A generalized framework to construct third order WENO weights using weight limiter functions
For researchers in numerical methods for hyperbolic PDEs, this provides a new, principled approach to designing WENO weights, though it is an incremental improvement over existing methods.
This work presents a generalized framework for constructing third-order WENO weights using weight limiter functions, ensuring non-oscillatory behavior and exact third-order accuracy in smooth regions. Numerical tests confirm the scheme achieves third-order accuracy while maintaining shock-capturing ability.
The main aim of this work is not to improve any existing non-linear weight but to give a generalized framework for the construction of non-linear weights to get non-oscillatory third order WENO schemes. It is done by imposing necessary conditions on weights to get non-oscillatory WENO reconstruction which give further insight on the structure of weights to ensure non-occurrence of oscillations and characterize the solution region for third order accuracy. This framework for WENO weights is new and completely different from the prevailing existing approach. New non-linear weights are designed using a function of smoothness parameter termed as weight limiter functions. Many such weight limiter functions are given and analyzed. These new weights are simple and by construction guarantee for exact third order accuracy in smooth solution region including smooth extrema away from critical point. Numerical results for various test problems are given and compared. Results show that proposed weights give third order accuracy without loosing the non-oscillatory shock capturing ability of the resulting scheme.