Wavefront Marching Methods: A Unified Algorithm to Solve Eikonal and Static Hamilton-Jacobi Equations.

Area of Science:

• Computational mathematics
• Numerical analysis
• Wave propagation modeling

Background:

• Classic Fast Marching Methods (FMM) solve the Eikonal equation with O(M log M) complexity.
• Solving more general static Hamilton-Jacobi equations typically requires higher computational complexity.
• Existing methods face challenges in balancing accuracy and computational efficiency for these equations.

Purpose of the Study:

• To present a unified propagation method applicable to both Eikonal and static Hamilton-Jacobi equations.
• To maintain a computational complexity of O(M log M) for both equation types.
• To enhance the accuracy of solutions compared to current state-of-the-art methods.

Main Methods:

• Development of a unified framework for solving Eikonal and static Hamilton-Jacobi equations.
• Introduction of 'mini wave-fronts' for interpolating solutions.
• Minimization of discretization error through interpolation techniques.

Main Results:

• The proposed method achieves O(M log M) complexity for both Eikonal and Hamilton-Jacobi equations.
• Demonstrated higher accuracy compared to existing state-of-the-art techniques.
• Experimental results show superior precision and reduced computational cost.

Conclusions:

• The unified propagation method offers an efficient and accurate solution for wave propagation problems.
• The 'mini wave-fronts' technique effectively reduces discretization errors.
• This approach advances the numerical solution of differential equations in computational mathematics.