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A New Class of Efficient and Robust Energy Stable Schemes for Gradient Flows
A new numerical technique to deal with nonlinear terms in gradient flows by introducing a scalar auxiliary variable (SAV), which can construct unconditionally second-order energy stable schemes and can easily construct even third or fourth order BDF schemes, although not unconditionally stable, which are very robust in practice.
Convergence and Error Analysis for the Scalar Auxiliary Variable (SAV) Schemes to Gradient Flows
  • Jie Shen, Jie Xu
  • Mathematics, Computer Science
    SIAM J. Numer. Anal.
  • 25 September 2018
We carry out convergence and error analysis of the scalar auxiliary variable (SAV) methods for $L^2$ and $H^{-1}$ gradient flows with a typical form of free energy. We first derive $H^2$ bounds,
Averages of unlabeled networks: Geometric characterization and asymptotic behavior
The lack of vertex labeling results in a nontrivial geometry for the space of unlabeled networks, which in turn is found to have important implications on the types of probabilistic and statistical results that may be obtained and the techniques needed to obtain them.
Strong averaging principle for slow-fast SPDEs with Poisson random measures
This work concerns the problem associated with an averaging principle for two-time-scales stochastic partial differential equations (SPDEs) driven by cylindrical Wiener processes and Poisson random
An Averaging Principle for Multivalued Stochastic Differential Equations
The averaging principle for multivalued stochastic differential equations (MSDEs) driven by Brownian motion with Brownian noise is investigated. An averaged MSDEs for the original MSDEs is proposed,
Solving the Yamabe Problem by an Iterative Method on a Small Riemannian Domain
We introduce an iterative scheme to solve the Yamabe equation −a∆gu+Su = λu p−1 on small domains (Ω, g) ⊂ R equipped with a Riemannian metric g. Thus g admits a conformal change to a constant scalar
Stabilized Predictor-Corrector Schemes for Gradient Flows with Strong Anisotropic Free Energy
Gradient flows with strong anisotropic free energy are difficult to deal with numerically with existing approaches. We propose a stabilized predictor-corrector approach to construct schemes which are
Unconditionally positivity preserving and energy dissipative schemes for Poisson-Nernst-Planck equations
It is proved that the schemes are mass conservative, uniquely solvable and keep positivity unconditionally, and the first-order scheme is proven to be unconditionally energy dissipative.