28 research outputs found
Highly efficient schemes for time fractional Allen-Cahn equation using extended SAV approach
In this paper, we propose and analyze high order efficient schemes for the
time fractional Allen-Cahn equation. The proposed schemes are based on the L1
discretization for the time fractional derivative and the extended scalar
auxiliary variable (SAV) approach developed very recently to deal with the
nonlinear terms in the equation. The main contributions of the paper consist
in: 1) constructing first and higher order unconditionally stable schemes for
different mesh types, and proving the unconditional stability of the
constructed schemes for the uniform mesh; 2) carrying out numerical experiments
to verify the efficiency of the schemes and to investigate the coarsening
dynamics governed by the time fractional Allen-Cahn equation. Particularly, the
influence of the fractional order on the coarsening behavior is carefully
examined. Our numerical evidence shows that the proposed schemes are more
robust than the existing methods, and their efficiency is less restricted to
particular forms of the nonlinear potentials
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 convergent stochastic scalar auxiliary variable method
We discuss an extension of the scalar auxiliary variable approach which was
originally introduced by Shen et al.~([Shen, Xu, Yang, J.~Comput.~Phys., 2018])
for the discretization of deterministic gradient flows. By introducing an
additional scalar auxiliary variable, this approach allows to derive a linear
scheme, while still maintaining unconditional stability. Our extension augments
the approximation of the evolution of this scalar auxiliary variable with
higher order terms, which enables its application to stochastic partial
differential equations. Using the stochastic Allen--Cahn equation as a
prototype for nonlinear stochastic partial differential equations with
multiplicative noise, we propose an unconditionally energy stable, linear,
fully discrete finite element scheme based on our stochastic scalar auxiliary
variable method. Recovering a discrete version of the energy estimate and
establishing Nikolskii estimates with respect to time, we are able to prove
convergence of appropriate subsequences of discrete solutions towards pathwise
unique martingale solutions by applying Jakubowski's generalization of
Skorokhod's theorem. A generalization of the Gy\"ongy--Krylov characterization
of convergence in probability to quasi-Polish spaces finally provides
convergence of fully discrete solutions towards strong solutions of the
stochastic Allen--Cahn equation
High Order Schemes for Gradient Flows
First, two new classes of energy stable, high order accurate Runge-Kutta schemes for gradient flows in a very general setting are presented: a class of fully implicit methods that are unconditionally energy stable and a class of semi-implicit methods that are conditionally energy stable. The new schemes are developed as high order analogs of the minimizing movements approach for generating a time discrete approximation to a gradient flow by solving a sequence of optimization problems. In particular, each step entails minimizing the associated energy of the gradient flow plus a movement limiter term that is, in the classical context of steepest descent with respect to an inner product, simply quadratic. A variety of existing stable numerical methods can be recognized as (typically just first order accurate in time) minimizing movement schemes for their associated evolution equations, already requiring the optimization of the energy plus a quadratic term at every time step. Therefore, our methods give a painless way to extend the existing schemes to high order accurate in time schemes while maintaining their stability. Additionally, we extend the schemes to gradient flows with solution dependent inner product. Here, the stability and consistency conditions of the methods are given and proved, specific examples of the schemes are given for second and third order accuracy, and convergence tests are performed to demonstrate the accuracy of the methods.
Next, two algorithms for simulating mean curvature motion are considered. First is the threshold dynamics algorithm of Merriman, Bence, and Osher. The algorithm is only first order accurate in the two-phase setting and its accuracy degrades further to half order in the multi-phase setting, a shortcoming it has in common with other related, more recent algorithms.
As a first, rigorous step in addressing this shortcoming, two different second order accurate versions of two-phase threshold dynamics are presented.
Unlike in previous efforts in this direction, both algorithms come with careful consistency calculations.
The first algorithm is consistent with its limit (motion by mean curvature) up to second order in any space dimension.
The second achieves second order accuracy only in dimension two but comes with a rigorous stability guarantee (unconditional energy stability) in any dimension -- a first for high order schemes of its type.
Finally, a level set method for multiphase curvature motion known as Voronoi implicit interface method is considered. Here, careful numerical convergence studies, using parameterized curves to reach very high resolutions in two dimensions are given.
These tests demonstrate that in the unequal, additive surface tension case, the Voronoi implicit interface method does not converge to the desired limit.
Then a variant that maintains the spirit of the original algorithm is presented. It appears to fix the non-convergence and as a bonus, the new variant extends the Voronoi implicit interface method to unequal mobilities.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162894/1/azaitzef_1.pd
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018
This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described