The Effect of the Sensitivity Parameter in Weighted Essentially Non-oscillatory Methods

Weighted essentially non-oscillatory methods (WENO) were developed to capture shocks in the solution of hyperbolic conservation laws while maintaining stability and without smearing the shock profile. WENO methods accomplish this by assigning weights to a number of candidate stencils, according to the smoothness of the solution on the stencil. These weights favor smoother stencils when there is a significant difference while combining all the stencils to attain higher order when the stencils are all smooth. When WENO methods were initially introduced, a small parameter \(\varepsilon \) was defined to avoid division by zero. Over time, it has become apparent that \(\varepsilon \) plays the role of the sensitivity parameter in stencil selection. WENO methods allow some oscillations, and it is well known that these oscillations depend on the size of \(\varepsilon \). In this work, we show that the value of \(\varepsilon \) must be below a certain critical threshold \(\varepsilon _c\) and that this threshold depends on the function used and on the size of the jump discontinuity captured. Next, we analytically and numerically show the size of the oscillations for one time-step and over long time integration when \(\varepsilon < \varepsilon _c\) and their dependence on the size of \(\varepsilon \), the function used, and the size of the jump discontinuity.