377 research outputs found
Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs
In this paper, we present a novel strategy to systematically construct
linearly implicit energy-preserving schemes with arbitrary order of accuracy
for Hamiltonian PDEs. Such novel strategy is based on the newly developed
exponential scalar variable (ESAV) approach that can remove the
bounded-from-blew restriction of nonlinear terms in the Hamiltonian functional
and provides a totally explicit discretization of the auxiliary variable
without computing extra inner products, which make it more effective and
applicable than the traditional scalar auxiliary variable (SAV) approach. To
achieve arbitrary high-order accuracy and energy preservation, we utilize the
symplectic Runge-Kutta method for both solution variables and the auxiliary
variable, where the values of internal stages in nonlinear terms are explicitly
derived via an extrapolation from numerical solutions already obtained in the
preceding calculation. A prediction-correction strategy is proposed to further
improve the accuracy. Fourier pseudo-spectral method is then employed to obtain
fully discrete schemes. Compared with the SAV schemes, the solution variables
and the auxiliary variable in these ESAV schemes are now decoupled. Moreover,
when the linear terms are of constant coefficients, the solution variables can
be explicitly solved by using the fast Fourier transform. Numerical experiments
are carried out for three Hamiltonian PDEs to demonstrate the efficiency and
conservation of the ESAV schemes
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 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
A novel class of linearly implicit energy-preserving schemes for conservative systems
We consider a kind of differential equations d/dt y(t) = R(y(t))y(t) +
f(y(t)) with energy conservation. Such conservative models appear for instance
in quantum physics, engineering and molecular dynamics. A new class of
energy-preserving schemes is constructed by the ideas of scalar auxiliary
variable (SAV) and splitting, from which the nonlinearly implicit schemes have
been improved to be linearly implicit. The energy conservation and error
estimates are rigorously derived. Based on these results, it is shown that the
new proposed schemes have unconditionally energy stability and can be
implemented with a cost of solving a linearly implicit system. Numerical
experiments are done to confirm these good features of the new schemes
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
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