For Zoom Link please contactÌýddelreyfernandez@uwaterloo.caÌýÌý
Speaker
Alina Chertock,ÌýProfessor and Head Department of Mathematics NC State University
Title
Structure Preserving Numerical MethodsÌýfor Hyperbolic Systems of Conservation and Balance Laws
Abstract
ManyÌýphysical models, while quite different in nature, can be described by nonlinearÌýhyperbolic systems ofÌýconservation and balance laws. The main source of difficultiesÌýone comes across when numerically solvingÌýthese systems is lack of smoothnessÌýas solutions of hyperbolic conservation/balance laws may develop veryÌýcomplicated nonlinear wave structures including shocks, rarefaction waves andÌýcontact discontinuities. TheÌýlevel of complexity may increase even further whenÌýsolutions of the hyperbolic system reveal a multiscaleÌýcharacter and/or theÌýsystem includes additional terms such as friction terms, geometrical terms,ÌýnonconservativeÌýproducts, etc., which are needed to be taken into account in order to achieve aÌýproperÌýdescription of the studied physical phenomena. In such cases, it isÌýextremely important to design a numericalÌýmethod that is not only consistentÌýwith the given PDEs, but also preserves certain structural and asymptoticÌýproperties of the underlying problem at the discrete level. While a variety ofÌýnumerical methods for such modelsÌýhave been successfully developed, there areÌýstill many open problems, for which the derivation of reliable high-resolutionÌýnumerical methods still remains to be an extremely challenging task.
In this talk, I will discussÌýrecent advances in the development of two classes of structure preservingÌýnumericalÌýmethods for nonlinear hyperbolic systems of conservation and balanceÌýlaws. In particular, I will present (i) well-balanced and positivity preservingÌýnumerical schemes, that is, the methods which are capable of exactlyÌýpreservingÌýsome steady-state solutions as well as maintaining the positivity of theÌýnumerical quantities when itÌýis required by the physical application, and (ii)Ìýasymptotic preserving schemes, which provide accurate andÌýefficient numericalÌýsolutions in certain stiff and/or asymptotic regimes of physical interest.