  The aim of this study is to develop a novel WENO scheme that improves the
performance of the well-known fifth-order WENO methods. The approximation space
consists of exponential polynomials with a tension parameter that may be
optimized to fit the the specific feature of the data, yielding better results
compared to the polynomial approximation space. However, finding an optimal
tension parameter is a very important and difficult problem, indeed a topic of
active research. In this regard, this study introduces a practical approach to
determine an optimal tension parameter by taking into account the relationship
between the tension parameter and the accuracy of the exponential polynomial
interpolation under the setting of the fifth-order WENO scheme. As a result,
the proposed WENO scheme attains an improved order of accuracy (that is,
sixth-order) better than other fifth-order WENO methods without loss of
accuracy at critical points. A detailed analysis is provided to verify the
improved convergence rate. Further, we present modified nonlinear weights based
on $L^1$-norm approach along with a new global smoothness indicator. The
proposed nonlinear weights reduce numerical dissipation significantly, while
attaining better resolution in smooth regions. Some experimental results for
various benchmark test problems are presented to demonstrate the ability of the
new scheme.
