185 research outputs found
Arbitrarily High-Order Unconditionally Energy Stable Schemes for Thermodynamically Consistent Gradient Flow Models
We present a systematic approach to developing arbitrarily high-order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization method, we formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic energy, arriving at an arbitrarily high-order scheme for gradient flow models. Schemes derived using both approaches are proved rigorously to be unconditionally energy stable. The proposed schemes are then implemented in four gradient flow models numerically to demonstrate their accuracy and effectiveness. Detailed numerical comparisons among these schemes are carried out as well. These numerical strategies are rather general so that they can be readily generalized to solve any thermodynamically consistent PDE models
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 thermodynamically consistent formulation of generalized thermoelasticity at finite deformations
A thermodynamically consistent model of non-classical coupled non-linear thermoelasticity capable of accounting for thermal wave propagation is proposed. The heat flux is assumed to consist of both additive energetic and dissipative components. Constitutive relations for the stress, the entropy and the energetic component of the heat flux are derived in a thermodynamically consistent manner. A Lyapunov function for the dynamics is obtained for the case in which the surface of the continuum body is maintained at a reference temperature. It is shown that the system is non-linearly stable. The linearized model is shown to be similar to the type III model of Green and Naghdi, except for some minor differences in the interpretations of some of the parameters
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