WENO methods

{{short description|Scheme used in the numerical solution of hyperbolic partial differential equations}}

In numerical solution of differential equations, WENO (weighted essentially non-oscillatory) methods are classes of high-resolution schemes. WENO are used in the numerical solution of hyperbolic partial differential equations. These methods were developed from ENO methods (essentially non-oscillatory). The first WENO scheme was developed by Liu, Osher and Chan in 1994.{{cite journal |doi=10.1006/jcph.1994.1187 |title=Weighted Essentially Non-oscillatory Schemes |journal=Journal of Computational Physics |volume=115 |pages=200–212 |year=1994 |last1=Liu |first1=Xu-Dong |last2=Osher |first2=Stanley |last3=Chan |first3=Tony |bibcode=1994JCoPh.115..200L |citeseerx=10.1.1.24.8744 }} In 1996, Guang-Shan Jiang and Chi-Wang Shu developed a new WENO scheme{{cite journal |doi=10.1006/jcph.1996.0130 |title=Efficient Implementation of Weighted ENO Schemes |journal=Journal of Computational Physics |volume=126 |pages=202–228 |year=1996 |last1=Jiang |first1=Guang-Shan |last2=Shu |first2=Chi-Wang |issue=1 |bibcode=1996JCoPh.126..202J |citeseerx=10.1.1.7.6297 }} called WENO-JS.{{cite journal |doi=10.1016/j.jmaa.2012.04.040 |title=Mapped WENO schemes based on a new smoothness indicator for Hamilton–Jacobi equations |journal=Journal of Mathematical Analysis and Applications |volume=394 |issue=2 |pages=670–682 |year=2012 |last1=Ha |first1=Youngsoo |last2=Kim |first2=Chang Ho |last3=Lee |first3=Yeon Ju |last4=Yoon |first4=Jungho |doi-access=free }} Nowadays, there are many WENO methods.{{cite journal |doi=10.1137/10080960X |title=Strong Stability Preserving Two-step Runge–Kutta Methods |journal=SIAM Journal on Numerical Analysis |volume=49 |issue=6 |pages=2618–2639 |year=2011 |last1=Ketcheson |first1=David I. |last2=Gottlieb |first2=Sigal |last3=MacDonald |first3=Colin B. |arxiv=1106.3626 |s2cid=16602876 }}