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David Shirokoff, New Jersey Institute of Technology Department of Mathematical Sciences
Implicit-Explicit (IMEX) Stability and Applications to the Dispersive Shallow Water Equations
In this talk we will introduce a new stability theory for implicit-explicit (IMEX) time integration schemes—which treat some terms in a differential equation implicitly (for stability) and others explicitly (for efficiency). Our focus will be on devising new efficient stable schemes for several fluid equations ranging from the incompressible Navier-Stokes equations, nonlinear diffusion equations, and a recent application to the dispersive shallow water equations (DSWE)—which are fluid models, applicable to coastal regions that include additional physics (such as dispersion) to the well-known shallow water equations. The new time-stepping approaches will avoid a fully implicit treatment of nonlinear terms, and in the case of the DSWE, avoid the difficulties created by mixed space and time derivatives while ensuring stability. Throughout the talk we will make use of conformal mappings, the numerical range, and the variational structure of the PDEs. Time permitting, we will also discuss recent work on devising diagonally implicit Runge-Kutta methods that avoid order reduction. Order reduction occurs when an RK scheme converges at a rate lower than the formal order of the scheme thereby seriously limiting the applicability of the method; DIRKs are often desirable for PDEs as they avoid the simultaneous solve of coupled RK stages.