High-order approximation techniques for nonlinear hyperbolic PDEs

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Many applications are based on nonlinear partial differential equations in which stability is not a result of an energy estimate. This is the case in nonlinear conservation laws, convection-dominated or multiphase flows, and free-boundary problems, where shocks fronts and discontinuities are important features and pose significant difficulties for numerical methods. The natural setting for these problems involves the physical notion of entropy and requires the positivity of quantities like mass, temperature or density. The investigators propose to continue the development of a new nonlinear approximation technique for solving the above class of differential equations. This new approach consists of computing the so-called entropy residual and use it to design a stabilization mechanism to the Galerkin formulation of the problem at hand. This is a different point of view than that of standard stabilization techniques. The investigators propose to design a nonlinear viscosity based on the second principle of thermodynamics and respect positivity/boundedness of the relevant quantities at the same time. Even though the nonlinear algorithms are more complicated and difficult to analyze, they yield great benefits when working with rough solutions, complicated geometry, and strong nonlinearities. In the past several decades, a large amount of work has been dedicated to the development of robust numerical methods modeling nonlinear phenomena. Significant advances have been made in many areas, but the current state of the art is far from providing accurate and faithful numerical representations of the complex physical processes. For instance, accurate approximation of interfaces, sharp fronts, and shock formations is still an enormous challenge. The proposed project has a broad impact in many fields. In mechanical and aerospace engineering, the proposed method improves on numerical models for simulating high velocity gas dynamics, nonlinear elasticity and phase transition problems. In petroleum engineering the new set of methods is beneficial for more accurate simulation of multiphase flows in reservoirs with complicated geometry. Moreover, the project will also have significant impact in other fields such as geophysics, nanotechnology, and environmental problems where reliable simulations for resolving shocks, sharp interfaces, and other nonlinear phenomena are needed.