Title

Energy-Stable Runge–Kutta Schemes for Gradient Flow Models Using the Energy Quadratization Approach

Document Type

Article

Journal/Book Title/Conference

Applied Mathematics Letters

Volume

94

Publisher

Elsevier Ltd

Publication Date

2-13-2019

Award Number

NSF, Division of Mathematical Sciences 1816783

Funder

NSF, Division of Mathematical Sciences

First Page

224

Last Page

231

Abstract

In this letter, we present a novel class of arbitrarily high-order and unconditionally energy-stable algorithms for gradient flow models by combining the energy quadratization (EQ) technique and a specific class of Runge–Kutta (RK) methods, which is named the EQRK schemes. First of all, we introduce auxiliary variables to transform the original model into an equivalent system, with the transformed free energy a quadratic functional with respect to the new variables and the modified energy dissipative law is conserved. Then a special class of RK methods is employed for the reformulated system to arrive at structure-preserving time-discrete schemes. Along with rigorous proofs, numerical experiments are presented to demonstrate the accuracy and unconditionally energy-stability of the EQRK schemes.

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