High-Order Invariant Domain Preserving-Numerical Methods for Nonlinear Hyperbolic Systems-
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Many important physical phenomena are modeled by nonlinear systems of hyperbolic conservation laws. When approximating such problems one encounters sharp interfaces, contact discontinuities, shocks and other nonlinear wave interactions. In such regions high order approximation methods are not stable and exhibit oscillations. It is very important that these oscillations be controlled because preservation of mass, positivity or boundedness of the solution is critical in numerical simulations. Moreover, spurious oscillations are known to promote convergence to nonphysical weak solutions or simply lead to failure to produce an approximation. Any advance in this direction will have a broad impact insofar the class of problems we want to address touches many fields in engineering (mechanical, aerospace, nuclear, ocean, etc.), in environmental sciences, in geophysics, in petroleum engineering, etc. Proposing a novel robust approximation technique for solving nonlinear hyperbolic problems developing shocks or sharp interfaces will have impact in every areas of science and engineering where controlling or dealing with this type of phenomenon is still an enormous challenge. The main focus of this proposal to investigate and develop new high-order approximation techniques for nonlinear hyperbolic systems on unstructured meshes in any space dimension. The project will be organized around three main objectives: (1) Development of high-order maximum principle preserving methods for scalar conservation equations. The emphasis will be on the design and analysis of numerical methods that are at least third-order in space and time. Convergence estimates for some of these methods will be established; (2) Design of invariant domain preserving methods for any hyperbolic system in any space dimension on non-uniform meshes. The goal is to construct invariant domain preserving methods that are at least formally second-order accurate both in space and time. The objective is to have methods that preserve all the invariant domains of the underlying physical system; (3) The last part of the project will consists of extending the new methodology to systems with source terms like the shallow water equations and radiative transport.