Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Hyperbolic partial differential equations

Background:

• Traditional ADER-WENO schemes use conserved variables for spatial reconstruction and time evolution.
• This can lead to increased oscillations and computational cost, particularly for complex systems like relativistic hydrodynamics.
• Efficient and accurate numerical methods are crucial for simulating fluid dynamics and related phenomena.

Purpose of the Study:

• To develop a novel conservative ADER-WENO finite volume scheme utilizing primitive variables for spatial reconstruction and time evolution.
• To reduce the number of variable conversions between conserved and primitive forms within the numerical scheme.
• To enhance solution accuracy and reduce computational cost compared to existing methods.

Main Methods:

• A new ADER-WENO finite volume scheme is proposed, performing spatial reconstruction twice: first on conserved variables, then on primitive variables.
• A novel space-time finite element predictor evolves reconstruction polynomials in primitive form, directly using the primitive variable PDE.
• The scheme is applied to Euler equations, relativistic hydrodynamics (RHD), ideal magnetohydrodynamics (RMHD), and the Baer-Nunziato model.

Main Results:

• The new ADER schemes yield less oscillatory solutions compared to schemes reconstructing in conserved variables, notably for RMHD and Baer-Nunziato equations.
• For RHD and RMHD equations, accuracy is improved, and CPU time is reduced by approximately 25%.
• The approach was successfully extended to ADER-Discontinuous Galerkin (DG) schemes on space-time adaptive grids (AMR).

Conclusions:

• The proposed ADER-WENO scheme using primitive variables offers significant advantages in accuracy and computational efficiency.
• It is recommended as a standard method for relativistic fluid dynamics due to its improved performance and reduced oscillations.
• The method's flexibility is demonstrated by its extension to adaptive mesh refinement (AMR) scenarios.