High-Order Invariant Domain Preserving Approximations of Multiphysics Systems of Conservation Equations

The objective of this project is to construct robust approximation techniques for nonlinear conservation systems on unstructured meshes in one, two, and three space dimensions. This class of problems touches many fields in engineering (mechanical, aerospace, nuclear, ocean, etc.) and in sciences (geophysics, astrophysics). A new set of novel robust approximation techniques for solving complex nonlinear conservation equations in realistic settings will also benefit numerous applications that involve models mixing hyperbolicity with other physical effects such as diffusion and dispersion.

The results of this project will be disseminated through graduate classes, mentoring of students, seminars, conference presentations, publications, and direct collaborations with colleagues working at various US institutions. The material developed in this project will be incorporated in an advanced class on conservation equations that is given every two years in the Department of Mathematics at Texas A&M.

In this project, robustness means that the proposed methods are guaranteed to deliver solutions that satisfy physical and thermodynamical constraints (in general based on quasi-concave or concave functionals). Robustness also means that some asymptotic properties of the solution may be preserved by the approximation even if the mesh is not fine enough to be in some asymptotic range (i.e., locking must be avoided).

These types of methods are often said to be asymptotic preserving in the literature. Finally, the algorithms we have in mind must depend very little on the space discretization at hand and be simple enough to be programed by users with very little know-how in numerical analysis and on the mathematical structure of the nonlinear system. The project will heavily rely on the solid theoretical foundations recently established by the PIs and will be organized around three objectives: (i) construct algorithms for nonlinear hyperbolic systems that are robust with respect to the polynomial degree, the mesh structure, and have very few (if any) tuning parameters; (ii) construct approximation techniques that are robust for models mixing hyperbolicity with other physical effects such as diffusion and dispersion; (iii) construct techniques that will guarantee that realistic physical bounds and thermodynamical inequalities are satisfied for the discrete approximation even when incomplete knowledge of the physics is available.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.