Mimetic Metrics for the DGSEM.

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Scientific computing

Background:

• Free-stream preservation is vital for accurate numerical simulations on curvilinear grids.
• Existing methods struggle to maintain this property due to challenges with metric terms.
• Divergence-free metric terms are essential for both free-stream preservation and entropy stability.

Purpose of the Study:

• To introduce a novel approach for calculating metric terms in discontinuous Galerkin spectral element methods (DGSEMs).
• To ensure that these computed metric terms are divergence-free.
• To enhance the stability and accuracy of numerical solvers on complex geometries.

Main Methods:

• A mimetic approach is employed for computing metric terms.
• The method utilizes projections compatible with de Rham Cohomology.
• This ensures discrete metric identities are satisfied, leading to divergence-free terms.

Main Results:

• The proposed method guarantees divergence-free metric terms for DGSEMs.
• This directly addresses the key requirement for free-stream preservation.
• The approach contributes to improved entropy stability on curvilinear grids.

Conclusions:

• The new mimetic approach provides a robust way to compute divergence-free metric terms.
• This significantly advances the capabilities of numerical solvers on curvilinear grids.
• The method offers a pathway to more accurate and stable computational simulations.