94 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
An enhanced and highly efficient semi-implicit combined Lagrange multiplier approach with preserving original energy law for dissipative systems
Recently, a new Lagrange multiplier approach was introduced by Cheng, Liu and
Shen in \cite{cheng2020new}, which has been broadly used to solve various
challenging phase field problems. To design original energy stable schemes,
they have to solve a nonlinear algebraic equation to determine the introduced
Lagrange multiplier, which can be computationally expensive, especially for
large-scale and long-time simulations involving complex nonlinear terms. This
paper presents an essential improved technique to modify this issue, which can
be seen as a semi-implicit combined Lagrange multiplier approach. In general,
the new constructed schemes keep all the advantages of the Lagrange multiplier
method and significantly reduce the computation costs. Besides, the new
proposed BDF2 scheme dissipates the original energy, as opposed to a modified
energy for the classical Lagrange multiplier approach in \cite{cheng2020new}.
We further construct high-order BDF schemes based on the new proposed
approach. In addition, we establish a general framework for extending our
constructed method to dissipative systems. Finally several examples have been
presented to demonstrate the effectiveness of the proposed approach
A novel high-order linearly implicit and energy-stable additive Runge-Kutta methods for gradient flow models
This paper introduces a novel paradigm for constructing linearly implicit and
high-order unconditionally energy-stable schemes for general gradient flows,
utilizing the scalar auxiliary variable (SAV) approach and the additive
Runge-Kutta (ARK) methods. We provide a rigorous proof of energy stability,
unique solvability, and convergence. The proposed schemes generalizes some
recently developed high-order, energy-stable schemes and address their
shortcomings.
On the one other hand, the proposed schemes can incorporate existing SAV-RK
type methods after judiciously selecting the Butcher tables of ARK methods
\cite{sav_li,sav_nlsw}. The order of a SAV-RKPC method can thus be confirmed
theoretically by the order conditions of the corresponding ARK method. Several
new schemes are constructed based on our framework, which perform to be more
stable than existing SAV-RK type methods. On the other hand, the proposed
schemes do not limit to a specific form of the nonlinear part of the free
energy and can achieve high order with fewer intermediate stages compared to
the convex splitting ARK methods \cite{csrk}.
Numerical experiments demonstrate stability and efficiency of proposed
schemes
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