Spectral Analysis of High Order Continuous FEM for Hyperbolic PDEs on Triangular Meshes: Influence of Approximation,...

Area of Science:

• Computational mathematics
• Numerical analysis
• Scientific computing

Background:

• Continuous finite element discretizations are crucial for solving hyperbolic partial differential equations.
• Extending one-dimensional findings to multi-dimensional frameworks presents significant challenges, particularly in stability analysis.
• Existing methods require evaluation for efficiency, stability, and dispersion error in two-dimensional contexts.

Purpose of the Study:

• To investigate and rank various continuous finite element discretization schemes for 2D hyperbolic PDEs.
• To analyze the impact of polynomial spaces, stabilization techniques, and time discretizations on scheme performance.
• To identify the most efficient and stable numerical methods for practical applications.

Main Methods:

• Comparison of Lagrangian (equispaced and Cubature) and Bernstein polynomial spaces.
• Evaluation of streamline-upwind Petrov-Galerkin, continuous interior penalty, and orthogonal subscale stabilization.
• Assessment of Runge-Kutta, strong stability preserving RK, and deferred correction time discretizations.
• Fourier analysis on periodic triangular meshes with varying advection angles for stability analysis.
• Introduction of high-order viscosity for discontinuity stabilization.

Main Results:

• Most schemes evaluated are mass-matrix free, enhancing efficiency.
• Performance ranking based on stability and dispersion error is provided.
• Optimal CFL and stabilization coefficients are determined.
• Theoretical findings are validated through numerical experiments on linear and non-linear problems.
• Error-CPU time curves illustrate computational efficiency.

Conclusions:

• Cubature elements combined with Strong Stability Preserving Runge-Kutta (SSPRK) and Orthogonal Subscale (OSS) stabilization demonstrate the most promising performance.
• This combination offers a robust and efficient approach for solving two-dimensional hyperbolic partial differential equations.
• The study provides a comprehensive guide for selecting numerical methods in scientific computing.