENOZ-AO. prove it attains order accuracy large stencil polynomial approximation when solution smooth, drops to small approximations there jump discontinuity solution. Numerical examples one two dimensions on general meshes verify properties reconstruction. They also show about 10 times faster than reconstructions using classic The new applied define finite volume approximate hyperbolic conservation laws. tests results same quality indicator, but an overall speedup time 3.5–5 2D tests. Moreover, computational efficiency (CPU versus error) noticeably improved.