Development and Application of Modern Numerical Methods for Nonlinear Hyperbolic Systems of Partial Differential Equations

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The goal of this project is to develop new mathematical and computational tools for a large class of hyperbolic conservation and balance laws and related problems. Such systems arise in a wide variety of applications ranging from classical fluid dynamics (gas dynamics including multicomponent and multiphase compressible flows, many shallow water models including rotating shallow water equations, thermal rotating shallow water equations, shallow magnetohydrodynamic equations, and others), astrophysics, meteorology, oceanography, atmospheric sciences, to electromagnetism and modern biological models. While the PI is planning to work on several particular applications, the main focus of the research will be in the development of novel numerical methods and computational techniques that can be applied to a wide class of applied problems arising in today’s science. The project has also a potential to contribute to the emergence of accurate, robust and efficient algorithms and will overall increase the practical applicability on numerical methods. This project involves the training of graduate students. Many practical applications, especially in the cases of high space dimensions, require development and implementation of special numerical methods that are not only consistent with the governing system of partial differential equations, but also preserve certain structural and asymptotic properties of the underlying problem at the discrete level. The project is aimed at the development of efficient high-order methods for systems of conservation and balance laws whose basic properties go beyond consistency, stability and convergence. This will be achieved by designing special numerical techniques for (i) finding a delicate balance between the numerical diffusion and dispersion to ensure sharp—yet non-oscillatory—resolution of shock and contact waves, while achieving a high order accuracy in smooth regions, (ii) exactly preserving physically relevant steady-state solutions and involution constraints, (iii) establishing asymptotic preserving properties in certain stiff regimes, and (iv) analyzing the influence of uncertainties in problems with random data. The design and implementation of the new numerical schemes will be based on high-order shock-capturing finite-volume and finite-difference methods, accurate and efficient time integrators, and stochastic Galerkin and collocation methods, utilizing the main advantages of each of these methods in the context of the studied problems. The derivation of such numerical techniques is fundamental for understanding many physical phenomena and will contribute to their quantitative and qualitative study.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.