Improvement of weighted essentially non-oscillatory schemes near discontinuities

• The standard fifth-order weighted essentially non-oscillatory (WENO-5) scheme has accuracy order decreased at the transition point from smooth region to a discontinuity point or vice versa.
• A new method is proposed to overcome the drawback by introducing fourth-order fluxes combined with high order smoothness indicator.
• Numerical examples show that the new method is more accurate near discontinuities.

In this article, we analyze the fifth-order weighted essentially non-oscillatory (WENO-5) scheme and show that, at a transition point from smooth region to a discontinuity point or vice versa, the accuracy order of WENO-5 is decreased to third order. A new method is proposed to overcome this drawback by introducing fourth-order fluxes combined with high order smoothness indicator. Numerical examples show that the new method is more accurate near discontinuities with accuracy improved to fourth order.