NSERC grants at Laurentian University Subventions du CRSNG `a l ...

97Nicolas RobidouxCochain-based numerical modelsA grant of $8,000 per year.Une subvention de 8 000 $ par année.Discovery Grant – Subvention à la découverteThis research proposal concerns the development andtesting of computer software for the modeling of physicalreality (numerical modeling), the interpretation ofdata arising from experiments or computer simulations(scientific visualization), and the simulation of physicalreality (computer graphics).Consider the cooling of a piece of metal, the diffusionof a ground contaminant, the emission of electric andmagnetic fields by a cell phone, or the flow of a fluid ina cavity. The physically relevant quantities associatedwith such problems are fluxes (“How much fluid goes through this surface?”);circulations (related to “What is the potential difference between these twopoints?”); masses (“How much contaminant is contained in this volume?”);and point values (“What is the temperature at this point?”). A natural wayto translate such physically grounded quantities into computer code is withthe help of cochains, a data structure borrowed from algebraic topology.One of the benefits of using cochains in a computer model is that conservationlaws – fundamental laws of physics which characterize many systems– are then directly translated into computer code. This property is expectedto lead to superior algorithms and methods.Specifically, we will develop software for the automatic detection of vortices,sources and sinks in a fluid flow; we will formulate cochain-based “averagedphysical laws” to be used when a computer modeller cannot affordto resolve the fine structure of a material in a simulation; and we will useso-called “superconvergent” cochains to solve partial differential equationscheaply and accurately.