| Abstract: | In this paper, we combine the nonlinear HWENO reconstruction in 43] and
the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static
Hamilton-Jacobi equations in a novel HWENO framework recently developed in 22].
The proposed HWENO frameworks enjoys several advantages. First, compared with
the traditional HWENO framework, the proposed methods do not need to introduce
additional auxiliary equations to update the derivatives of the unknown function $\phi$.
They are now computed from the current value of $\phi$ and the previous spatial derivatives of $\phi$. This approach saves the computational storage and CPU time, which greatly
improves the computational efficiency of the traditional HWENO scheme. In addition,
compared with the traditional WENO method, reconstruction stencil of the HWENO
methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping method
is used to update the numerical approximation. It is an explicit method and does
not involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves some
known issues, including that the iterative number is very sensitive to the parameter $ε$ used in the nonlinear weights, as observed in previous studies. Finally, to further
reduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate the
good performance of the proposed fixed-point fast sweeping HWENO methods. |
