Stochastic and Multiscale Modeling and Computation Seminar by Cheng Wang: Global in Time Energy Estimate for the Exponential Time Differencing Runge-Kutta (ETDRK) Numerical Scheme for the Phase Field Crystal Equation

Speaker: Cheng Wang, professor of mathematics, University of Massachusetts at Dartmouth

Title: Global in Time Energy Estimate for the Exponential Time Differencing Runge-Kutta (ETDRK) Numerical Scheme for the Phase Field Crystal Equation

Abstract: The global in time energy estimate is derived for the ETDRK2 numerical scheme for the phase field crystal (PFC) equation, a sixth order parabolic equation to model crystal evolution. The energy stability is available for the exponential time differencing Runge-Kutta (ETDRK) numerical scheme to the gradient flow equation, under an assumption of global Lipschitz constant. To recover the stabilization constant value, some local-in-time convergence analysis has been reported, so that the energy stability becomes available over a fixed final time. In this work, we develop a global in time energy estimate for the ETDRK2 numerical scheme to the PFC equation, so that the energy dissipation property is valid for any final time. An a-priori assumption at the previous time step, combined with a single-step H^2 estimate of the numerical solution, turns out to be the key point in the analysis. Such an H^2 estimate recovers the maximum norm bound of the numerical solution at the next time step, so that the stabilization parameter value could be theoretically justified. This justification in turn ensures the energy dissipation at the next time step, so that the mathematical induction could be effectively applied, and the global-in-time energy estimate is accomplished. This technique is expected to be available for many other Runge-Kutta numerical schemes.

Stochastic and Multiscale Modeling and Computation Seminar