58 research outputs found
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A HYBRID METHOD FOR STIFF REACTION-DIFFUSION EQUATIONS.
The second-order implicit integration factor method (IIF2) is effective at solving stiff reaction-diffusion equations owing to its nice stability condition. IIF has previously been applied primarily to systems in which the reaction contained no explicitly time-dependent terms and the boundary conditions were homogeneous. If applied to a system with explicitly time-dependent reaction terms, we find that IIF2 requires prohibitively small time-steps, that are relative to the square of spatial grid sizes, to attain its theoretical second-order temporal accuracy. Although the second-order implicit exponential time differencing (iETD2) method can accurately handle explicitly time-dependent reactions, it is more computationally expensive than IIF2. In this paper, we develop a hybrid approach that combines the advantages of both methods, applying IIF2 to reaction terms that are not explicitly time-dependent and applying iETD2 to those which are. The second-order hybrid IIF-ETD method (hIFE2) inherits the lower complexity of IIF2 and the ability to remain second-order accurate in time for large time-steps from iETD2. Also, it inherits the unconditional stability from IIF2 and iETD2 methods for dealing with the stiffness in reaction-diffusion systems. Through a transformation, hIFE2 can handle nonhomogeneous boundary conditions accurately and efficiently. In addition, this approach can be naturally combined with the compact and array representations of IIF and ETD for systems in higher spatial dimensions. Various numerical simulations containing linear and nonlinear reactions are presented to demonstrate the superior stability, accuracy, and efficiency of the new hIFE method
Low regularity integrators for semilinear parabolic equations with maximum bound principles
This paper is concerned with conditionally structure-preserving, low
regularity time integration methods for a class of semilinear parabolic
equations of Allen-Cahn type. Important properties of such equations include
maximum bound principle (MBP) and energy dissipation law; for the former, that
means the absolute value of the solution is pointwisely bounded for all the
time by some constant imposed by appropriate initial and boundary conditions.
The model equation is first discretized in space by the central finite
difference, then by iteratively using Duhamel's formula, first- and
second-order low regularity integrators (LRIs) are constructed for time
discretization of the semi-discrete system. The proposed LRI schemes are proved
to preserve the MBP and the energy stability in the discrete sense.
Furthermore, their temporal error estimates are also successfully derived under
a low regularity requirement that the exact solution of the semi-discrete
problem is only assumed to be continuous in time. Numerical results show that
the proposed LRI schemes are more accurate and have better convergence rates
than classic exponential time differencing schemes, especially when the
interfacial parameter approaches zero.Comment: 24 page
An Exponential Time Differencing Scheme with a Real Distinct Poles Rational Function for Advection-Diffusion Reaction Equations
A second order Exponential Time Differencing (ETD) scheme for advection-diffusion reaction systems is developed by using a real distinct poles rational function for approximating the underlying matrix exponential. The scheme is proved to be second order convergent. It is demonstrated to be robust for reaction-diffusion systems with non-smooth initial and boundary conditions, sharp solution gradients, and stiff reaction terms. In order to apply the scheme efficiently to higher dimensional problems, a dimensional splitting technique is also developed. This technique can be applied to all ETD schemes and has been found, in the test problems considered, to reduce computational time by up to 80%
Exponential integrators for second-order in time partial differential equations
Two types of second-order in time partial differential equations (PDEs),
namely semilinear wave equations and semilinear beam equations are considered.
To solve these equations with exponential integrators, we present an approach
to compute efficiently the action of the matrix exponential as well as those of
related matrix functions. Various numerical simulations are presented that
illustrate this approach.Comment: 19 pages, 10 figure
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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