Colloquium


DATE2021-04-29 16:10-17:00

PLACE數學館3173教室

SPEAKER薛名成 教授(國立陽明交通大學

TITLEPressure-Correction Projection Methods for The Incompressible Viscous Flow Based on the Scalar Auxiliary Variable Approach

ABSTRACT In this talk, the pressure-correction projection methods based on the scalar auxiliary variable approach are proposed and studied for the 2d Navier-Stokes equations and Boussinesq equations. In the literature, enormous amounts of work have contributed to the study of numerical schemes for computing the Navier-Stokes equations. In general, two of the main numerical difficulties for solving Navier-Stokes equations are the incompressible condition and the nonlinear term. One of the approaches to deal with the incompressible condition is the so-called projection method. The typical projection method only needs to solve the Poisson type of equations depending on the nonlinear term's treatment, which is efficient. However, the pressure-correction projection methods suffer from the splitting error, leading to spurious numerical boundary layers and the limitation of accuracy in time. In the literature, an iterated pressure-correction projection method has been proposed to overcome the difficulty. As for the nonlinear term treatment, which is the main issue of the talk, it is better to treat the nonlinear term explicitly so that one only requires to solve the corresponding linear system with constant coefficients at each time step. However, such treatment often results in a restricted time step due to the stable issue. Recently, the scalar auxiliary variable approach has been constructed to have an unconditional energy stable numerical scheme. In this talk, we will begin with introducing the stability of the time-stepping schemes for ODEs and some basic physical equations. A new pressure-correction projection method based on the scalar auxiliary variable's simple choice for the incompressible viscous flow is proposed. We find that this new scheme can still enjoy unconditional energy stability with this simple choice of the scale auxiliary variable. The proofs of the energy stability and error convergence are provided and analyzed. Finally, numerical examples illustrate the theoretical work, and these methods are applied to efficiently solve parametrized flows that play an important role in uncertainty quantification. This is joint work with Tony Chang.