SIAM Student Chapter Seminar

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Fall 2020

date speaker title
9/29 Yu Feng (Math) Phase separation in the advective Cahn--Hilliard equation
9/29 Dongyu Chen (WPI) A Half-order Numerical Scheme for Nonlinear SDEs with one-sided Lipschitz Drift and H\:{o}lder Continuous Diffusion Coefficients

Abstracts

9/29, Yu Feng (Math)

Phase separation in the advective Cahn--Hilliard equation

The Cahn--Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn--Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing then no phase separation occurs, and the solution instead converges exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified in terms of the dissipation time of the associated advection-hyperdiffusion equation, and we produce examples of velocity fields with a small dissipation time. We also study the relationship between this quantity and the dissipation time of the standard advection-diffusion equation.


10/14, Yu Feng (Math)

A Half-order Numerical Scheme for Nonlinear SDEs with one-sided Lipschitz Drift and Hölder Continuous Diffusion Coefficients

We consider positivity-preserving explicit schemes for one-dimensional nonlinear stochastic differential equations. The drift coefficients satisfy the one-sided Lipschitz condition, and the diffusion coefficients are Hölder continuous. To control the fast growth of moments of solutions, we introduce several explicit schemes including the tamed and truncated Euler schemes. The fundamental idea is to guarantee the non-negativity of solutions. The proofs rely on the boundedness for negative moments and exponential of negative moments. We present several numerical schemes for a modified Cox-Ingersoll-Ross model and a two-factor Heston model and demonstrate their half-order convergence rate.



Past Semesters