386 research outputs found
High-efficiency and positivity-preserving stabilized SAV methods for gradient flows
The scalar auxiliary variable (SAV)-type methods are very popular techniques
for solving various nonlinear dissipative systems. Compared to the
semi-implicit method, the baseline SAV method can keep a modified energy
dissipation law but doubles the computational cost. The general SAV approach
does not add additional computation but needs to solve a semi-implicit solution
in advance, which may potentially compromise the accuracy and stability. In
this paper, we construct a novel first- and second-order unconditional energy
stable and positivity-preserving stabilized SAV (PS-SAV) schemes for and
gradient flows. The constructed schemes can reduce nearly half
computational cost of the baseline SAV method and preserve its accuracy and
stability simultaneously. Meanwhile, the introduced auxiliary variable is
always positive while the baseline SAV cannot guarantee this
positivity-preserving property. Unconditionally energy dissipation laws are
derived for the proposed numerical schemes. We also establish a rigorous error
analysis of the first-order scheme for the Allen-Cahn type equation in
norm. In addition we propose an energy
optimization technique to optimize the modified energy close to the original
energy. Several interesting numerical examples are presented to demonstrate the
accuracy and effectiveness of the proposed methods
A scalar auxiliary variable unfitted FEM for the surface Cahn-Hilliard equation
The paper studies a scalar auxiliary variable (SAV) method to solve the
Cahn-Hilliard equation with degenerate mobility posed on a smooth closed
surface {\Gamma}. The SAV formulation is combined with adaptive time stepping
and a geometrically unfitted trace finite element method (TraceFEM), which
embeds {\Gamma} in R3. The stability is proven to hold in an appropriate sense
for both first- and second-order in time variants of the method. The
performance of our SAV method is illustrated through a series of numerical
experiments, which include systematic comparison with a stabilized
semi-explicit method.Comment: 23 pages, 12 figure
REMARKS ON THE ASYMPTOTIC BEHAVIOR OF SCALAR AUXILIARY VARIABLE (SAV) SCHEMES
We introduce a time semi-discretization of a damped wave equation by a SAV scheme with second order accuracy. The energy dissipation law is shown to hold without any restriction on the time step. We prove that any sequence generated by the scheme converges to a steady state (up to a subsequence). We notice that the steady state equation associated to the SAV scheme is a modified version of the steady state equation associated to the damped wave equation. We show that a similar result holds for a SAV fully discrete version of the Cahn-Hilliard equation and we compare numerically the two steady state equations
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