Many engineering applications involve partial differential equations where stability is not the result of an energy estimate. This is the case for nonlinear conservation laws, advection-dominated flows, multi-phase flows, and free-boundary problems, where shocks fronts and discontinuities are driving features and pose significant difficulties for numerical methods. The natural stability setting for these problems involves integrability, bounded variations, or boundedness. This kind of stability naturally occurs when one wants to preserve quantities like mass or when one wishes to preserve the positivity or the boundedness of quantities like temperature or density. The investigators propose to develop a new nonlinear approximation technique for solving the above class of differential equations. This new approach consists of computing the best approximation in the natural stability norm of the problem, which is a radically different point of view than that of standard techniques. The investigators trade a linear non-optimal perspective (working in energy spaces) for an optimal nonlinear one (working in bounded-variation-like spaces). Even though the nonlinear algorithms are more complicated and difficult to analyze, they yield great benefits when working with rough data, complicated boundary, and stiff nonlinearities.
A large amount of work has been dedicated in the past to the development of robust numerical methods. Significant progress has been made in some areas, but the current state of the art is far from providing accurate and faithful numerical representations of complex processes. For instance, simulating interfaces, shocks, and sharp fronts is still an enormous challenge. The proposed project has a broad impact in many fields. In mechanical and aerospace engineering, the proposed method improves numerical models for simulating high velocity gas dynamics, nonlinear elasticity problems, and phase transition in new materials like shape memory alloys. In petroleum engineering the new set of methods is beneficial for simulating multi-phase flows in reservoirs. In general, the project will also have significant impact on environmental sciences, geophysics, and nanotechnologies where robust approximation techniques for solving shocks, sharp interfaces, and nonlinear phenomena are needed.
