Algorithm Development, Analysis, and Application of High Order Schemes

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In this project the PI will perform research in algorithm design and analysis of high order accurate and efficient numerical methods for solving partial differential equations. These algorithms are used to solve scientific and engineering problems arising from diverse application fields such as aerospace engineering, semi-conductor device design, astrophysics, and biological problems. Even with today's fast computers, it is still essential to design efficient and reliable algorithms which can be used to obtain accurate solutions to these application problems. The broader impacts resulting from the proposed activity will be a suite of powerful computational tools, suitable for various applications mentioned above. These tools are expected to make positive contributions to computer simulations of the complicated solution structure in these applications.The algorithms the PI plans to investigate include the finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes and discontinuous Galerkin finite element methods, for solving hyperbolic and other convection dominated partial differential equations (PDEs). While the emphasis of this project is on algorithm design and analysis, close attention will be paid to applications. Topics of proposed investigations will include the study on an inverse Lax-Wendroff procedure for high order numerical boundary conditions for finite difference schemes on Cartesian meshes solving problems in general geometry, Lagrangian type finite volume schemes for multi-material flows, a simple weighted essentially non-oscillatory limiter for discontinuous Galerkin methods with strong shocks, high order stable conservative methods on arbitrary point clouds, discontinuous Galerkin methods for weakly coupled hyperbolic multi-domain and network problems, efficient time-stepping techniques for discontinuous Galerkin schemes, high order accurate bound-preserving schemes and applications, bound-preserving high order discontinuous Galerkin schemes for radiative transfer equations, energy-conserving DG methods for Maxwell's equations in Drude metamaterials, efficient discontinuous Galerkin method for front propagation problems with obstacles, superconvergence analysis of discontinuous Galerkin methods and its applications, multi-scale methods based on the discontinuous Galerkin framework, and applications in areas including traffic and pedestrian flow models and aggregation, coordinated movement and cell proliferation in computational biology. Problems in applications will motivate the design of new algorithms or new features in existing algorithms; mathematics tools are used to analyze these algorithms to give guidelines for their applicability and limitations; practical considerations including parallel implementation issues are addressed to make the algorithms competitive in large scale calculations; and collaborations with engineers and other applied scientists enable the efficient application of these new algorithms or new features in existing algorithms.