Implicit Weighted Essentially Non-Oscillatory (WENO) Schemes for Advection-Diffusion-Reaction Systems

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Computational modeling is used in science and engineering to simulate how physical and biological systems work, so that we can better understand them and how they may be modified for societal benefit. Many of these systems mix advective (or transport), diffusive, and reactive processes. We have good numerical techniques for simulating a single such process, but only a few of these can handle multiple processes at once. This project concerns theoretical and algorithmic development of an alternate category of numerical techniques for simulation of systems of nonlinear advection-diffusion-reaction equations. These new numerical techniques show great promise, and they are likely to lead to better accuracy and computational efficiency. Applications to geoscience problems important to energy production and environmental protection will be pursued. Assessment, design, and monitoring of human activities involving reservoirs and aquifers in the Earth's subsurface require large-scale simulation of advective, diffusive, and reactive processes over long time periods. There is a potential societal benefit in energy production and environmental protection. The project may also have an impact on broad areas of science and engineering that use models consisting of nonlinear, coupled advection-diffusion-reaction equations. The project is expected to have an impact on the STEM workforce and its diversity through the education and training of two Ph.D. graduate students (both female, one a native citizen). Such students are in high demand in industrial and governmental labs, as well as in academia.The mathematical structure of physical or biological models governed by nonlinear advection-diffusion-reaction partial differential equations is often poorly understood, and solutions can develop shocks or very steep fronts. This project concerns theoretical and algorithmic development of high order, implicit, weighted essentially non-oscillatory (iWENO) schemes for numerical approximation of systems of such equations, because this type of scheme has the potential to handle all three processes well. The development will including finite volume and finite difference schemes, Eulerian-Lagrangian approaches, and a multi-moment variant. The objectives are to (1) develop a suitable smoothness indicator and time integrator for the problem; (2) develop a general procedure to handle possibly degenerate diffusive processes; (3) make advances on space discretization and related issues, such as handling boundary conditions and satisfying local maximum principles; (4) test the approach on applications to porous media; and (5) educate and train students in an interdisciplinary setting. The project will lead to a very general computational framework can approximate all the necessary physics in a locally mass conservative way. It will be simple to implement, handle general computational meshes in two and three space dimensions, be high order accurate in both space and time, maintain local mass conservation properties, be robust (i.e., unconditionally linearly stable), and maximize mesh resolution. The schemes will be efficient on high performance computers, which are memory bandwidth limited, because local information that can fit in cache memory will dominate the computations, and the global system of discrete equations will have about as small a number of degrees of freedom as possible. The project is expected to have a broader impact on the STEM workforce and its diversity, and on the geosciences through applications of the schemes, and it may impact broad areas of science and engineering, especially those that use models of complex, coupled problems for which the mathematical structure of the application may not be well understood.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.