  In this paper, we propose a novel Hermite weighted essentially
non-oscillatory (HWENO) fast sweeping method to solve the static
Hamilton-Jacobi equations efficiently. During the HWENO reconstruction
procedure, the proposed method is built upon a new finite difference fifth
order HWENO scheme involving one big stencil and two small stencils. However,
one major novelty and difference from the traditional HWENO framework lies in
the fact that, we do not need to introduce and solve any additional equations
to update the derivatives of the unknown function $\phi$. Instead, we use the
current $\phi$ and the old spatial derivative of $\phi$ to update them. The
traditional HWENO fast sweeping method is also introduced in this paper for
comparison, where additional equations governing the spatial derivatives of
$\phi$ are introduced. The novel HWENO fast sweeping methods are shown to yield
great savings in both computational time and storage, which improves the
computational efficiency of the traditional HWENO scheme. In addition, a hybrid
strategy is also introduced to further reduce computational costs. Extensive
numerical experiments are provided to validate the accuracy and efficiency of
the proposed approaches.
