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Towards Robust and Efficient High-Order Methods for Unsteady Nonlinear Partial Differential Equations

David Del Rey Fernandez, University of Waterloo
Brian Vermeire, Concordia University
Siva Nadarajah, McGill University
Modern and future computational architectures promise unprecedented power that could enable the simulation of unsteady nonlinear partial differential equations in realistic scenarios (e.g., complex geometry) at unprecedented scale and accuracy. High-order methods are excellent candidates for such systems as a result of their dense compute kernels, particularly in the context of unsteady problems requiring high accuracy. However, for such problems high-order methods have traditionally suffered from stability issues making them impractical. In the context of computational fluid dynamics, there have been huge advances towards developing high-order schemes with provable properties (e.g., entropy-stability, positivity preservation, etc) and the required mechanics to make them efficient (e.g., adaptation) and practical. In this minisymposium we look broadly at nonlinear partial differential equations and the mathematics required to develop efficient high-order schemes with provable properties. Example PDEs of interest include but are not limited to, incompressible and compressible flow equations, multiphase equations, nonlinear wave equations, and nonlinear reaction diffusion equations. Moreover, numerical techniques of interest include but are not limited to, provably stable schemes, positivity preservation, time stepping, stabilization, adaptation, space-time methods, unstructured schemes, and mechanics for the efficient deployment of high-order methods on modern and future compute architectures.